13.0 Questions and Answers

Arguments have been the object of philosophical interest for a long time. Logicians and philosophers have studied the formal properties of arguments at least since Aristotle and have long discussed the logical sense of arguments as sets of premises and conclusions (Hamblin Reference Hamblin1970; Walton Reference Walton1990; Parsons Reference Parsons1996; Rumfitt Reference Rumfitt2015). The structure of arguments has been investigated by epistemologists (e.g. Pollock Reference Pollock1987, Reference Pollock1991a, Reference Pollock1991b, Reference Pollock2010), and has given rise to formal argumentation theory, which has developed into a branch of computer science in its own right (e.g. Dung Reference Dung1995; Wan et al. Reference Wan, Grosof, Kifer, Fodor and Liang2009; Prakken Reference Prakken2010). Philosophers of mind have contemplated the nature of reasoning and inference as mental acts and theorize about the relation between those mental acts and doxastic states, such as beliefs and credences (e.g. Longino Reference Longino1978; Broome Reference Broome2013; Neta Reference Neta2013; Boghossian Reference Boghossian2014). By contrast, comparatively little attention has been paid to arguments as a distinctive kind of discourse, with its own semantics and pragmatics. Most work on speech act theory fails to discuss arguments as a kind of speech act (cf. Austin Reference Austin1975; Searle Reference Searle1969; Searle & Vanderveken Reference Searle and Vanderveken1985). Even recent discussions of speech acts tend to focus primarily on assertions, orders, imperatives, and interrogatives (cf. Murray & Starr Reference Murray, Starr, Fogal, Harris and Moss2018, Reference Murray and Starr2020; Fogal et al. Reference Fogal, Harris and Moss2018). Though arguments have not been widely studied qua linguistic constructions, they are central to linguistic theory and to philosophy (Dutilh Novaes Reference Dutilh Novaes and Zalta2021). Just like we use language for exchanging information, for raising questions, for issuing orders, for making suppositions, etc., we also use language to give arguments, as when we argue on behalf of a certain conclusion and when we share our reasonings. Indeed, giving arguments is one of philosophers’ favorite speech acts; and it is quite remarkably widespread outside the philosophy classroom.

Recent developments in linguistics provide ample new resources for providing a semantics and pragmatics argumentation. We make arguments through constructions of the form:

1. (1)

   a.	$P_1,\dots ,P_n.\text{Therefore}/\text{thus}/\text{hence}/\text{so}\mathrm{C};$
   b.	$\text{Suppose}{P}_1,\dots ,P_n.\text{Then}\mathrm{C}.$

These constructions are sets of sentences – or discourses. It is therefore natural to study these constructions by looking at semantic approaches that take discourses rather than sentences to be the main unit of semantic analysis. Because of this, dynamic approaches to the semantics of arguments will be at the center of my discussion. In particular, I will discuss the resources that discourse coherence approaches (Hobbs Reference Hobbs1985; Asher Reference Asher1993; Asher & Lascarides Reference Asher and Lascarides2003; Kehler Reference Kehler2002) as well as dynamic semantic approaches to the study of language (Veltman Reference Veltman1985, Reference Veltman1996; Beaver Reference Beaver2001; Kaufmann Reference Kaufmann, Faller, Kaufmann and Pauly2000; Brasoveanu Reference Brasoveanu2007; Gillies Reference Gillies2009; Murray Reference Murray2014; Willer Reference Willer2013; Starr Reference Starr2014a, Reference Starr2014b; Pavese Reference Pavese2017, Reference Pavese2021; Kocurek & Pavese Reference Kocurek and Pavese2021) have to understand the semantics and dynamics of arguments.

Speech acts tend to be conventionally associated with certain linguistic features. For example, assertions are associated with the declarative mood of sentences; suppositions with the subjunctive mood, orders with the imperative mood, questions with interrogative features, etc. Like other speech acts, giving an argument is conventionally associated with certain grammatical constructions of the form as (1a) and (1b) above. In order to study the speech act of giving an argument, I will therefore look at the semantics and pragmatics of words such as ‘therefore’, ‘thus’, ‘so’, ‘hence’, and ‘then’ – argument connectives, as Beaver (Reference Beaver2001: 209) calls them – which are used in natural languages to signal the presence of arguments and to express relations between premises and conclusions. These argument connectives exhibit a distinctive anaphoric behavior. Their anaphoric component enables arguments to make use of multiple bodies of information at once. They often consist of multiple suppositions (as in proof by cases), suppositions within suppositions (as in conditional proofs), and so on. As we will see, in order to model these anaphoric relations, discourses have to be thought not simply as a sequences of sentences, but as sequences of labeled sentences – which can track different information states as different sets of premises and suppositions. It also requires thinking of contexts as more structured as usually required in dynamic semantics – not simply as information states or sets of possible worlds, but as having a distinctive layered (indeed, tree-like) structure (Kocurek & Pavese Reference Kocurek and Pavese2021).

Here are a few outstanding questions pertaining the semantics and pragmatics of argumentations: what does the speech act of arguing and making an argument amount to? In particular, how does it affect the context set? What relations do argument connectives express (if any) between premises and conclusions? In virtue of what mechanisms (i.e. presupposition, implicature, etc.) do they get to express those relations? How does the semantics of these words compare to their counterparts in formal languages? How are we to think of the syntax of argumentative discourses and how are we to model contexts in order to model the dynamics of argumentative discourses? Can a unified semantics of argument connectives be provided across their deductive, practical, causal, and inductive usages? How are we to think of the syntax of argumentative discourses and how are we to model contexts in order to model the dynamics of argumentative discourses? What do argument connectives such as ‘therefore’ contribute to the arguments where it occurs? What is the nature of the support relation tested by argument connectives? How are we to model the subtle differences between argument connectives – between ‘therefore’, ‘then’, ‘so’, ‘thus’, and ‘hence’? What makes a discourse an argument, rather than an explanation? How are we to characterize the distinctive utterance force of arguments versus explanations? Are there such things as zero-premises arguments in natural languages? How do deductive arguments in natural language differ, if at all, from proofs in natural deduction systems – such as Fitch’s proofs?

13.1 Introduction

This chapter overviews recent work on the semantics and pragmatics of arguments. In natural languages, arguments are conventionally associated with particular grammatical constructions, such as:

1. (2)

   a.	$P_1,\dots ,P_n.\text{Therefore},\mathrm{C};$
   b.	$\text{Suppose}{P}_1,\dots ,P_n.\text{Then},\mathrm{C}.$

These constructions involve argument words such as ‘therefore’, ‘thus’, ‘so, ‘hence’ and ‘then’ – entailment words (cf. Brasoveanu Reference Brasoveanu2007) or, as I will call them, following Beaver (Reference Beaver2001: 209), argument connectives – which are used in natural languages to signal the presence of arguments. It is, therefore, natural to study the speech act of giving an argument by looking at semantics and pragmatics of argument connectives.Footnote ¹

The first six sections of this chapter look at the semantics of argument connectives. Because arguments typically stretch through discourse, and argument connectives are kinds of discourse connectives, it is natural to start with semantic approaches that take discourses rather than sentences to be the main unit of semantic analysis. Recent developments in linguistics provide ample new resources for a semantics of argumentation. In particular, I will discuss the resources that discourse coherence approaches as well as dynamic approaches to the study of language have to understand the semantics of argument connectives. Section 13.2 compares argument connectives in English to their formal counterparts in proof theory. Section 13.3 explores thinking of argument connectives as expressing discourse coherence relations (e.g. Asher Reference Asher1993; Asher & Lascarides Reference Asher and Lascarides2003; Bras et al. Reference Bras, Le Draoulec and Vieu2001a, Reference Bras, Le Draoulec and Vieu2001b; Le Draoulec & Bras Reference Le Draoulec, Bras, de Saussure, Moeschler and Puskas2007; Bras et al. Reference Bras, Le Draoulec and Asher2009; Jasinskaja & Karagjosova Reference Jasinskaja, Karagjosova, Matthewson, Meier, Rullmann, Zimmermann and Gutzmann2020). Section 13.4 discusses Grice’s view according to which argument connectives come with an associated conventional implicature and compares it to the competing analysis on which ‘therefore’ is a presupposition trigger (Pavese Reference Pavese2017, Reference Pavese2021; Stokke Reference Stokke2017). Section 13.5 discusses Brasoveanu (Reference Brasoveanu2007)’s proposal that semantically ‘therefore’ works as a modal, akin to epistemic ‘must’. Section 13.6 examines dynamic analyses of argument connectives (Pavese Reference Pavese2017; Kocurek & Pavese Reference Kocurek and Pavese2021), with an eye to highlight the scope and the advantages of these sorts of analyses. Section 13.7 looks at the pragmatics of argument connectives and at the difference between arguments and explanations. Section 13.8 concludes.

13.2 Arguments in Logic and in Natural Languages

Consider Argument Schema, with the horizontal line taking a list of premises and a conclusion into an argument:

Argument Schema

$\frac{{\varphi }_1,\dots ,{\varphi }_n}{\psi }$

Now, compare Argument Schema to the following arguments in English:

1. (3)

   a.	There is no on going epidemic crisis. Therefore, there is no need for vaccines.
   b.	It is raining. Therefore, the streets are wet.
   c.	I am smelling gas in the kitchen. Therefore, there is a gas leak.
   d.	This substance turns litmus paper red. Therefore, this substance is an acid.

These arguments all have the form “$\mathrm{\Phi }$, Therefore $\psi$” where $\mathrm{\Phi }$ is the ordered set of premises ${\varphi }_1,\dots ,{\varphi }_n$ and $\psi$ is the conclusion. Because of the syntactic resemblance of Argument Schema and (3a)–(3d), it is tempting to think of ‘therefore’ and other argument connectives such as ‘thus’, ‘so, ‘hence’ and ‘then’ as having the same meaning as the horizontal line (e.g. Rumfitt Reference Rumfitt2015: 53).

However, Argument Schema is not perfectly translated by the construction “$\mathrm{\Phi }$. Therefore/ Thus/ Hence/ Then $\psi$”; nor is the horizontal line perfectly translated by the argument connectives available in English. First of all, the horizontal line does not require premises, for it tolerates conclusions without premises, as in the case of theorems:

Theorem

$\overline{\psi \vee \neg \psi }$

By contrast, ‘therefore’, ‘thus’, ‘so’, ‘hence’, ‘then’, etc. do require explicit premises:Footnote ²

1. (4)

   a.	?? Therefore/hence, we should leave (looking at one’s partner’s uncomfortable face).
   b.	?? Therefore/hence, the streets are wet (looking at the rain pouring outside).
   c.	?? Therefore/hence, either it is raining or it is not raining.

A plausible explanation for this contrast is that ‘therefore’, ‘thus’, ‘so’, ‘hence’, and ‘then’ differ from the horizontal line in that they contain an anaphoric element (cf. Brasoveanu Reference Brasoveanu2007: 296; Kocurek & Pavese Reference Kocurek and Pavese2021). Like anaphors, argument connectives require not just an antecedent but its explicit occurrence.Footnote ³

That is the first difference between ‘therefore’ and the horizontal line. Here is a second difference (cf. Pavese Reference Pavese2017: 95–96; Reference Pavese2021). In Argument Schema, the premises can be supposed, rather than asserted. By contrast, ‘therefore’ (and ‘hence’, ‘thus’, ‘so’) is not always allowed in the context of a supposition:

1. (6)

   a.	It is raining. Therefore/so/hence, the streets are wet.
   b.	?? Suppose it is raining; therefore/so/hence the streets are wet.
   c.	If it is raining, therefore/so/hence the streets are wet.
   d.	??? If Mary is English, therefore/so/hence she is brave.
   e.	??? Suppose Mark is an Englishman. Therefore/so/hence, he is brave.

Under supposition, connectives like ‘then’ are much preferred to ‘therefore’:

1. (7)

   a.	Suppose Φ; then, Ψ.
   b.	Suppose it is raining. Then, the streets are wet.
   c.	If it is raining, then the streets are wet.
   d.	If Mary is English, then she is brave.
   e.	Suppose Mark is an Englishman. Then, he is brave.

For this reason, Pavese (Reference Pavese2017) speculates that the slight infelicity of (6b) may indicate that ‘therefore’ is more similar to the square – i.e. ‘$\square$’ – that ends proofs than to the horizontal line in Argument Schema:

Proof of Theorem

Theorem $\dots \square$

Just like ‘$\square$’, ‘therefore’ would require its premises having been discharged and not conditionally dependent on other premises.

