3.1 Proof theory

The proof theory of $\mathsf {EML}$ is formulated by means of natural deduction rules. We briefly discuss the rules as described in Incurvati & Schlöder (Reference Incurvati and Schlöder2019).

For conjunction, we simply take the standard rules and prefix each sentence with the strong assertion sign.

The rules say that from two strongly asserted propositions one can infer the strong assertion of their conjunction, and that from a strongly asserted conjunction one can infer the strong assertion of either conjunct.

Linguistic analysis reveals that weakly asserting a proposition has the same consequences as rejecting its negation (see Incurvati & Schlöder, Reference Incurvati and Schlöder2019). Hence, the weak assertion of A and the rejection of not A should be interderivable. Linguistic analysis also shows that rejecting a proposition has the same consequences as weakly asserting its negation. Thus, the rejection of A and the weak assertion of not A should be interderivable too. This licenses the following rules for negation.

The rules for the epistemic possibility operator are based on two observations. The first is that uttering perhaps A has the same consequences as uttering it might be that A. It follows that the weak assertion of A and the strong assertion of $\Diamond A$ should be interderivable.

The second observation is that uttering perhaps A has the same consequences as uttering perhaps it might be that A. It follows that the weak assertion of A and the weak assertion of $\Diamond A$ should be interderivable too.

The rules for $\neg $ and $\Diamond $ are clearly harmonious, since the elimination rules are the inverses of the corresponding introduction rules. They are also simple and pure in Dummett’s (Reference Dummett1991, p. 257) sense: only one logical constant features in them and this constant occurs only once.

We have presented the operational rules of $\mathsf {EML}$ —that is, rules for the introduction and elimination of its operators. But we are not quite finished yet. For in multilateral systems (just as in bilateral systems) we also need rules that specify how the speech-act markers interact. Such rules are known as coordination principles and are needed to validate mixed inferences—inferences involving propositions that are uttered with different forces and are therefore prefixed by different signs. One example, involving strong assertion and rejection, is the seemingly valid pattern of inference that allows one to conclude the rejection of p from the strong assertion of if p, then q and the rejection of q (see Smiley, Reference Smiley1996). Another example, involving weak and strong assertion, is the inference pattern of weak modus ponens, which allows one to conclude the weak assertion of q from the weak assertion of p and the strong assertion of if p, then q (see Incurvati & Schlöder, Reference Incurvati and Schlöder2019). That is, from if p, then q and perhaps p one may infer perhaps q.

As in standard bilateral systems, the rules coordinating strong assertion and rejection characterize these speech acts as contraries.

(Rejection) states that strong assertion and rejection are incompatible: it is absurd to both propose and prevent the addition of the same proposition to the common ground. (SR $_1$ ) says that if, in the presence of one’s extant commitments, strongly asserting a proposition leads to absurdity, then one is already committed to preventing the addition of that proposition to the common ground. A similar reading of (SR $_2$ ) is available. The conjunction of (SR $_1$ ) and (SR $_2$ ) is known as the Smileian reductio principle, after Smiley (Reference Smiley1996).

The remaining coordination principles characterize weak assertion as subaltern to its strong counterpart. We write $+\vdots$ for a derivation in which all premisses and undischarged assumptions are strongly asserted sentences, i.e. formulae of the form $+\hspace {-0.1em} A$ . Since $\bot $ is treated as a punctuation mark, in (Weak Inference) we distinguish between the case in which one infers $+\hspace {-0.1em} B$ in the subderivation and the case in which one infers $\bot $ . In the former case, (Weak Inference) allows one to conclude $+\hspace {-0.1em} B$ ; in the latter case, it allows one to conclude $\bot $ .Footnote ³

(Assertion) ensures that in performing the strong assertion of a proposition, one is committed to dissenting from its negation. (Weak Inference) ensures that weak assertion is closed under strongly asserted implication and hence that inferences like if p then q; perhaps p; therefore perhaps q are valid. For if one’s evidential situation sanctions perhaps p and one knows that any situation where p is the case is also a situation where q is the case, then one is entitled to conclude perhaps q.