However, the data are more complex than Pavese (Reference Pavese2017) recognizes and should be assessed with caution. ‘Therefore’ can be licensed in the context of supposition. For example, consider:

1. (8)

   a.	If it were raining, the streets would, therefore, be wet.
   b.	Suppose it were raining; the streets would, therefore, be wet.
   c.	If Mary were English, she would, therefore, be brave.
   d.	Suppose Mark were anEnglishman. He would, therefore, be brave.

‘Therefore’ is licensed in this construction, where the mood of the linguistic environment is subjunctive. In this respect, ‘therefore’, ‘thus’, ‘so’, and ‘hence’ differ from ‘then’, for ‘then’ is permitted within the scope of a supposition whether or not the mood is indicative:Footnote ⁴

1. (9)

   a.	Suppose it were raining. Then, the streets would be wet.
   b.	If it were raining, then the streets would be wet.
   c.	If Mary were English, then she would brave.
   d.	Suppose Mark were an Englishman. Then, he would be brave.

Moreover, ‘therefore’ is at least tolerated with so-called ‘advertising conditionals’ – interrogatives that play a role in discourse similar to that of antecedents of conditionals:

1. (10)

   a.	Single? (Then) You have not visited Match.com. (Starr Reference Starr2014a: 4)
   b.	Single? Therefore, you have not visited Match.com.
   c.	Still looking for a good pizzeria? Therefore you have not tried Franco’s yet.

This suggests that at least under certain conditions, ‘therefore’ can appear in suppositional contexts (cf. Pavese Reference Pavese2021).

Another respect under which argument connectives in English differ from the horizontal line in Argument Schema is that while their premises have to be declarative, their conclusion does not need to be.Footnote ⁵ Several philosophers have observed that imperatives can appear as conclusions of arguments (e.g. Parsons Reference Parsons2011, Reference Parsons2013; Charlow Reference Charlow2014; Starr Reference Starr2020):

In addition to allowing imperative conclusions, argument connectives can also have interrogative conclusions:

The final important observation is that argument connectives in English differ from the horizontal line in that they can also appear in nondeductive arguments, both in inductive arguments such as (13a)–(13c), in abductive arguments such as (13c)(13d), in causal arguments as in (14a)–(14d), as well as practical arguments, such as (14e):

1. (13)

   a.	It happened, therefore it can happen again: this is the core of what we have to say. It can happen, and it can happen everywhere. (from Primo Levi, The Drowned and the Saved, Vintage: New York, 1989, p. 199). [Inductive argument]
   b.	Almost every raven is black, and the animal that we are about to observe is a raven. Therefore, it will be black too. [Inductive argument]
   c.	Mark owns a Bentley. Therefore, he must be rich (Douven et al. 2013) [Abductive argument]
   d.	The victim has been killed with a screwdriver. Therefore, it must have been the carpenter. [Abductive Argument]

1. (14)

   a.	John pushed Max. Therefore, Max fell. [Causal Argument]
   b.	John was desperate for financial reasons. Therefore, he killed himself. [Causal Argument]
   c.	Mary qualified for the exam. Therefore, she could enroll. [Causal Argument]
   d.	Reviewers are usually people who would have been poets, historians, biographers, etc., if they could; they have tried their talents at one or the other, and have failed; therefore they turn into critics. (Samuel Taylor Coleridge, Lectures on Shakespeare and Milton) [Causal Argument]
   e.	We cannot put the face of a person on a stamp unless said person is deceased. My suggestion, therefore, is that you drop dead (attributed to J. Edward Day; letter, never mailed, to a petitioner who wanted himself portrayed on a postage stamp). (Brasoveanu Reference Brasoveanu2007: 279) [Practical Argument]

To sum up, there are at least four dimensions along which argument connectives differ from the horizontal line in deductive logic. First, they differ in that they have an anaphoric component; second, they are mood-sensitive, in that whether they allow embedding under supposition and subarguments might depend on the mood of the linguistic environment. Thirdly, argument connectives can allow for nondeclarative conclusions and, fourthly, they can occur with logical, causal and practical flavors, as well as in inductive and abductive arguments.

13.3 Argument Connectives within Discourse Coherence Theory

Giving an argument is a speech act that stretches through a discourse – i.e. from its premises to its conclusion. It is therefore natural to start an analysis of arguments by looking at the resources provided by discourse coherence analysis – an approach to the study of language and communication that aims at interpreting discourses by uncovering coherence relations between their segments (Asher Reference Asher1993; Asher & Lascarides Reference Asher and Lascarides2003). The crucial question behind a coherence discourse theoretic approach to the meaning of argument connectives is, then, what kind of coherence relation they express. The most notable discourse relations studied by discourse coherence theorists are Narration, Elaboration, Background, Continuation, Result, Contrast, and Explanation.

Although this literature has focused much more on temporal discourse connectives than on argument connectives, the general tendency in this literature is to assimilate the meaning of ‘therefore’ to the meaning of ‘then’ in its temporal uses and to its French counterpart ‘alors’ (cf. Bras et al. Reference Bras, Le Draoulec and Vieu2001a, Reference Bras, Le Draoulec and Vieu2001b, Reference Bras, Le Draoulec and Asher2009). According to the prevailing analysis, ‘therefore’ would then introduce Result (Hobbs Reference Hobbs1985; Asher Reference Asher1993; Asher & Lascarides Reference Asher and Lascarides2003; Asher & Gillies Reference Asher and Gillies2003; Kehler Reference Kehler2002).Footnote ⁶ If the relation of Result is a causal relation: if it holds between two constituents, then the former causes the latter.

While this account captures well causal uses of ‘therefore’ as in (14a)–(14c), not every use of ‘therefore’ is plausibly causal in this fashion. For example, in the following arguments, the truth of the premises does not cause the truth of the conclusion:Footnote ⁷

1. (15)

   a.	All the girls have arrived. Therefore, Mary has also arrived.
   b.	Mary has arrived. Therefore, somebody has arrived.
   c.	2 is even. Therefore either 2 is even or 3 is.

In order to extend their discourse coherence analysis to uses of ‘therefore’ that are recalcitrant to the causal analysis, Bras et al. (Reference Bras, Le Draoulec and Asher2009) proposes we appeal to Inferential Result – i.e. a relation holding between two events or propositions just in case the latter is a logical consequence of the former ($K_{\alpha }$ indicates a constituent’s way of describing an event $\alpha$ and the arrow stands for the material conditional):

$\text{I}\mathrm{\text{NFERENTIAL}}\text{R}\mathrm{\text{ESULT}}\left(\alpha ,\beta \right)\text{ iff }\square \left(K_{\alpha }\to K_{\beta }\right).$

However, not every nonnarrative use of argument connectives can be analyzed in terms of Inferential Result. For example, consider the use of ‘therefore’ in inductive, abductive, or practical arguments, as in (13c)–(14e). None of these arguments plausibly express Inferential Result. Even if we restrict Inferential Result to the deductive uses of argument connectives, the problem remains that this approach would result in a rather disunified theory of the meaning of argument connectives. We are told that sometimes discourses involving ‘therefore’ express the causal relation of Result, sometimes they express a different discourse relation altogether – i.e. Inferential Result or classical entailment in deductive uses, and maybe some other discourse relations in practical and inductive uses.

Here is a unifying proposal, one that preserves the discourse coherence theorists’ important insight that ‘therefore’ is a discourse connector expressing some or other discourse relation. Suppose we understand Result in terms of a restricted notion of entailment. For example, we might understand Result in terms of nomological entailment – entailment given the laws of nature – or default entailment, as in Asher and Morreau (Reference Asher, Morreau and van Eijck1990) and Morreau (Reference Morreau1992) (cf. Meyer & van der Hoek Reference Meyer and van der Hoek1993; Weydert Reference Weydert1995; Veltman Reference Veltman1996). Quite independently of the consideration of argument connectives, Altshuler (Reference Altshuler2016) has proposed that we understand Result in terms of enthymematic nomological entailment.Footnote ⁸ $\varphi$ enthymematically entails the proposition $\psi$, if and only if there is a nonempty set of propositions $\mathrm{\Phi }$ such that $\mathrm{\Phi }\cup \left\{\varphi \right\}$ logically entails $\psi$. For example, consider again (14a). While John’s having pushed Max does not entail that Max fell, Altshuler (Reference Altshuler2016: 70–71) proposes John’s having pushed Max might enthymematically entail that Max fell, for John’s having pushed Max in conjunction with an appropriate set of background propositions might entail that Max fell.Footnote ⁹

Following and extending this proposal, we might then take argument connectives in their deductive uses to express a nonrestricted form of entailment – i.e. classical (or relevantist) entailment; by contrast, in their causal uses, they express nomological entailment and in their practical uses practical entailment – entailment given the prudential/ practical/ moral laws. Inductive uses might be understood in terms of a restricted form of entailment as well, where the restriction comes from the general principle of uniformity of nature or a specific version thereof (cf. Kocurek & Pavese Reference Kocurek and Pavese2021 for this unifying idea). On this proposal, every use of argument connectives expresses some more or less general relation of entailment. We thereby reach unification across uses of argument connectives while preserving the differences.

In conclusion, discourse coherence theory provides us with the resources to study the semantics and pragmatics of arguments from the correct methodological standpoint: because arguments are discourses, this approach analyzes argument connectives as discourse connectors and thus as expressing discourse relations. From our discussion, however, it emerges that argument connectives appear with a variety of different flavors (narrative, causal, inferential, etc.), and so the question arises of what unified discourse relation they express. In order to capture what is common to all of these uses, it seems promising to think of the relevant discourse relations in terms of more or less restricted relations of entailment.

13.4 Conventional Implicature or Presupposition?

In “Logic and Conversation,” Grice (Reference Grice, Cole and Morgan1975: 44–45) uses the case of ‘therefore’ to illustrate the notion of a conventional implicature. Grice observes that in an argument such as (16a) and in a sentence such as (16b), ‘therefore’ contributes the content that the premise entails the conclusion – in other words, it contributes Target Content:

1. (16)

   a.	Jill is English. Therefore, she is brave. (‘therefore’-argument)
   b.	Jill is English and she is, therefore, brave. (‘therefore’-sentence)
   c.	Jill is English and she is brave.
   d.	Her being brave follows from her being English. (Target Content)

Grice points out that in an argument such as (16a) or in a sentence such as (16b), Target Content is communicated without being asserted, for by saying (16b), one commits to Target Content’s being true but whether Target Content is true does not contribute to what is said by (16b). Grice took this to indicate that Target Content is only conventionally implicated by ‘therefore’, for he further thought that (16b) would not be false if Target Content were false. It is customary for linguists and philosophers to follow Grice here. For example, Potts (Reference Potts2007: 2) tells us that the content associated with ‘therefore’ is a relatively uncontroversial example of a conventional implicature (see also Neta Reference Neta2013 and Wayne Reference Wayne and Zalta2014: section 2). Whether the conventional implicature analysis of ‘therefore’ best models the behavior of ‘therefore’ is, however, questionable. Some have argued that several considerations suggest that the explanatory category of presuppositions, rather than that of conventional implicatures, might actually better capture the status of the sort of content that is conveyed by argument connectives (see Pavese Reference Pavese2017, Reference Pavese2021; Stokke Reference Stokke2017).

The first kind of evidence for this claim is that ‘therefore’ satisfies the usual tests for presupposition triggers: Projectability and Not-At-Issuedness. Start with Projectability. Like standard presupposition triggers, Target Content projects out of embeddings – i.e. out of negation (17a), out of questions (17b), in the antecedents of conditionals (17c), out of possibility modals (17d) and out of evidential modal and probability adverbs (17e), as can be seen from the fact that all of the following sentences still convey that Mary’s braveness follows from her being English:

1. (17)

   a.	It is not the case that Mary is English and, therefore, brave. (Negation)
   b.	Is Mary English and, therefore, brave? (Question)
   c.	If Mary is English and, therefore, brave, she will act as such. (Antecedent of a conditional)
   d.	It might be that Mary is English and, therefore, brave. (Possibility Modal)
   e.	Presumably Mary is English and therefore brave. (Evidential modal, probability adverb)

Some speakers also hear a nonprojective reading for Negation (17a). On this projective reading, we are not simply denying that Mary is English. We are denying that her braveness follows from her being English. However, the claim that ‘therefore’ works as a presupposition trigger in (17a) is compatible with (17a) also having a nonprojective reading. For example, consider (18):

Clearly, the definite article in “the knave” must have a nonprojective reading in “The tarts were not stolen by the knave,” for else (18) would have to be infelicitous. Presumably, whatever explains the nonprojective reading in (18) can explain the nonprojective reading in (17a) (cf. Abrusán Reference Abrusán2016, Reference Abrusán and Altshuler2022). The standard explanations for nonprojective readings under negation are available here: maybe we are dealing with two different kinds of negation (metalinguistic negation versus negation simpliciter (cf. Horn Reference Horn1972, Reference Horn1985); or we might be dealing with an example of local accommodation (cf. Heim Reference Heim, Portner and Partee1983); or we might appeal to Bochvar (Reference Bochvar1939)’s A operator (cf. Beaver Reference Beaver1985; Beaver & Krahmer Reference Beaver and Krahmer2001).