The restrictions on the subderivation of (Weak Inference) are due to the specificity problem (Incurvati & Schlöder, Reference Incurvati and Schlöder2017). As noted by Imogen Dickie (Reference Dickie2010), a correct strong assertion of A requires there being evidence for A. By contrast, the speech act of weak rejection is ‘messy’ in that it makes unspecific demands about evidence: a weak rejection of A may be correct because there is evidence against A, but may also be correct because of the (mere) absence of evidence for A (Incurvati & Schlöder, Reference Incurvati and Schlöder2017). This means the following for the justification of (Weak Inference). Suppose that in the base context one’s evidential status sanctions perhaps A. To apply (Weak Inference), one then considers the hypothetical situation in which A is the case and attempts to derive B. In the hypothetical situation in which A is the case, there may be evidence that is not available in the base context (e.g. evidence for A). Thus, in the subderivation of (Weak Inference), one may not use rejected premisses from the base context that one rejects for lack of evidence. Due to the unspecificity of weak rejection, this means that one may not use any premisses that are weakly rejected. Since $\oplus $ switches with $\ominus $ and $\oplus $ switches with $+\hspace {-0.1em}\Diamond $ , one may also not use any weakly asserted premisses or apply $\Diamond $ -Eliminations to strongly asserted premisses.Footnote ⁴

3.2 Derived rules

This concludes the exposition of the proof theory of $\mathsf {EML}$ , and we use $\vdash $ to denote derivability in $\mathsf {EML}$ . To simplify the presentation in the remainder of the paper, however, it will be useful to present some additional derived rules of $\mathsf {EML}$ , characterizing the behavior of the primitive connectives under some of the speech acts not covered by the basic rules and the behavior of the defined connectives. This will also serve to give a flavour of how derivations in $\mathsf {EML}$ work.

We begin with a rule which specifies how to introduce the strong assertion of a negation.

Proposition 3.1. The following rule is derivable in $\mathsf {EML}$ .

Proof. $\mathsf {EML}$ derives ( $+\hspace {-0.1em}\neg $ I.) as follows.

□

Next, we have rules specifying the behavior of conjunction under weak assertion.Footnote ⁵

Proposition 3.2. The following rules are derivable in $\mathsf {EML}$ .

Proof. $\mathsf {EML}$ derives ( $\oplus \hspace {-0.08em}\land $ I. $_1$ ) as follows.

The case of ( $\oplus \hspace {-0.08em}\land $ I. $_2$ ) is analogous. ( $\oplus \hspace {-0.08em}\land $ E. $_1$ ) follows from (Weak Inference):

( $\oplus \hspace {-0.08em}\land $ E. $_2$ ) is analogous. □

Note that these derived rules are applicable in arbitrary proof contexts, even when their premisses are derived using $\Diamond $ -Elimination rules. While the derivations make use of ( $+\hspace {-0.1em}\neg $ I.) and (Weak Inference)—rules that disallow $\Diamond $ -Eliminations—the applications of ( $+\hspace {-0.1em}\neg $ I.) and (Weak Inference) occur within self-contained subderivations and hence are correct irrespective of the wider proof context.

3.2.1 The material conditional

As mentioned, we are taking the conditional $A \rightarrow B$ to be defined as $\neg (A \wedge \neg B)$ . Given the classicality of our calculus (to be shown below), this means that the conditional will be material. The following rules for strongly asserted conditionals can be derived.

Proof. ( $+\hspace {-0.1em}\rightarrow $ I.) is derivable as follows.

And ( $+\hspace {-0.1em}\rightarrow $ E.) is derivable as follows:

□

In addition, one can derive the rule of weak modus ponens (WMP).

Proof.

□

By ( $+\hspace {-0.1em}\Diamond $ I.) and ( $+\hspace {-0.1em}\Diamond $ E.), (WMP) entails the derivability of epistemic modus ponens (EMP).