Hence, Target Content is projectable to the extent to which presuppositions are usually taken to be projectable. Moreover, Target Content satisfies the second standard set of tests for spotting presupposition triggers – i.e. the not-at-issuedness tests. Target Content also cannot be directly challenged – i.e. (19a) and (19b) – in striking contrast to when it is instead made explicit – i.e. (19c)–(19d):

1. (19)

   a.	Jill is English and, therefore, she is brave.
   	*That is false/That is not true.
   b.	Jill is English. Therefore, she is brave.
   	*That is false/That is not true.
   c.	Jill is English and from that it follows that she is brave.
   	That is false/That is not true.
   d.	Jill is English. It follows from that that she is brave.
   	That is false/That is not true.
   e.	Jill is English and, therefore, she is brave. Hey, wait a minute! Not all English people are brave!
   f.	Jill is English. Therefore, she is brave. What? Not all English people are brave!

While the Target Content cannot be directly challenged, it can be indirectly challenged, by taking some distance from the utterance, as evidenced by (19e) and (19f), through locutions such as ‘wait a minute!’ and ‘what?’. Note that this phenomenon is not just observable for inferential uses of ‘therefore’. The same pattern is observable for narrative uses of ‘therefore’ too:

1. (20)

   a.	John was desperate for financial reasons. Therefore, he killed himself.
   b.	*That is false/*That is not true. He did not kill himself for financial reasons.
   c.	Wait a moment!!! He did not kill himself for financial reasons.
   d.	What?? He did not kill himself for financial reasons.

That suggests that whether the relation expressed by ‘therefore’ is classical entailment (in inferential uses of ‘therefore’) or some restricted notion of entailment (as in narrative uses of ‘therefore’), such relation is backgrounded in the way presuppositions are.

Like presuppositions, Target Content also cannot be cancelled when unembedded, on pain of Moorean paradoxicality:

1. (21)

   a.	?? Jill is English. Therefore, she is brave. But her braveness does not follow from her being English.
   b.	?? Jill is English. Therefore, she is brave. But I do not believe/know that her being brave follows from her being English.

And like other strong presupposition triggers, which cannot felicitously follow retraction (cf, Pearson Reference Pearson2010), ‘therefore’ cannot follow retraction either, as evidenced by (22a) and (22b)

1. (22)

   a.	?? Well, I do not know if her braveness follows from her being English. But Mary is English. And therefore, she is brave.
   b.	?? Well, I do not know if her being from the North follows from her being progressive. But Mary is a progressive. And therefore, she is from the North.

Finally, just like presuppositions issued by strong presupposition triggers Target Content cannot even be suspended, as evidenced by (23c) (Abrusán Reference Abrusán2016, Reference Abrusán and Altshuler2022):

Do these tests suffice to show that ‘therefore’ is a presupposition trigger? Now, the boundaries between conventional implicatures and presuppositions are notoriously hard to draw. And many supposed examples of conventional implicatures also satisfy many of the aforementioned tests. However, there are some additional considerations that suggest that the presuppositional analysis is more explanatory of the behavior of argument connectives. Conventional implicatures project even more massively than presuppositions (Potts Reference Potts2015: 31). For example, additive articles such as ‘too’ and ‘also’ project out of standard plugs such as attitude reports (cf. Karttunen Reference Karttunen1973). By contrast, the presupposition associated with ‘therefore’ can be plugged by belief reports:

Also under epistemic modals and negation, not-projective readings are sometimes available for ‘therefore’ (cf. Pavese Reference Pavese2021; Kocurek & Pavese Reference Kocurek and Pavese2021 for discussion).

Moreover, it seems a necessary condition for presuppositions that a sentence s presupposes p only if s does not warrant an inference to p when s is in an entailment-cancelling environment and when p is locally entailed (cf Mandelkern Reference Mandelkern2016). This condition is satisfied also by discourses featuring ‘therefore’. For example, the following conditionals do not entail Target Content:

1. (25)

   a.	If being brave follows from being English, Mary is English and, therefore, brave.
   b.	If liking the Steelers follows from being from Pittsburgh, then Mary likes the Steelers and, therefore, she is from Pittsburgh.

In conclusion, the presuppositional analysis seems to capture the projective behavior associated with ‘therefore’ better than the conventional implicature analysis.Footnote ¹⁰ I take it, however, that the real interesting question – and the one I will focus on going forward – is not how to label ‘therefore’ (whether as a presuppositional trigger or as a conventional implicature trigger) but rather how best to formally model its projective and nonprojective behavior. It is to this question which I turn next.

13.5 ‘Therefore’ as a Modal

Another important observation about the meaning of ‘therefore’ is that it closely resembles that of necessity modals. For example, (26) is very close in meaning to the modalized conditional (27):

1. (26)

   a.	Sarah saw a puppy. Therefore, she petted it.
   b.	If Sarah saw a puppy, she (obviously/necessarily/must have) petted it.

provided that we add to (26b) the premise (27):

Moreover, as we have seen in (13a)–(14e), ‘therefore’ comes in different flavors (logical, causal, practical, inductive, abductive). So in this respect too it resembles modals (cf. Kratzer Reference Kratzer1977, Reference Kratzer, Portner and Partee2002). On these bases, following Kratzer’s analysis of modals, Brasoveanu (Reference Brasoveanu2007) proposes we understand different flavors of ‘therefore’ as resulting from a restriction of the corresponding ‘modal base’. A modal base is a variable function from a world to a set of propositions, modeling the nature of the contextual assumptions – whether causal, practical, or epistemic. Its intersection returns the set of possible words in which all the propositions in the modal base are true. The logical consequence flavor of ‘therefore’ derives from an empty modal base, whose intersection is the universe. This formally captures the fact that logical consequence is the unrestricted flavor of ‘therefore’.

This approach captures both the similarity between ‘therefore’ and ‘must’ and several possible flavors with which ‘therefore’ is used. However, it is unclear that this approach resorting to modal bases can effectively model inductive and abductive uses of ‘therefore’. Inductive arguments are notoriously nonmonotonic. For example, consider:

1. (28)

   a.	The sun has risen every day in the past. Therefore, the sun will rise again tomorrow.
   b.	The sun has risen every day in the past. And today is the end of the world. ??Therefore, the sun will rise again tomorrow.

If we apply the modal base approach to (28a), we get that in any context where (28a) is felicitious, (28b) should be, too. For suppose in our current state s, when we update s with the premises in (28a), each world in the resulting state $s^{\prime }$ is assigned by the modal base a set of propositions whose intersection supports the conclusion. Let $s″$ be the result of updating s with the premises in (28-b). Since every world in $s″$ is a world in $s^{\prime }$, when we apply the modal base to a world in $s″$, it also supports the conclusion. One way Brasoveanu’s approach could be extended to model the nonmonotonicity of inductive arguments is by appeal to some context shift. But it is difficult to see how the sort of context shifts needed could be motivated. This observation does not undermine the important similarity between ‘therefore’ and ‘must’ observed by Brasoveanu (Reference Brasoveanu2007), for ‘must’ seems to be amenable to inductive uses too, as in:

However, it does raise the issue of how to model inductive and abductive uses of both ‘therefore’ and modals. (For promising work in this respect, see Del Pinal Reference Del Pinal2021).

13.6 Dynamic Treatments of Argument Connectives

13.6.1 A Simple Semantics

So far, we have observed that argument connectives appear to behave as presupposition triggers and that they also resembles modals. Any semantic analysis ought to capture these two sets of data. Pavese (Reference Pavese2017) suggests that dynamic semantics offers the tools to develop an analysis that meets this desiderata. Kocurek and Pavese (Reference Kocurek and Pavese2021) improve on Pavese (Reference Pavese2017)’s analysis and develop this proposal in some detail. Here, I review some of the most important aspects of these dynamic analyses.

In dynamic semantics, a test is an expression whose role is to check that the context satisfies certain constraints, as Veltman (Reference Veltman1996)’s ‘might’ or von Fintel and Gillies (Reference von Fintel, Gillies, Gendler and Hawthorne2007)’s ‘must’. These expressions check that the context supports their prejacent: so “It might be raining” checks that the context supports the sentence that it is raining.

Define an information state as a set $s\subseteq W$ of worlds. We define the update effect of a sentence on an information state recursively, as follows:

$\begin{array}{l}s\left[p\right]=\left\{w\in s,,|,,w\left(p\right)=1\right\}\\ s\left[\neg \varphi \right]=s-s\left[\varphi \right]\\ s\left[\varphi \wedge \psi \right]=s\left[\varphi \right]\left[\psi \right]\\ s\left[\varphi \vee \psi \right]=s\left[\varphi \right]\cup s\left[\psi \right]\\ s\left[\square \varphi \right]=\left\{w\in s,,|,,s\left[\varphi \right]=s\right\}\\ s\left[\diamond \varphi \right]=\left\{w\in s,,|,,s\left[\varphi \right]\ne \varnothing \right\}\\ s\left[\varphi \to \psi \right]=\left\{w\in s,,|,,s\left[\varphi \right]\left[\psi \right]=s\left[\varphi \right]\right\}.\\ s\left[\therefore \varphi \right]=\left\{\begin{array}{ll}s& \text{if}s\left[\varphi \right]=s\\ \text{undefined}& \text{otherwise}\end{array}\right.\end{array}$

In the above definition, $\square$, $\diamond$, $\to$ $\therefore$ are all tests. $\diamond$ (corresponding to Veltman Reference Veltman1996’s ‘might’) tests whether the context is compatible with its prejacent; if not, it returns the empty set. $\square$ (corresponding to von Fintel & Gillies Reference von Fintel, Gillies, Gendler and Hawthorne2007 and von Fintel & Gillies Reference von Fintel and Gillies2010’s ‘must’) tests that the context supports its prejacent – i.e. that $s\left[\varphi \right]=s$. If not, it returns the empty set. Notice that $\therefore$ (corresponding to our ‘therefore’) is similar to $\square$ — like $\square$ it checks that the current context (augmented with ‘$\therefore$’s antecedents) supports the conclusion. $\therefore$ also closely resembles $\to$ (corresponding to Veltman Reference Veltman1985’s conditional): the latter tests whether the context augmented with the antecedent supports the consequent; $\therefore$ tests whether the context augmented with the premises support the conclusion. One respect in which discourses containing ‘therefore’ differ from Veltman (Reference Veltman1985)’s conditional is that Veltman (Reference Veltman1985) conditionals return the initial context after the test. But intuitively, an argument updates the context with the premises. For example, an argument with assertoric premises P after the checking must return the context updated with P. To see why this must be so, consider:

(30)

Paolo is from Turin_i. Therefore_i he is from Piedmont_j. And, therefore_j he is from Italy.

If in (30), ‘therefore_i he is from Piedmont_j ’ returned the context antecedent to the update with ‘Paolo is in Turin_i ’, the output context might not support the proposition that Paolo is from Italy. So we cannot explain why (30) is a good argument. This observation motivates taking the entry for $\therefore$ to model this feature of ‘therefore’: $\therefore$ takes the current context (already updated with its antecedents) and returns that context if the test is positive. This explains why successive ‘therefore’ can test the context so updated with the earlier premises (see Kocurek & Pavese Reference Kocurek and Pavese2021 for a proposal on which the conditional test also returns the context updated with the antecedents, motivated by the need to model modal subordination under conditionals).

These entries allow to capture the similarities between necessity modals such as ‘must’ and ‘necessarily’ and ‘therefore’ that we have observed in the previous section. On this proposal, one notable difference between ‘therefore’ and ‘must’ that is relevant for our purposes is that if the test fails, the former returns an undefined value rather than the empty set. This feature is needed to account for the different projective behavior of ‘therefore’, ‘must’ and the conditional. Conditionals and ‘must’ are not plausibly presupposition triggers. ‘Must’-sentences, and in general sentences containing modals, do not need to presuppose that the context supports their prejacent. Consider:

1. (31)

   a.	It is not the case that Mark is a progressive and must be from the North.
   b.	Is Mark a progressive and must be from the North?
   c.	If Mark is a progressive and must be from the North, he will not vote for Trump.
   d.	It might be that Mark is a progressive and must be from the North.

None of these convey that Mark’s being from the North follows in any way from him being a progressive. Conditionals also do not project out when embedded in antecedent:

(32)

If Jen gets angry if irritated, you should not mock her.