3.2.2 Necessity modal

$\Box A$ is defined as $\neg \Diamond \neg A$ . The following are derived rules for $\Box $ :

Proof. ( $+\hspace {-0.1em}\Box $ I.) is derivable as follows:

( $+\hspace {-0.1em}\Box $ E.) is derivable as follows:

□

Note that the derivability of ( $+\hspace {-0.1em}\Box $ I.) does not imply that $+\hspace {-0.1em} (A\rightarrow \Box A)$ , since the derivation of $+\hspace {-0.1em} \neg \Diamond \neg A$ from $+\hspace {-0.1em} A$ uses a $\Diamond $ -Elimination rule, which rules out an application of ( $+\hspace {-0.1em}\rightarrow $ I.). In fact, there is no derivation of $+\hspace {-0.1em} (A\rightarrow \Box A)$ in $\mathsf {EML}$ , as shown by the soundness of $\mathsf {EML}$ with respect to $\mathbf {S5}$ modulo an appropriate translation (see §4).

3.3 Classicality

We now show that, in a defined sense, the logic of strong assertion extends classical logic (in much the same way that normal modal logics extend classical logic).

Let $\sigma : \text {At} \rightarrow \text {wff}_{\text {EML}}$ be any mapping from propositional atoms to formulae in the language $\mathcal {L}_{EML}$ of $\mathsf {EML}$ . If A is a formula in the language $\mathcal {L}_{PL}$ of propositional logic, write $\sigma [A]$ for the $\mathcal {L}_{EML}$ -formula that results from replacing every atom p in A with $\sigma (p)$ . Moreover, let $\models ^{\text {CPL}}$ be the consequence relation of classical propositional logic. We have:

Proof. Since $\mathsf {EML}$ validates modus ponens, it suffices to show that the strongly asserted versions of the axioms of the propositional calculus are theorems of $\mathsf {EML}$ . That is, for arbitrary A, B and C in $\mathcal {L}_{EML}$ , we have:

$$ \begin{align*} \begin{array}{l} \vdash +\hspace{-0.1em}(A \rightarrow \neg\neg A),\\ \vdash +\hspace{-0.1em}(\neg\neg A\rightarrow A),\\ \vdash +\hspace{-0.1em}((A \rightarrow B) \rightarrow (\neg B \rightarrow \neg A)),\\ \vdash +\hspace{-0.1em}((A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) \rightarrow (A \rightarrow C))),\\ \vdash +\hspace{-0.1em}((A \rightarrow (B \rightarrow C)) \rightarrow (B \rightarrow (A \rightarrow C))),\\ \vdash +\hspace{-0.1em}(A \rightarrow (B \rightarrow A)). \end{array} \end{align*} $$

These are easy to check (see Incurvati & Schlöder, Reference Incurvati and Schlöder2019). □

Together with the Soundness result (Theorem 4.1) we will prove in §4, this yields the following corollary.

3.4 $\mathsf {EML}$ extends $\mathbf {S5}$

We now show that $\mathsf {EML}$ extends $\mathbf {S5}$ , i.e. that the logic of strong assertion validates every $\mathbf {S5}$ -valid argument. To this end, we shall prove that if A is an $\mathbf {S5}$ -tautology, then its strongly asserted counterpart is a theorem of $\mathsf {EML}$ . That is:

This will yield the desired result.

Towards the proof of Theorem 3.5, we first prove a technical lemma.

Lemma 3.7. The following rule is derivable in $\mathsf {EML}$ .

Proof.

□

We are now ready to prove the main result of this section.

Proof of Theorem 3.5

$\mathsf {EML}$ proves all substitution-instances of classical tautologies (Theorem 3.3). Moreover, by the ( $+\hspace {-0.1em}\Box $ I.) rule, we have that if $\vdash +\hspace {-0.1em} A$ then also $\vdash +\hspace {-0.1em}\Box A$ . Thus, it suffices to show that the $\mathbf {KT5}$ axioms are derivable in $\mathsf {EML}$ .

Axiom $\mathbf {K}$ can be written on the signature $\{\neg ,\land ,\Diamond \}$ as

$$ \begin{align*} (K)\ \neg (\neg \Diamond (A\land \neg B) \land \neg \Diamond \neg A \land \Diamond \neg B). \end{align*} $$

The following derivation witnesses $\vdash +\hspace {-0.1em} \neg (\neg \Diamond (A\land \neg B) \land \neg \Diamond \neg A \land \Diamond \neg B)$ :

Axiom $\mathbf {T}$ follows immediately from ( $+\hspace {-0.1em}\Box $ E.) and ( $+\hspace {-0.1em}\rightarrow $ I.), as the derivation of ( $+\hspace {-0.1em}\Box $ E.) does not involve $\Diamond $ -Elimination rules.