(32) does not presuppose that Jen will get angry follows from her being irritated. ‘Therefore’ seems to differ from other tests such as conditionals and ‘must’ in that the checking is done by the presupposition triggered by ‘therefore’.

‘Therefore’-discourses are infelicitous if the checking is not positive, like in the case of ‘must’-sentences and Veltman (Reference Veltman1985)’s conditional. But in the case of ‘therefore’, the infelicity is due to presupposition failure. Because of its behavior as a presupposition trigger, it is more accurate to give ‘therefore’ a semantic entry similar to the one that Beaver (Reference Beaver2001: 156–162) assigns to the presuppositional operator $\delta$:

$s\left[\mathit{\delta \varphi }\right]=\left\{\begin{array}{ll}s& \text{if}s\left[\varphi \right]=s\\ \text{undefined}& \text{otherwise}\end{array}\right.$

Compare $\square$ on one hand and $\delta$ and $\therefore$ on the other. They only differ in that the former returns the empty set if the context does not support $\varphi$, whereas the latter returns an undefined value. The difference between these two ‘fail’ values – undefinedness versus the empty set – is important. A semantic entry that returns the empty set receives a nonfail value under negation. But in order to account for the projection of the presupposition from a sentence containing ‘therefore’ to its negation, the negation of that sentence must also receive a fail value if the sentence does. Choosing ‘undefined’, rather than the empty set, gives the desired result here – i.e. that the negation of the sentence containing ‘therefore’ will also be undefined.

This analysis can be illustrated with the following example. Consider:

(33)

It’s not the case that Mark is progressive and, therefore, from the North.

$\neg \left(p\wedge \therefore n\right)$

Compositionally, we get that the meaning of (33) is the following function:

$\begin{array}{l}s\left[\neg \left(p\wedge \therefore n\right)\right]=s-s\left[p\wedge \therefore n\right]\\ =s-s\left[p\right]\left[\therefore n\right]\\ =\left\{\begin{array}{ll}s-s\left[p\right]& \text{if}s\left[p\right]\left[n\right]=s\left[p\right]\\ \text{undefined}& \text{otherwise}\end{array}\right.\end{array}$

13.6.2 Refining the Analysis: Supposition, Parenthetical, and Subarguments

While this analysis might be a good starting point, it is oversimplified in several ways. One way in which it is oversimplified is that it says nothing about how to model arguments that have not premises but other arguments as antecedents, such as conditional proofs:

(34)

Suppose Paolo is from Turin, Then he is from Piedmont. Therefore, if Paolo is from Turin he is from Piedmont.

Moreover, argumentative discourses seem to have a layered structure: suppositions introduce new states of information, at a different level from categorical states of information, and suppositions can be embedded to add further levels. For example, consider:

(35)

Paolo is either from Turin or from Madrid. Suppose₁, on the one hand, that he is from Turin. Then₁ either he did his PhD there or he did it in the US. Suppose_1.1 he did his PhD in Turin. Then_1.1, he studied Umberto Eco’s work. Suppose_1.2 instead he did his PhD in the US. Then_1.2 he studied linguistics. Therefore₁, he either did continental philosophy or philosophy of language. Now on the other hand, suppose₂ he is from Madrid. Then₂ he definitely did his PhD in the US. Therefore₂, he studied linguistics. Either way, therefore, he did either continental philosophy or philosophy of language.

As the indexes indicate, in (35), supposition₁ introduces a new layer, over and above the categorical context where ‘Paolo is either from Turin or from Madrid’. Moreover, suppositions can be embedded one after the other (as supposition 1 and supposition 1.1) or might be independent (as supposition 1 and supposition 2). ‘therefore’ and ‘then’ might test the context introduced by the most recent premises or suppositions (as ‘then₂’ and ‘therefore₂’) or refer back to suppositions introduced earlier (as ‘therefore₁’). Finally, after a supposition, parentheticals can be used to add information to the categorical level and to every level above. For example, consider:

(36)

Suppose Mary went to the grocery store this morning. [Have you been? It’s a great store with great fruit.] She bought some fruit. Therefore, she can make a fruit salad.

To model the discourse in (36), we need to be able to exit the suppositional context, update the categorical context, and then return back to that suppositional context. In (36), however, the information added by the parenthetical to the categorical content seems to percolate up to the suppositional context too. Ideally, a theory of argumentative discourse ought to be able to account for these complexities. It seems that in order to model discourses such as (36), we need to refine Pavese (Reference Pavese2017)’s analysis in some important ways.

Kocurek and Pavese (Reference Kocurek and Pavese2021) propose we can model these data by adding structure both to the syntax of discourses as well as to the contexts used to interpret them. In order to capture the syntax of argumentative discourses such as the above, they propose we take discourses not just as sequences of sentences but rather as sequences of labeled sentences. A labeled sentence is a pair of the form $〈n,\varphi 〉$, which we write as $n:\varphi$ for short (Throughout, we use $\varnothing$ to stand for the empty tuple $〈〉$). So parts of discourses are labeled sentences. Here, n is a label, which is a sequence of numbers (where, for shorthand, we write $〈n_1,\dots ,n_k〉$ as $n_1.n_2.\dots .n_k$) that represents which suppositions are active, and $\varphi$ is a sentence. Labels enable to keep track of which suppositions are active when and to model the function of parentheticals of going back to the categorical contexts. So for example, the following is a representation of (36) with labeled sentences (where m = ‘Mary went to the grocery this morning’; g = ‘Have you been? It’s a great store with great fruit’; b = ‘She bought some fruit’; f = ‘She can make a fruit salad’).

$1:m,\varnothing :g,1:b,1:\therefore f$

The second move is to distinguish between the meaning of a sentence and the meaning of a part of a discourse – or labeled sentence. The meaning of a sentence is simply its update effect on information states – i.e. a function from information states to information states, as outlined in Section 13.6.1. This semantics would suffice if argumentative discourse did not have the layered structure we have seen it does have and if argument connectives did not license different anaphoric relations towards their antecedents. This further information is captured by parts of discourses or labeled sentences. So, in order to capture suppositional reasoning as well as the anaphoric relations that argument connectives establish in discourse, we ought to interpret labeled sentences as well. While the meaning of sentences is a function from information states to information states, the meaning of parts of discourses is its update effects on a context. Instead of modeling contexts as information states, Kocurek and Pavese (Reference Kocurek and Pavese2021) model contexts rather as labeled trees – i.e. a tree where each node is an information state which is given its own label. Labeled trees contain much more structure than simple information states. They also contain more structure than stacks of information states of the sort proposed by Kaufmann (Reference Kaufmann, Faller, Kaufmann and Pauly2000) to model suppositional reasoning. Labeled trees differ from stacks of information states in that (1) they allow nonlinear branching, so that independent suppositions can be modeled at the same “level” as well as at different levels and (2) can model anaphoric relations, which will allow us to temporarily exit a suppositional context and later to return to that context. This also allows us to capture the distinctive ability of ‘therefore’ to be anaphoric on different suppositional contexts. A context is a partial function $c:{\mathrm{\mathbb{N}}}^{<\omega }\to \wp W$ from labels (i.e. sequences of numbers) to information states, where:

• $\varnothing \in dom\left(c\right)$ (i.e. the categorical state is always defined);

• if $〈n_1,\dots ,n_{k+1}〉\in dom\left(c\right)$, then $〈n_1,\dots ,n_k〉\in dom\left(c\right)$ (i.e. a subsuppositional state is defined only when its parent suppositional state is defined).

The value of a context applied to the empty sequence is the categorical state, denoted by $c_{\varnothing }$. The value of a context applied to a nonempty sequence is a suppositional state. So for example, $n:\varphi$ will tell us to update c_n with $\varphi$. However, when we introduce a new supposition in a discourse, we don’t simply update the current information state with that supposition (suppositions are not just assertions). Rather, we create a new information state updated with that supposition so that subsequent updates concern this new state as opposed to (say) the categorical state. The new supposition effectively copies the information state of its parent and then updates that state with the supposition.

Formalizing, where $n=〈n_1,\dots ,n_{k+1}〉$ is a label, let $n^{-}=〈n_1,\dots ,n_k〉$ (${\varnothing }^{-}$ is undefined). This will allow us to keep track of which information state gets copied when a new supposition is introduced. For labels n and k, we write $n⊑k$ just in case n is an initial segment of k and $n⊏k$ just in case n is a proper initial segment of k (i.e. k is “above” n in the labeled tree). Where c is a context, let $c{\uparrow }_n\varphi$ be the result of replacing c_k with $c_k\left[\varphi \right]$ for each $k\in dom\left(c\right)$ such that $k⊒n$ (i.e. $c{\uparrow }_n\varphi$ updates c_n and all information states “above” c_n in the tree with $\varphi$). Finally, where s is an information state, let $c\left[n\mapsto s\right]$ be just like c except that $c_n=s$:

$c\left[n:\varphi \right]=\left\{\begin{array}{ll}c{\uparrow }_n\varphi & \text{if}{c}_n\text{is defined}\\ c\left[n\mapsto c_{n^{-}}\left[\varphi \right]\right]& \text{if}{c}_n\text{is}\text{not}\text{defined}\text{but}{c}_{n^{-}}\text{is defined}\\ \text{undefined}& \text{otherwise}\end{array}\right.$

Unpacking this semantic clause: If c_n is defined, we update c_n and all subsequent states above it with $\varphi$. If $n=\varnothing$ (the categorical state), then every state that’s currently defined is updated with $\varphi$. If $n=〈n_1,\dots ,n_k〉$, then we only update states assigned to a label that starts with $n_1,\dots ,n_k$. If c_n is undefined, that means we’re creating a new suppositional state:

• First, find the state whose label is right below n (so, e.g. if $n=〈1〉$, then the label right below n is $〈〉$, i.e. the label of the categorical state).

• Next, copy the state with that label and assign n to that state. Finally, update that copied state with $\varphi$.

This semantics for parts of discourses can be illustrated by considering two examples. Under a plausible interpretation, the following discourse is represented as the following sequence of labeled sentences:

(37)

Either it is raining or not. Suppose it’s raining. Then better to take the umbrella. Suppose it is not raining. Then, taking the umbrella will do no harm. Therefore, you should take the umbrella.

$\varnothing :\left(r\vee \neg r\right),1:r,1:\therefore u,2:\neg r,2:\therefore u,\varnothing :\therefore u$

The dynamics of this discourse can be summarized as follows: First, we update the categorical state s with the trivial disjunction $r\vee \neg r$ (so no change). Next, $1:r$ requires setting $c_1=s\left[r\right]$. Then $1:\therefore u$ tests $s\left[r\right]\left[u\right]=s\left[r\right]$. If it passes, it returns $s\left[r\right]$ as $c_1$. Otherwise, the context is undefined. Assuming $s\left[r\right]$ passes the test, $2:\neg r$ requires defining a new information state $c_2=s\left[\neg r\right]$. Then $2:\therefore u$ tests $s\left[\neg r\right]\left[u\right]=s\left[\neg r\right]$. If it passes, it returns $s\left[\neg r\right]$ as $c_2$. Otherwise, the context is undefined. Assuming $s\left[\neg r\right]$ passes the test, $\therefore u$ tests $s\left[u\right]=s$. Since $s\left[r\right]$ and $s\left[\neg r\right]$ have passed this test, s will, too. Or consider the following example with a parenthetical:

(38)

Suppose Mary went to the grocery store this morning. [Have you been? It’s a great store.] Then she bought some fruit. Therefore, she can make a fruit salad.

This is represented as:

$1:m,\varnothing :g,1:\therefore b,1:\therefore f$

First, we introduce a suppositional context c₁ by copying s and updating it with $s\left[m\right]$. Next, $\varnothing :g$ updates both the categorical context s and the suppositional context $s\left[m\right]$ with g. Then $1:\therefore b$ tests $s\left[m\right]\left[g\right]\left[b\right]=s\left[m\right]\left[g\right]$. If it passes, it returns $s\left[m\right]\left[g\right]$ as c₁. Otherwise, the context crashes. Likewise for $1:\therefore f$.

13.7 The Pragmatics of Arguments

So much for the semantics of arguments. Onto the pragmatics. How are we to model the speech act of giving an argument? To begin, compare the following two discourses:

1. (44)

   a.	It is raining. I conclude that the streets are wet.
   b.	It is raining. Therefore, the streets are wet.

Prima facie, these two discourses are equivalent. The locution “I conclude that …” seems to mark the speech act of concluding. It is tempting, then, to assimilate the meaning of ‘therefore’ to the meaning of “I conclude that …”. On this analysis, argument connectives such as ‘therefore’ work as a speech act modifier – taking pairs of sentence types, into a distinctive kind of speech act – i.e. the speech act of giving an argument for a certain conclusion.Footnote ¹²

One issue with this analysis is that argument connectives are not always used to make arguments. Consider again (45a)–(45d) from Section 13.2:

1. (45)

   a.	John pushed Max. Therefore, Max fell.
   b.	John was desperate for financial reasons. Therefore, he killed himself.
   c.	Mary qualified for the exam. Therefore, she enrolled.
   d.	Max passed his A-levels. Therefore, he could go to the university.

While superficially, these discourses have the same form of an argument, they can be used to make other speech acts too. For example, one may utter, say, (45a) without arguing for the conclusion that Max fell. In fact, the most common use of (45a) is simply to explain what happened when John pushed Max (suppose (45a) is used in the process of reporting what happened yesterday). In this use, the discourse does not necessarily have argumentative force. Rather, it uses ‘therefore’ narratively or explanatorily. Similarly for (45b). Arguments and explanations are different kinds of speech acts. That can be seen simply by observing that while an explanation might presuppose the truth of its explanandum, an argument cannot presuppose the truth of its conclusion, on pain of being question-begging. For example, one might use (45a) in the course of an explanation of how Max fell, in a context where it is already common ground that Max fell. As used in this explanation, (45a) is not the same as an argument.

It is also tempting to think that the causal uses are explanatory and not argumentative whereas the logical uses are argumentative but not explanatory. However, this cannot be correct, as there are causal and yet argumentative uses of ‘therefore’. For example, consider TRIAL:

TRIAL In a trial where John is accused of murdering his wife, the prosecutor argues for his conviction, as follows:

The discourse (46) in TRIAL can undeniably be used in an argument – for example, an argument aiming to convince the jury of the fact that John has killed his wife. And yet the relation expressed by this use of ‘therefore’ is causal, if anything is.

There are also deductive uses of ‘therefore’ in explanations. For example, consider the following (Hempel Reference Hempel1962; Railton Reference Railton1978):

1. (i) Whenever knees impact tables on which an inkwell sits and further conditions K are met (where K specifies that the impact is sufficiently forceful, etc.), the inkwell will tip over. (Reference to K is necessary since the impact of knees on table with inkwells does not always result in tipping.)

2. (ii) My knee impacted a table on which an inkwell sits and further conditions K are met.

Explanandum Therefore, the inkwell tipped over.

In this explanation of why the inkwell tipped over, that the inkwell tipped over deductively follows from the premises. In this sense, there are logical uses of ‘therefore’ in explanations too.

The conclusion is that the distinction between argumentative uses of ‘therefore’ and explanatory uses of ‘therefore’ cuts across the distinction between causal and logical meaning of ‘therefore’. How are we to capture this distinction between argumentative uses of ‘therefore’ and explanatory uses of ‘therefore’? This distinction might have to be captured not at the level of the semantics of arguments but rather at the level of the pragmatics of arguments. Chierchia and McConnell-Ginet (Reference Chierchia and McConnell-Ginet2000) have introduced an important distinction (then defended and elaborated by Murray & Starr Reference Murray, Starr, Fogal, Harris and Moss2018, Reference Murray and Starr2020) between conventional force and utterance force. The conventional force of a sentence type consists in the distinctive ways different sentence types are used to change the context – e.g. declaratives are used to change the common ground, by adding a proposition to the common ground (Stalnaker Reference Stalnaker, Portner and Partee1978); interrogatives affect the questions under discussion (e.g. Groenendijk & Stokhof Reference Groenendijk and Stokhof1982; Roberts Reference Roberts, Yoon and Kathol1996) and imperatives the to do list (e.g. Portner Reference Portner2004, Reference Portner2007; Starr Reference Starr2020; Roberts Reference Roberts, Yoon and Kathol1996). Utterance force, by contrast, consists in the distinctive ways utterance types change the context. This is the total force of an utterance, while the conventional force is the way a sentence’s meaning constrains utterance force. Crucially, as Murray and Starr (Reference Murray and Starr2020) argue, conventional force underdetermines utterance force. For example, assertions are conventionally associated with declarative sentences. However, declarative sentences can also be used to make conjectures, to lie, to pretend, etc. So, while the conventional force of a speech act is conventionalized and can be modeled by looking at its invariant conversational effects on a public scoreboard, the utterance force of a speech act might vary depending on the effects of the speech act on the private mental states of the participants to the conversations as well as on the mental state of the utterer.

Suppose we apply this distinction between conventional force and utterance force to the case of argument connectives and discourses that feature them. The proposal then is that across all of its uses – causal, explanatory, as well as practical, inductive, deductive – argument connectives have the same conventional force. As we have seen, following Kocurek and Pavese (Reference Kocurek and Pavese2021), the core meaning of argument connectives might be dynamic across the board: all uses of ‘therefore’ express that the premises in the context (logically, causally, nomologically, probabilistically) support the conclusion. However, in addition to argument connectives’ having this dynamic meaning, uses of discourses with argument connectives come with a distinctive utterance force – in some cases with the force of an argument, in others with the force of an explanation. If that is correct, then the distinctive force of arguing versus explaining can be recovered at the level of argument connectives’ utterance force.

13.8 Conclusions

This chapter has overviewed recent studies on the semantics and pragmatics of arguments. From this discussion several issues emerge for further research. These include: How are we to think of the syntax of argumentative discourses and how are we to model contexts in order to model the dynamics of argumentative discourses? What consequences does the presuppositional nature of ‘therefore’ have on how to think of arguments? What is the nature of the support relation tested by argument connectives? How do we define entailment for arguments understood as sequences of labeled sentences? What makes a discourse an argument, rather than an explanation? At which level of linguistic analysis lies the difference between arguments and explanations? How are we to characterize the utterance force distinctive of arguments? Are there such things as zero-premises arguments in natural languages? How do deductive arguments in natural language differ, if at all, from proofs in natural deduction systems – such as Fitch’s proofs? Although many of the issues pertaining the semantics and pragmatics of argumentation are left open for further research, I hope to have made a plausible case that they deserve attention since foundational questions concerning the nature of context and discourse, as well as their dynamics, turn on them.

14.0 Questions and Answers

Rejection, as a speech act, is in some sense a foil to assertion. As such, the study of assertion – an early mainstay in both philosophy and semantics – should take seriously the role of rejection.Footnote * However, ever since Frege’s (Reference Frege1919) influential paper Die Verneinung, many linguists and philosophers have supposed that rejection is just a chimera created by negation: proper analysis, such as Frege’s, reveals rejection to be nothing more than negative assertion. The last 20 years saw the rise of an opposition to this view, united under Timothy Smiley’s slogan that ‘assertion and rejection [are] distinct activities on all fours with one another.’ Logicians should take heed of the fact that in a systematic inquiry, one might assert some hypotheses only to later reject some of them – which seems to call for logics of assertion and rejection. In addition, the recent rise of disagreement data in semantics and philosophy similarly calls for rigorous analyses that make sense of rejection alongside assertion.

One key result in the study of assertion and rejection is the observation that rejection is neither reducible to negative assertion, nor equivalent to negative assertion. On the one hand, this observation has helped sharpen the debate on the nature of assertion (as understood in interplay with rejection). On the other hand, recent work in semantics and philosophy of language has seen an increase of interest in disagreement data in such wide-ranging areas as norms of assertion, epistemic modals (cf. Mandelkern, this volume), negation, predicates of personal taste (cf. Anand & Toosarvandani, this volume; Borg, this volume), imperative semantics, discourse (cf. Pavese, this volume), aesthetics and metaethics. A crucial consequence of considering rejection for the study of assertion is that it appears to be mistaken to characterize the effect of assertion as immediately updating a context. This has led some dynamic semanticists to adopt commitment slate models that allow one to treat context updates as pending until accepted or rejected.

I contend – and will defend here at some length – that one needs both assertion and rejection as primitives in one’s semantics and one’s logic. This, in turn, entails that one needs formal models that respect the fact that assertions may be rejected; prominent examples include the various kinds of commitment semantics (see above) or bilateral logics (Smiley Reference Smiley1996; Rumfitt Reference Rumfitt2000).

The Big Question is how exactly rejection relates to assertion. This question can be cashed out in terms of various smaller questions, such as how some formal semantics need to be adjusted to make room for rejection, or how any given account of assertion can make room for – and is improved by – an account of rejection. For instance, one might ask if knowledge is the norm of assertion, what is the norm of rejection? I will outline some possible answers to such questions, with a particular focus on the last one.

14.1 Introduction

Speech acts can be correct or incorrect in that they adhere to or violate some normative component of the conventions surrounding their use. A lot hinges on which assertions are correct. If assertions are presentations of truth-apt contents, then we can draw conclusions about truth-functional semantics from which sentences can be correctly asserted in which circumstances; if, as Frege (Reference Frege1879) suggested, logic tells us which conclusions we may correctly assert given previously asserted premisses, the study of correct assertion elucidates what logic is (Dummett Reference Dummett1991); and if correct assertion is intimately connected to knowledge, we can draw conclusions about the nature of knowledge from the nature of assertion (Williamson Reference Williamson2000).

None of these analyses of correct assertion, or their purported import, are uncontroversial. But few doubt that an analysis of correct assertion plays a pivotal role in both linguistics and philosophy. I contend that it is equally important to provide an analysis of rejection to say something about which sentences can be correctly rejected.Footnote ¹ As a case in point, consider how disagreement data like in (1) and (2) can elucidate the semantics of epistemic modals.

1. (1)

   a.	A:	The keys might be in the car.
   b.	B:	(No,) you’re wrong! I checked the car.

1. (2)

   a.	A:	For all I know, the keys are in the car.
   ??b.	B:	(No,) you’re wrong! I checked the car.

The contrast in (1)/(2) suggests that epistemic might cannot be paraphrased as for all I know, since (1a) is correctly rejectable by (1b), whereas (2a) is not (von Fintel & Gillies Reference von Fintel, Gillies, Gendler and Hawthorne2007). Arguably, (1a) and (2b) are correctly assertible in the same contexts, so do not differ in their assertibility conditions. However, they apparently differ in their rejectability conditions. Considering rejections reveals something that would be lost if we had only considered assertibility. Similar observations can be made in other domains, e.g. disagreement about taste, morals, or aesthetics.Footnote ²

The jury is still out on what precisely we learn from such data. Surely, however, the jury will benefit from studying what it means to correctly reject something. Yet, the speech act of rejection has been somewhat neglected (particularly when compared to assertion). This is one of Frege’s many legacies. In his Die Verneinung, he argued that it is useless to consider rejection on its own terms, as the job of rejection is done by negative assertion. On this view, rejection is a shadow thrown by assertion – to not reduce it to negative assertion would be a ‘futile complication’ that ‘cometh of evil’Footnote ³ (Geach Reference Geach1965: 455). I will argue that this view is mistaken.

Beyond addressing the Fregean point, the focus of this chapter will be on the notion of correctness. Following Williamson (Reference Williamson1996, Reference Williamson2000), it has become popular to characterize speech acts by the norms that are essential for their correct performance. (But many of those who do not accept that such norms characterize speech acts still accept that there are norms governing speech acts.) I am sympathetic to this and in Section 14.4 will spell out in some more detail to what this view amounts and respond to some of its critics.

However, the speech act of rejection seems to catch this normative conception of speech acts in a dilemma. The argument, in brief, goes as follows. On the one hand, rejection is clearly also a norm-governed act and thus should also be characterized by a norm.Footnote ⁴ But on the other hand, rejection appears to be the device by which one registers that some other speech act is in violation of its norm. Such a device is needed in order to even start telling a story about how discourse is a norm-governed activity.

Thus, we need to stipulate rejection as a primitive that is not itself definable by a norm. I argue that the following characterization of rejection will get the normative conception off the ground.

Much of the appeal of the normative conception of assertion stems from the fact that it allows one to characterize assertion without having to complete the sentence “to assert is to …”. However, one nevertheless has to characterize rejection by completing “to reject is to …” as in (Mistake). While this may appear to be a reductio of the normative conception, I argue that it is not.

Indeed, some version of the problem that requires the adoption of (Mistake) will likely afflict any proposal to define what assertion is. In brief, the problem of the normative conception is that we expect a norm-governed activity to come with a mechanism à la (Mistake) for registering rule violations. Hence talk of norms requires talk of rejection, so rejection is not itself characterized by a norm. Other conceptions of assertion likewise explain what assertions are by appealing to certain in-place frameworks (e.g. related to context updates or the undertakings of commitments). Such frameworks, I argue, must likewise include rejection as a primitive, so one cannot give an explanation of rejection itself from within the framework.