Axiom $\mathbf {5}$ can be written on the signature $\{\neg ,\land ,\Diamond \}$ as $\neg (\Diamond A \land \Diamond \neg \Diamond A)$ . The following derivation witnesses $\vdash +\hspace {-0.1em} \neg (\Diamond A \land \Diamond \neg \Diamond A)$ :

□

The derivations of Axioms K and 5 make use of $\Diamond $ -Elimination rules, but we can apply these axioms within (Weak Inference) by assuming them and then discharging them. To wit, suppose that from $+\hspace {-0.1em} A$ we can infer $+\hspace {-0.1em} B$ using Axiom K. Then the following application of (Weak Inference) is, strictly speaking, incorrect.

But we may still derive $\oplus \hspace {-0.08em} B$ from $\oplus \hspace {-0.08em} A$ as follows.

Here, the application of (Weak Inference) is licit since it uses $+\hspace {-0.1em} (K)$ as a dischargeable assumption. If this application of (Weak Inference) occurs within another application of (Weak Inference), one can defer the derivation of $+\hspace {-0.1em} (K)$ to the outermost proof level. Clearly, this method generalizes to uses of Axiom 5 within applications of (Weak Inference). So we will use the S5 axioms freely from now on. In general, if $+\hspace {-0.1em} A$ is a theorem—if one can show it from no side premisses, even when using $\Diamond $ -Eliminations—one may use $+\hspace {-0.1em} A$ even in proof contexts that disallow $\Diamond $ -Eliminations by assuming $+\hspace {-0.1em} A$ , conditionalizing on A and deriving $+\hspace {-0.1em} A$ at the outermost proof level.

4.1 Soundness

We begin with the translation. The idea is to translate strong assertion with necessity, rejection with possible falsity, and weak assertion with possibility, but note that this is a translation only in the technical sense: it is not intended to provide the intended interpretation of the force-markers. Formally, we define a mapping $\tau $ from $\mathcal {L}_{EML}$ -formulae to formulae in the language $\mathcal {L}_{ML}$ of modal logic.

$$ \begin{align*}\tau(\varphi) = \begin{cases} \Box\psi\text{, if }\varphi = +\hspace{-0.1em}\psi\\ \Diamond\neg\psi\text{, if }\varphi = \ominus\hspace{-0.08em}\psi\\ \Diamond\psi\text{, if }\varphi = \oplus\hspace{-0.08em}\psi \end{cases}\end{align*} $$

If $\Gamma $ is a set of $\mathcal {L}_{EML}$ -formulae, we write $\tau [\Gamma ]$ for $\{\tau (\varphi ): \varphi \in \Gamma \}$ .

We now prove that, under this translation, $\mathsf {EML}$ is sound with respect to $\mathbf {S5}$ . That is:

The main challenge is to show that the restrictions on (Weak Inference) are effective in ensuring the soundness of the calculus. To this end, let $\mathsf {EML}^+$ be the calculus of $\mathsf {EML}$ plus the derivable rules for conjunction under weak assertion, which we recall for convenience.

Since these are derivable in $\mathsf {EML}$ , the soundness of $\mathsf {EML}$ is equivalent to the soundness of $\mathsf {EML}^+$ . We use $\vdash ^+$ to denote derivability in $\mathsf {EML}^+$ and write $\Gamma \vdash ^+_D \varphi $ to indicate that the derivation D witnesses the existence of this derivability relation between $\Gamma $ and $\varphi $ .

Now let $\mathsf {EML}^-$ be the calculus of $\mathsf {EML}^+$ without (Weak Inference) and write $\vdash ^-$ for the resulting derivability relation. By inspecting the proofs of ( $+\hspace {-0.1em}\rightarrow $ E.) and (WMP), one can see that ( $+\hspace {-0.1em}\rightarrow $ E.) and (WMP) are derivable in $\mathsf {EML}^-$ . (The conditional proof rule ( $+\hspace {-0.1em}\rightarrow $ I.) is not derivable in $\mathsf {EML}^-$ , but this does not matter for present purposes.)