This chapter is structured as follows. In the next section, I argue that the study of rejection must free itself from Frege’s grasp because rejections can be weak, i.e. there are rejecting speech acts that are not reducible to assertions of negatives. Afterwards, in Section 14.3, I argue that rejections cannot be reduced to assertions at all. The discussion there suggests an account of rejection as pointing out norm violations. To investigate this further, I elaborate in Section 14.4 a story about how speech acts can be characterized by their essential norms by comparing discourse to another rule-governed activity: the game of chess. With these preliminaries in place, I further investigate in Section 14.5 what the normative conception would have to say about rejection. I argue that rejection is subject to some essential norm, but cannot be defined solely by that norm, instead requiring the principle (Mistake). I conclude with some further commentary in Section 14.6.

14.2 Rejection and Negative Assertion

The seminal work on rejection is Frege’s Die Verneinung (Reference Frege1919). In what is today known as the Frege–Geach Argument (Geach Reference Geach1965; Schroeder Reference Schroeder2008), Frege considers valid inferences like (3) and two possible analyses in (4) and (5).

1. (3)

   a.	If the accused was not in Berlin, he did not commit the murder.
   b.	The accused was not in Berlin.
   $\therefore$ c.	He did not commit the murder.

1. (4)

   a.	Assert:	If not p, then not q.
   b.	Assert:	Not p.
   $\therefore$ c.	Assert:	Not q.

1. (5)

   a.	Assert:	If not p, then not q.
   b.	Reject:	p.
   $\therefore$ c.	Reject:	q.

The analysis (4) straightforwardly explains the validity of (3) as an application of modus ponens. The analysis (5) however is less straightforward. Frege stresses that the embedded use of not in (3a) cannot be an expression of a negative judgment (i.e. a rejection) but must be a operator that modifies a sentence, not express a judgment about it. But this means that if we analyse (3) as (5), we need a lot of machinery to explain the validity of (3) – machinery that was not needed to explain (3) as (4). At least we will need some principle that establishes a connection between rejection and the embeddable negation operator.

Frege does not deny that such principles can be found. He merely thinks it would be unparsimonious to have three primitives (assertion, rejection, and negation) where two would do (assertion and negation). By reducing rejection to negative assertion, Frege does away with the third primitive and, he contends, when we can make do with fewer primitives, ‘we must’ (Reference Frege1919: 154).Footnote ⁵

There is, however, an ambiguity in Frege’s argument. He assumes that rejections are linguistically realized by negatively answering polar questions (p. 153), but there are two possible ways to do so. To wit, one can answer with a sentence containing a negation as in (6a) and by using a polarity particle as in (6b).

1. (6)

   Is it the case that p?
   a.	It is not the case that p.
   b.	No.

The ambiguity exists in the original German; the word Verneinung (lit. ‘no-ing’) can denote both the act of responding with no (German, nein) and the negation operator in the logician’s sense. In Die Verneinung, Frege appears to use the forms in (6) interchangably, but in the later Gedankengefüge (Reference Frege1923: 34ff.) he is explicit that (6b) is a Verneinung of p. In fact, the form (6b) is more congenial to Frege’s discussion. When analyzing (3), an important part of Frege’s argument was that rejections cannot embed in the antecedent of a conditional. And indeed, negative answers to self-posed polar questions do not embed like this: if is it the case that p? No, then … is incomprehensible, as noted by Ian Rumfitt (Reference Rumfitt2000).

If one performs a rejection of some proposition p by putting the polar question is it the case that p? to oneself and answering negatively with No!, then it makes sense to say that one performs an assertion by answering positively with Yes! (Smiley Reference Smiley1996). This is already observed by Ludwig Wittgenstein in the Investigations, paragraph 22.

Thus, we may analyse assertions of p by considering utterances of the form in (7a); rejections of p by considering utterances of the form in (7b); and negative assertions by considering utterances of the form in (7c). This gives us a linguistic grip on investigating the relationship between rejection and negative assertion.

1. (7)

   a.	Is it the case that p? Yes!
   b.	Is it the case that p? No!
   c.	Is it the case that not p? Yes!

There is an imprecision. Here, one uses the polarity particles yes and no to respond to self-posed questions to perform assertions and rejections. But we typically think of these particles as responding to other speakers’ speech acts – and we also frequently think of rejections as responding to other speakers. This points towards another important distinction we need to untangle. On the one hand, we can do logic by asserting and rejecting some propositions to ourselves and investigating what follows from this. This is Frege’s concern. But on the other hand, we can also use the same speech acts in dialogue to, in particular, accept or reject another speaker’s contribution. Expressing a rejection by negatively responding to a self-posed question is not per se unsuitable for this second purpose. It is clumsy, but not incorrect, to reject (8a) with (8b).

1. (8)

   a.	A:	The accused was in Berlin.
   b.	B:	Was the accused in Berlin? No!

There is more to say about the difference between rejections in solipsistic deliberation and rejections in dialogue. I return to this in Section 14.5.

Frege claims that rejections are just negative assertions, i.e. that (7b) is to be analyzed as (7c). However, as incisively argued by Smiley (Reference Smiley1996), Frege only succeeds in showing that not is not an indicator for rejection, due to the fact that not embeds. That is, Smiley concedes that (5) is not the correct analysis of (3). But, he continues, one may now ask whether (5) is the correct analysis of (9).

1. (9)

   a.	If the accused was not in Berlin, he did not commit the murder.
   b.	Was the accused in Berlin? No!
   $\therefore$ c.	Did he commit the murder? No!

As before, this appears to be a valid inference and we should be able to give some explanation of its validity. Frege could insist that for reasons of parsimony the two forms of Verneinung in (6) are to be analyzed the same, their divergent embedding behavior notwithstanding. Thus, he could insist that it is most parsimonious to analyse (9) as (4) as well.

Smiley counters that there is nothing unparsimonious about introducing additional primitives (such as a conception of rejection distinct from negative assertion) if it accounts for additional data. Smiley appears to think that differences in embedding behavior suffice to establish that (9) is new data, distinct from (3), that Frege leaves unexplained. But this could be seen as begging the question against Frege. The difference between (3) and (9) is acknowledged by Frege and it is precisely this difference that he seeks to analyse away.

There is, however, additional data that gives succor to Smiley. Say that a rejection of some proposition p is strong if it is equivalent to the assertion of not p, i.e. if instead of rejecting p one had asserted not p nothing else would have been different. Call a rejection weak if it is not strong. Many accounts of rejection have it that all rejections are strong. For example, according to Frege (Reference Frege1919), all rejections of any proposition p are strong because they simply are assertions of not p; according to Rumfitt (Reference Rumfitt2000), from a rejection of p one can infer exactly what one can infer from the assertion of not p (in the same context);Footnote ⁷ according to Smiley (Reference Smiley1996), rejecting p is correct if and only if asserting not p is correct.

Enter Imogen Dickie (Reference Dickie2010), who argues that some rejections are weak. Some of her examples are in (10).

1. (10)

   a.	Did Homer write the Iliad? No! Actually Homer did not exist.
   b.	Was Homer a unicorn? No! There is no such property as the property of being a unicorn.

Dickie argues that such rejections cannot be strong, as, for example, if (10a) is interpreted as a strong rejection, the speaker has performed a speech act that is equivalent to the assertion of Homer did not write the Iliad. But this is not the case, as she would reject Homer did not write the Iliad on the same grounds that led her to reject Homer wrote the Iliad. The same goes for (10b). Such data seem to doom the Fregean project of reducing rejection to assertion – but they equally trouble Smiley (Reference Smiley1996) and Rumfitt (Reference Rumfitt2000), who still insist that all rejections are strong.

The Fregean has some room to maneuver. In a sentence like Homer did not write the Iliad because Homer did not exist, it appears that the speaker does assert that Homer did not write the Iliad, but the use of negation here is metalinguistic (see Horn Reference Horn1989). Thus, perhaps the Fregean can resist Dickie’s argument from (10) by claiming that such rejections are reducible to assertions of negatives, if the latter includes assertions of metalinguistic negatives. Thus, we might say that the rejections in (10) are metalinguistically strong in that they are equivalent to an assertion containing a metalinguistic negation.Footnote ⁸

But this will not do. Incurvati and Schlöder (Reference Incurvati and Schlöder2017) claim that one can reject out of ignorance, which suggests to them the existence of bona fide weak rejections. The following are cases in point.Footnote ⁹

1. (12)

   a.	Is it the case that X will win the election? No! Z might win!
   b.	This ticket has a one in a million chance to win. Will it lose? No! I don’t know that.
   c.	All I know is that the streets are wet. Is it raining? No! This doesn’t follow.

If we were to read the No! in (12a) as expressing strong rejection (in the nonmetalinguistic sense), it would follow that the speaker asserts X will not win. But the speaker claims something weaker than this. Namely, that this might be the case (see Bledin & Rawlins Reference Bledin, Rawlins, Moroney, Little, Collard and Burgdorf2016 and Mandelkern this volume for additional discussion and data related to how epistemic modals occur in disagreements). In (12b) and (12c), the speaker rejects a proposition since they are not in the position to assert the proposition in question; but they are clearly not in a position to assert its (nonmetalinguistic) negation either.

It seems far fetched to read such examples as ‘metalinguistically strong’. The following utterances sound odd.

1. (13)

   a.	It is not the case that X will win the election (because Z might win).
   b.	This ticket has a one in a million chance to win. It will not lose (because I don’t know that it will).
   c.	All I know is that the streets are wet. It is not raining (because this doesn’t follow).

It is difficult to read (or even coerce) a metalinguistic interpretation of the negations in these examples. Intuitively, none of (13a, b, c) have the same meaning as, respectively, (12a, b, c).

Hence, rejections such as in (12) are weak – they cannot be explained as being reducible or equivalent to negative assertion. As these cases are not to be explained away by sorting them as metalinguistic, we cannot give an account of the phenomenon of rejection that would reduce rejection to negative assertion.

Incurvati and Schlöder (Reference Incurvati and Schlöder2017) conclude that the most parsimonious explanation of all the data is to accept a primitive operation for rejection that encompasses both weak and strong instances of answering No! to a self-posed polar question. They characterize this as the speech act that expresses that one refrains from accepting some content and suggest that this is the fundamental function of all rejections. Strong rejections, they claim, arise as a pragmatic strengthening of this more basic function. A related proposal by Manfred Krifka (Reference Krifka and Snider2013, Reference Krifka, D’Antonio, Moroney and Little2015) and colleagues (Cohen & Krifka Reference Cohen and Krifka2014; Meijer et al. Reference Meijer, Claus, Repp and Krifka2015) is that responding no to an assertion is to foreclose continuing the conversation in a way where the assertion would have been accepted (which need not mean to continue the conversation in a way where a negative was accepted). They call this denegating the assertion.

Be that as it may. Negation and rejection have sometimes been recognized as multi-category phenomena (Geurts Reference Geurts1998; Schlöder & Fernández Reference Schlöder, Fernández, Howes, Hough and Kenning2019) and arguments from parsimony do not have the last word in this debate. An analysis of rejection in terms of assertion and two or more embeddable operators to cover the various weak cases has not been ruled out. However, in the next section I expand on an argument by Huw Price (Reference Price1990) to argue that we must admit a primitive operation for rejection, since rejection fulfills a need that cannot be fulfilled by any assertoric speech act.

14.3 Rejections Fulfill a Need

Price (Reference Price1990) argued that the negation operator not is to be explained by appealing to a primitive speech act of rejection (Price calls it denial, but this is a mere terminological difference). The purpose of this speech act, Price argues, is that it is a means of “registering … a perceived incompatibility.”

To see this need, imagine that we are members of a speech community that does not possess such a means. Then we could find ourselves in the following unfortunate situation. You might point to some berries, proclaim that these are edible and make motions to begin consuming some of them. I, however, see that the berries are lilac and know that all lilac food is highly poisonous. Your death would greatly trouble me, but I am not able to physically stop you – so I have a need to linguistically inform you of the mistake you are making. What sort of recourse do I have? I could tell you these are lilac! but you might not realize that edible and lilac are incompatible. Then all I have achieved is that you now believe that these berries are edible and lilac. Clearly, me telling you these are poisonous is equally hopeless, as you may not realize the incompatibility between edible and poisonous either.Footnote ¹⁰

As Price points out, even if you and I have a shared understanding of the truth-conditional semantics of negation (e.g. by knowing the truth-table for negation), I could not point out your mistake by uttering these berries are not edible, since you might not realize the incompatibility of truth and falsity (despite being a competent user of the language) and believe these berries are edible and not edible.Footnote ¹¹ It does not suffice for there to be an incompatibility (such as between truth and falsity), I also need to be able to inform you of it. Evidently, without you having an understanding of some relevant incompatibility, there is no assertion I could make that would make you realize that you are mistaken to believe that these berries are edible.

Of course, in our actual linguistic practices, competent speakers understand the incompatibility of edible and not edible, so that I can point out such mistakes by asserting these are not edible!. Thus, actual competent use of negation goes beyond the truth-table for negation: it includes an understanding of the fact that the use of not is registering an incompatibility (Price Reference Price1990). Price claims that the act of registering an incompatibility is rejection and claims that not is the expression of rejection. Taken literally, his suggestion of letting not directly express rejection falls prey to Frege–Geach problem discussed in the previous section. The problem can be resolved by assuming a primitive speech act for rejection and stating the meaning of not in terms of this speech act instead of the direct expression of that speech act (Smiley Reference Smiley1996; Rumfitt Reference Rumfitt2000; Incurvati & Schlöder Reference Incurvati and Schlöder2017, Reference Incurvati and Schlöder2019).

Either way, the thought experiment suggests that we need to stipulate a distinct expression of incompatibility. Although suggestive, such thought experiments may not be ultimately compelling. The need for a primitive mechanism for rejection can also be appreciated by probing into analyses of assertion, such as Stalnaker’s (Reference Stalnaker and Cole1978). On his account, the essential effect of an assertion of p is to propose to expand the common ground by adding p – and such proposals can be rejected. Sometimes, Stalnaker is misunderstood to claim that an assertion immediately updates the common ground.Footnote ¹² But this is a strawman. Stalnaker is explicit that not all assertions result in a common ground update, since they can be rejected.

Thus, an assertion does not expand the common ground immediately, but does so only in the absence of rejection. Put differently, asserting that p proposes to make p common ground and making it common ground is a further process that needs to be negotiated by the interlocutors (also see Clark Reference Clark1996).

Some have tried to prop up the strawman. Notably, Seth Yalcin (Reference Yalcin, Fogal, Harris and Moss2018)

His reason is that characterizing assertion by appealing to a speech act of proposing is just to ‘pass the buck to the question what proposing is.’ But any serious account of speech acts must face the question of what proposing is (and of how proposals are rejected) anyway. Consider the speech act of betting. It is clearly mistaken to say that a bet is automatically accepted – that is, the context is changed so that speakers are obliged to adhere to the rules of the bet – and that to reject it is to undo these changes. This is mistaken, because rejecting a bet – that is, not accepting it – is distinct from undoing a bet. Rejecting a bet is something I can do unilaterally, but to undo a bet that both sides have agreed on usually requires both speakers to agree to this.Footnote ¹³

Thus, if we want to make sense of betting, we need to make sense of a mechanism by which context changes are proposed and then either accepted or rejected. We can use the same mechanism to give the Stalnakerian account of assertion as proposing context updates. This is not passing the buck, but simplifying and unifying. Yalcin’s suggestion might simplify things if one is only interested in explaining assertion (which may indeed be Yalcin’s ambition), but as soon as other speech acts come into play, it becomes a needless complication to make assertion function differently from speech acts whose rejection and undoing are distinct activities.

The occurrence of the word “reject” here does not immediately entail that Stalnaker’s account must contain a primitive for rejection alongside assertion. Intuitively, one can reject by making a counterproposal. That is, if you have asserted that p, I may respond by asserting that q, where q is inconsistent with p. As the common ground must be consistent, p and q cannot simultaneously be in the common ground. Hence, a story might go, my assertion is rejecting yours. But this story is confused, as Price’s observations apply here as well. There being an incompatibility does not exempt us from needing a mechanism to register it. That is, one may not always realize that a proposal is a counterproposal.

Reducing rejection to counterproposing faces another problem. To explain a rejection of a proposal to update the common ground as another proposal to update the common ground may lead to a regress with proposals stacking up and not being resolved. There must be a mechanism to halt the regress. That is, there is a need for a response to an assertion is not itself an update proposal. This operation should register that one is rejecting an update proposal.Footnote ¹⁴

There are different ways of spelling out this operation. Price (Reference Price1990) argues that rejection signals an incompatibility between truth and falsity; so roughly, rejection would be the speech act that signals I am not accepting your assertion of p because p is false. But this fails to capture the data on weak rejection discussed in the previous section. If the incompatibility between X will win and no, Z might win is the incompatibility of truth and falsity, it would follow that no, Z might win means that X will not win, which it does not. By adopting the Stalnakerian picture, one can fix this issue: rejection registers that I do not wish some p to be in the common ground. I may have different reasons for this. I might not want p to be in the common ground because p is malformed, has an unmet presupposition, there is insufficient evidence for p, or indeed because p is false. This or a similar conception of rejection is available to the gamut of accounts that characterize speech acts by their context-update potential (e.g. Roberts Reference Roberts2012; Portner Reference Portner2018b; Murray & Starr Reference Murray, Starr, Fogal, Harris and Moss2018).

There is another option to spell out a sufficiently broad notion of rejection, not requiring a context-update framework. As said in the Introduction, not all assertions or rejections are correct. Some are incorrect in that they are violating some convention associated with these speech acts. A good explanation of how speech acts can be correct or incorrect is that they are, in some sense, governed by certain norms or rules. (This can be said while leaving open whether or not these norms are essential to linguistic activity or not; but see Williamson Reference Williamson1996, Reference Williamson2000 and below.) Roughly following Brandom (Reference Brandom1983, Reference Brandom1994), one may think of these norms as being enforced by a social order in which violating these norms makes one liable to social sanction. The relevant sanctions are social consequences such one’s partial exclusion from the practice of assertion; e.g. the boy who cried wolf is sanctioned for his misdeeds in that nobody heeds his assertions (Brandom Reference Brandom1994: 180).

This leads to an understanding of rejection as the device by which one informs one’s interlocutor that they have made themselves liable to sanction. Supposing that truth is a norm of assertion, if you assert that the berries are edible and I know they are not, I judge your speech act as violating a norm. By rejecting your claim, I inform you that you are liable to sanction (and, tangentially, death). On such a conception, instead of saying that rejection is registering incompatibilities, it is more apt to say that it registers mistakes.

Brandom himself does not consider the rules related to sanction to constitute the meaning of assertion and rejection. Rather, the rules and sanctions surrounding linguistic behavior constitute a framework in which speakers keep track of each other’s commitments to certain contents, from which certain permissions and obligations derive. Brandom then suggests to explain the act of asserting as the undertaking of a commitment to the asserted content. This suggestion is taken up by Asher and Lascarides (Reference Asher and Lascarides2003), Farkas and Bruce (Reference Farkas and Bruce2010), and Krifka (Reference Krifka, D’Antonio, Moroney and Little2015), among many others. Aside from Krifka’s denegations, however, not much has been said about the role of rejection in such a framework with a perspective towards rejections being possibly weak.

Recall that Price’s (Reference Price1990) original argument was based on the puzzle of how we can inform someone that they are mistaken. The normative story outlined above would suggest that a mistake is a violation of a norm and thus we may take rejection to register norm violations. This seems to hit the target. Price pointed out that speakers may not realize certain incompatibilities, so we need a device to explicitly point out an incompatibility. But we might equally wonder what would be required to point out a norm violation to someone who does not realize the appropriate norm. (I will continue this line of thought in Section 14.5.)

I will now turn my attention on the normative conception of speech acts according to which speech acts are characterized by the norms that essentially apply to them. Compared to Brandom, such an account cuts out the middle man: instead of characterizing commitment by norms and sanctions and then assertion in terms of commitment, we may characterize assertion directly in terms of norms. I first elaborate my preferred understanding of the normative conception of speech acts. Afterwards, I investigate the prospects of conceiving of the essential function of rejections as registering mistakes.

14.4 The Normative Conception

It has become popular to characterize speech acts by stating the norms (or rules) that essentially apply to them in the conversation game. A particular focus of recent debate are accounts of assertion that seek to characterize it by identifying the constitutive norm of assertion – the fundamental rule that governs assertions (conceived of as moves in the conversation game). One such rule is the knowledge norm of assertion (KNA), proposed by Williamson (Reference Williamson1996, Reference Williamson2000).

Other putative norms of assertion have been proposed (e.g. Lackey Reference Lackey2007; Weiner Reference Weiner2007), but it is not the purpose of this chapter to adjudicate between them. Aside from the vibrant debate on which putative norm is the essential norm of assertion, there is the attendant debate on whether a normative analysis of assertion is possible. Invariably, defenders of the normative conception draw a prima facie convincing analogy to games such as chess, rugby or baseball. But the dialectic suffers from there being insufficient clarity on how exactly the activity of asserting is like a game. In what follows, I elaborate my preferred understanding of how conversation is like playing chess. I use chess purely for familiarity. It should be easy to see how analogous arguments using any other game can be constructed.

It makes sense to say that the game of chess is made up by a number of rules: when we are asked to explain what chess is, we explain that it is a game subject to a particular set of rules. One of them may be written as (Rook).

It seems that the question What are moves of a rook (in chess?) has no more satisfactory answer than identifying among the rules of chess those rule(s) that specifically or essentially govern the movements of rooks. Namely, a move of a rook is a move that is subject to (Rook).Footnote ¹⁵ Then, analogously, the question What are assertions? has no more satisfactory answer than identifying those rule(s) among the rules of conversation that essentially govern assertion. To wit, an assertion is a speech act subject to the norm of assertion (be it the knowledge norm or another one).

This analysis of assertion is not troubled by the fact that there are further rules of conversation that govern assertion, but are not essential to assertion. For example, assertion – like any speech act – seems to be bound by general rules of relevance and informativeness (to name just two). Likewise, the movement of rooks – like other moves in chess – is bound by further rules as well. For instance, the rule (Check) applies to all pieces in chess.

But (Check) is not part of our understanding of rook moves. If someone knows the rule (Rook) without knowing (Check) we would still attribute to them the knowledge of what rook moves are. Say, if we are teaching chess to someone, we would be satisfied that they understood what rook moves are if they understood (Rook), even if we haven’t yet explained (Check). In this sense, (Rook) is essential to the understanding of moving rooks, whereas (Check) is not.

Again analogously, a proponent of a norm account of assertion claims that it is only the specific norm of assertion that constitutes the knowledge of what assertions are, regardless of other putative rules of the conversation game that are less intimately related to assertion. Such rules stand to the norm of assertion as (Check) stands to (Rook). Furthermore, there are broader behavioral rules that apply to assertion (such as politeness or general morals), just as there are broader rules of sporting behavior that apply in chess (e.g. that opponents shake hands). As the latter do not seem to contribute to our understanding of rook moves, we should not think of the former as contributing to our understanding of asserting.

However, there are some doubts about the true extent of such an analogy between conversation and everyday games and about how useful any such analogy is in characterizing a speech act (Hindriks Reference Hindriks2007; Maitra Reference Maitra, Brown and Cappelen2011; MacFarlane Reference MacFarlane, Brown and Cappelen2011). One salient criticism is that a rule like (KNA) might tell us under which conditions one may assert, but tells us nothing about how to assert, i.e. about how to complete the sentence “to assert is to …”. Consider, for instance, the following rules that seem to define the move of short castling in chess.

The rule (Short Castling 1) alone is not sufficient for us to know how to short castle. We need to know (Short Castling 2) as well. Now, it may appear as though (KNA) has the same form as (Short Castling 1). Thus, one may be inclined to conclude, (KNA) alone is insufficient to characterize assertion, just as (Short Castling 1) is insufficient to characterize short castling (MacFarlane Reference MacFarlane, Brown and Cappelen2011). It appears we require another rule of the form to assert is to (…). But appearances mislead here. There are many possible assertions and many possible rook moves, but there is only one move called ‘short castling’ (namely, what is stated in Short Castling 2). The phrase short castling is a mere abbreviation for this one move. Unabbreviating leads to the following rule for (Short Castling), which is properly analogous to (KNA).

Nothing more than knowledge of (Short Castling) is required to understand how to perform the move in chess that is known as short castling. If one knows (Short Castling), but not (Short Castling 1+2), one does not know that the move is called ‘short castling’. But such knowledge – knowledge of the names of certain moves or pieces – is not required to play a game of chess. Likewise, one need not know that assertions are called ‘assertions’ to partake in the conversation game (and few people use the term regularly).

But this does not fully address the objection that on the normative account one cannot complete the sentence “to assert is to …”. We have now seen that one can state the rule for short castling without completing the sentence “to short castle is to …”, but the rule (Short Castling) still contains an unanalyzed primitive: move. So, in explaining short castling by appealing to (Short Castling), one presupposes an understanding of move in chess; and in explaining assertion by appealing to a norm of assertion, one presupposes an understanding of move in conversation, i.e. of making a speech act. Shouldn’t we demand explanations like to move a piece in chess is to (…) and to make a speech act is to (…)?

There is a straightforward answer to this. We have no reason to suppose that there is any better explanation of move (in chess) than (Chess Move).

It appears to be hopeless to explain moves in chess by spelling out the form of an act that moves a piece. These forms vary vastly: one can make moves by physically moving pieces, by declaration (“E2 to E4”), by sending a letter, or even by entirely mental acts (some can play a full game of chess in their head). Moreover, one can perform any act that has the form of a move without playing a game of chess. I can, for example, idly move pieces on a board and by sheer circumstance happen to follow the rules of chess, but these idle moves are not moves in a game of chess. (Such observations about intentionality are of course familiar from the literature on speech acts.) What does and does not count as a move in chess is a social phenomenon. A move in chess is a sort of act that occurs in a particular setting that is understood by everyone in it to be subject to the rules of chess. That is, (Chess Move).

Then, we may explain what it means to move a rook as (Rook Move).

If we are happy with (Chess Move) and (Rook Move) characterizing what it means to make moves in chess, then we should be equally happy with (Speech Act) and (Assertion) being the explanations of what it means to assert and make speech acts.Footnote ¹⁶

Finally, another salient and frequent objection to the normative conception of assertion attacks the claim that a norm of assertion is constitutive of assertion. Defenders of the normative conception countenance that an assertion that violates the norm still counts as an assertion; e.g. Williamson (Reference Williamson1996), who defends the knowledge norm, explicitly allows that one can assert that p without knowing that p. This would be an incorrect assertion, but an assertion nonetheless. Some think that this is nonsense: according to Searle’s (Reference Searle1969) definition of constitutive rules, if a rule R is constitutive of an activity A, then one ceases to A when one violates R.

Ishani Maitra (Reference Maitra, Brown and Cappelen2011) offers the useful clarification that only flagrant violations of R result in a cessation of A, but argues further that this does not resolve the complaint, as there are speech acts that appear to be assertions despite flagrantly violating a norm of assertion (e.g. a defendant asserting their innocence in the face of definitive condemning evidence). The claim that any putative norm of assertion is constitutive of the speech act of assertion is apparently incompatible with the claim that speech acts that flagrantly violate that norm can still count as assertions (Hindriks Reference Hindriks2007).

The complaint has bite if we understand constitutive like Searle does. But this is not the definition that defenders of the normative conception have in mind. Williamson (Reference Williamson1996: 491) remarks that ‘[w]hen one breaks a rule of a game, one does not thereby cease to be playing that game.’ In (Assertion), I suggest to define assertions as those linguistic acts that are understood to be subject to some rules. This dovetails with Williamson’s argument. Certainly, an act can be understood to be subject to some rules despite violating them.

As a matter of fact, this is the case in chess. Plainly, one can speak of illegal moves in chess; the FIDE Laws of Chess do so in Article 7.4. Thus, if we agree that the notion of move in chess is defined by a set of rules, we must accept that there are acts that can be called moves in chess albeit violating one or more of these rules. Otherwise, the very term illegal move would be unintelligible. Making an illegal move does not end a game. Rather, if and once the violation becomes apparent, one would be requested to undo the move. (Which appears to be analogous to the request to retract an assertion made in violation of a norm of assertion.) Thus, if (Rook) is a constitutive rule, constitutive rules are violable.

Some have denied the antecedent of this conditional: Hindriks (Reference Hindriks2007), for instance, claims that the rules that define the legal moves of chess, like (Rook), are merely regulative. But now, the debate has shifted to the semantics of constitutive. Plainly, a rule like (Rook) is part of the rules that define the game of chess – that make up the game. If we are playing a game that is not subject to (Rook), but instead subject to, say, (Rook’), we are not playing chess.

We may insist on a particular, technical understanding of the term constitutive according to which rules like (Rook) are not constitutive of chess. But this would not change the fact that (Rook) is one of the rules that define what the game of chess is. Whether or not one is inclined to call such rules “constitutive” is besides the point. One also may want to say that it is constitutive of chess that (Rook) is a regulative rule. I wouldn’t object to this, though it strikes me as spurious.

In any case, there is no objection against the normative conception to be found in the observation that one can make assertions violating a norm of assertion. This is because, as shown by the example of chess, violable rules like (Rook) can have the status of definitions. But, as I will argue next, the fact that there are such violable rules entails that rejection has a central place within the normative conception.

14.5 Rejection in the Normative Conception

The dialectic in this section, in brief, is as follows. If you accept that conversation is a rule-governed activity like chess, you have to acknowledge the existence of illegal moves (as argued towards the end of the last section). That is, moves that are part of an activity (performing them does not end the activity), but are violating some of the rules that define the activity. But this means that such an activity must also have rules that determine what happens in such a situation – rules that govern how to proceed when an illegal move has been made. Based on the discussion in Section 14.3, this includes, at the very least, a device to register that an illegal move has been made: that device is rejection. However, I will argue, such a device cannot itself be characterized by a norm.

The need for rejection is particularly visible in learning scenarios. If the norms of the language game are part of the fabric of our social lives, newcomers to our community should learn them. Suppose that assertion is properly characterized by (KNA).Footnote ¹⁷ Some language learner might assert that p, i.e. make an act that is understood to be subject to certain social norms (even though the learner has not fully grasped these norms), but a competent speaker does not believe that the learner knows that p. She might point that out by saying you don’t know that p. If rejections do not register norm violations, nothing would stop the learner from assuming that they properly asserted that p and that, in addition, they do not know that p. To make her realize her mistake, a mistake must be registered by the rejection.Footnote ¹⁸ It does not matter whether one performs the rejection verbally or by intonation or body language etc. The point is just that this registering signal, however it is sent, is not explainable by appealing to an account of assertion.

Now, someone endorsing the normative conception of speech acts wants to characterize speech acts by their essential norms, e.g. characterizing assertion by the (KNA). Can this be done for rejection? Based on what we have seen about rejection so far, the following norm appears to be a good candidate.

There is a lot to like about (Rejection). Conceivably, asserting p is in violation of some norm if: some presupposition of p is not met (as in example (10a)); or p involves nondenoting properties (as in example (10b)); or the speaker has insufficient evidence for p (as in example (12)). Thus, (Rejection) appears to be broad enough to capture the data from Section 14.2.Footnote ¹⁹

The norm (Rejection) also accounts for the puzzle discussed in Section 14.3. The puzzle was that if you assert these berries are edible and I respond no, they are lilac, you may not realize that edible and lilac are incompatible, thus forming the belief that the berries are edible and lilac and proceed with consuming them. The solution was to say that rejection registers that what I said is incompatible with what you said. This incompatibility is in particular registered if we conceive of rejections as being governed by (Rejection). For then your claim that these berries are edible and my claim that no, they are lilac cannot both be correctly performed. Either your assertion was correct, in which case my rejection violated (Rejection). Or my rejection was correct, in which case your assertion violated the norm of assertion. But we cannot both be right, so you have no reason to believe that the berries are edible and lilac.

The norm (Rejection) also accounts for the fact that nonassertoric speech acts can be rejected as well. Supposedly, these speech acts are also explained by the norm that is essential to their correct performance. For example, supposing that it is (part of) the norm of questions that one may not ask questions to which one knows the answer, I may reject a question by You know that!. In general, then, (Rejection) entails that a speech act s and a speech act rejecting s can not be both correct. This is as it should be.

A particularly interesting case is to reject another rejection. If you perform some speech act and I reject it, you need not give in. You can reject my rejection (Schlöder et al. Reference Schlöder, Venant and Asher2017). The norm (Rejection) accounts for that fact. If I reject your speech act, then my rejection was correct if you violated a norm. But of course, if you think that you did not violate any norms, you may reject my rejection. According to (Rejection), your rejection of my rejection is correct if and only if your initial speech act was correct. This is also as it should be.

Finally, although we naturally think of rejections was being in response to other speech acts, there is a coherent notion of rejecting a proposition that is not in response to anything (see Section 14.2). We can extract the correct norm for these rejections from (Rejection). To wit, we may think of rejections of propositions as governed by (P-Rejection).Footnote ²⁰

Note that there is an asymmetry here. I can properly P-reject those propositions that I cannot properly assert; but I can properly reject your assertion of a proposition if you cannot properly assert them. This means that there are cases where you can properly assert a proposition p that I can properly P-reject. This is also as it should be, since you may have more information than I do. If you know p but I do not, I can correctly P-reject p. But, if you assert p, me rejecting this assertion would be incorrect. Moreover, your assertion grants me license to assert p to others based on your authority (Brandom Reference Brandom1983); thus after you properly assert p to me, I am no longer able to properly P-reject p.

All of this sounds good. And yet, if you accept the normative conception of speech acts elaborated in Section 14.4, you should not endorse (Rejection) as defining the speech act of rejection. I argued that rejection – as the device that points out mistakes – is required for language learners to acquire the right norms. I need to be able to register a norm violation even if you do not realize the norms of the speech acts we are using. This is analogous to Price’s story in which I need to be able to register an incompatibility even if you do not realize any incompatibilities. Saying that rejection is the speech act governed by (Rejection) does not fulfill this purpose, since if you do not yet understand (Rejection), and this is the norm that characterizes rejection, my rejections would fail to register with you that there was a mistake.

Hence, having an understanding of rejection as a mistake-registering device is prior to characterizing speech acts by their essential norms. Characterizing rejection by (Rejection) presupposes an understanding of mistakes and how to register them, so an understanding of rejection. This is a vicious regress. The point is quite simple: if our social fabric is (partially) made up by certain rules, I need to be able to point out which behavior is sanctionable so that a newcomer can sort good from bad behavior. Clearly, my method of pointing that out cannot itself be defined by a rule that needs to be learned this way.

Thus, there is at least one speech act – rejection – that cannot be characterized by appealing to an essential norm. Does this doom the normative conception? I think not. But someone endorsing this conception needs to acknowledge that the registration of mistakes is a fundamental and unanalysable part of norm-governed activities. That is, we should accept (Mistake).

With (Mistake) in place, we can then also adopt the norm (Rejection) to explain the data discussed in this paper. The situation is somewhat curious: I maintained that the speech act of assertion is governed by a permissive norm and that it is useless, possibly hopeless, to ask how to finish the sentence “to assert is to …” beyond saying that assertions are acts understood to be subject to certain norms. But for the speech act of rejection, we appear to require the more substantive principle (Mistake).

I don’t think this a reason to worry. The registration of mistakes seems to be a fundamental part of any rule-governed activity. In any game we play, we will at some point want to register that someone made a mistake. But we do not expect the rules of the game to explain to us what it is to register a mistake, only how to proceed once a mistake has been registered.Footnote ²¹ We simply understand that a way to register mistakes is part of the fact that there are rules. This means that the speech act of rejection is on the same conceptual level as the concept of a norm or rule.

In fact, it seems that some version of this problem – the need to stipulate a fundamental principle for rejection – occurs in any attempt to characterize speech acts. In Section 14.3, I outlined how rejection appears in the Stalnakerian account of assertion. I argued that rejection cannot be reduced to some version of the fundamental operation of updating the context, but needs to be taken as a primitive that governs such updates. This is analogous to the situation for the normative conception: rejection cannot be reduced to some version of the fundamental principle of a permissive norm, but must be taken as a primitive that governs the application of these norms. Similarly, Brandom (Reference Brandom1994), as anticipated in Section 14.3, also cannot explain rejection (as the act that points out that someone is liable to sanction) in terms of commitment but needs to take it as a fundamental operation that is part of the mechanisms surrounding commitment.

14.6 Conclusion

The purpose of this chapter is to win some repute for rejection as a sui generis speech act whose study should be of interest to linguists and philosophers. My main goal is to establish that rejection is not reducible to assertion by arguing (i) that there are rejections that are not equivalent to negative assertions; and (ii) that the speech act of rejection fulfills a particular purpose – registering mistakes – that cannot be met by assertoric speech acts. The most natural explanation of what it means to register a mistake is that it is to point out the violation of a norm. This supports the idea to explain speech acts by determining the norms that essentially apply to them.

Importantly (and curiously), however, the speech act of rejection cannot itself be defined by an essential norm, as the act of registering mistakes must be prior to the norm that governs when mistakes may be registered. I do not take this to refute the project of characterizing speech acts by their norm – rather, this seems to reveal the fundamentality of rejection in linguistic practice. The arguments I presented here suggest that rejection is similarly fundamental in other conceptions of speech acts, although I have not given them as much attention as the normative conception.