Now, it is straightforward to show that $\mathsf {EML}^-$ is sound with respect to $\mathbf {S5}$ modulo the translation $\tau $ .

The proof is a standard induction on the length of derivations and is therefore omitted. Next, we prove the soundness of the full calculus. First, we need an auxiliary definition and a technical lemma. The following definition provides the tools to rewrite a proof D not involving $\Diamond $ -Eliminations to a proof where $\Diamond $ s only occur in sentences that translate back to S5-tautologies.

Note that all formulae in $\Sigma ^D$ substitute to S5-tautologies under the map $c_X \mapsto X$ (i.e. the inverse of $\pi ^D$ ). The following lemma shows that these added assumptions suffice to rewrite the proof D under the translation $\pi ^D$ .

Note that no applications of (Weak Inference) were added when translating D to $D'$ , since (WMP) and ( $+\hspace {-0.1em}\rightarrow $ E.) can be derived in $\mathsf {EML}^+$ from ( $\oplus \hspace {-0.08em}\land $ E.) without using (Weak Inference).

Now we are ready to prove the Soundness of the full calculus.

Proof of Theorem 4.1

We prove the statement of the theorem for $\vdash ^+$ , which immediately entails the theorem. Without loss of generality, we may assume that $\Gamma $ contains only strongly asserted formulae: if it contains $\ominus \hspace {-0.08em} A$ , one can substitute with $+\hspace {-0.1em}\Diamond \neg A$ , and if it contains $\oplus \hspace {-0.08em} A$ , one can substitute with $+\hspace {-0.1em}\Diamond A$ .

The proof proceeds by induction on the number n of times that (Weak Inference) is applied in a derivation. The base case $n=0$ is exactly Theorem 4.2.

Suppose that Soundness holds for all derivations D in which (Weak Inference) is applied less than n times. We want to show that derivations with n applications are sound. Let D be a derivation with n applications of (Weak Inference) and consider any subderivation $D'$ that ends in one such application. Note that we do not need to treat applications of (Weak Inference) to conclude $\bot $ , since they are equivalent to the case in which B is $p\land \neg p$ for an arbitrary p. Thus, the local proof context is this:

In this situation, $D'$ does not use $\Diamond $ -Elimination rules, and there is a finite subset $\Gamma ' \subseteq \Gamma $ such that all formulae in $\Gamma '$ are signed by $+$ , such that $\Gamma ',+\hspace {-0.1em} A \vdash ^+_{D'} +\hspace {-0.1em} B$ and $\Gamma ' \vdash ^+ \oplus \hspace {-0.08em} A$ . To conclude the proof, it suffices to show that $\tau [\Gamma '] \models ^{\mathbf {S5}} \tau (\oplus B)$ .

For readability we will henceforth omit mentioning $\tau $ , so that, say, $\Gamma ' \models ^{\mathbf {S5}} +\hspace {-0.1em} A$ is understood to stand for $\tau [\Gamma ']\models ^{\mathbf {S5}} \tau (+\hspace {-0.1em} A)$ . Since $D'$ contains less than n applications of (Weak Inference), by the induction hypothesis we have that

(I1) $$ \begin{align} \Gamma', +\hspace{-0.1em} A \models^{\mathbf{S5}} +\hspace{-0.1em} B \end{align} $$

and that

(I2) $$ \begin{align} \Gamma' \models^{\mathbf{S5}} \oplus\hspace{-0.08em} A. \end{align} $$

The proof that $\Gamma ' \models ^{\mathbf {S5}} \Diamond B$ now proceeds in two steps.

1. i. We show that $\Gamma ' \models ^{\mathbf {S5}} \oplus \hspace {-0.08em} B$ if A and B are $\mathcal {L}_{PL}$ -formulae and $\Gamma '$ can be split in $\Gamma ' = \Delta \cup \Theta $ such that: for all $+\hspace {-0.1em} C \in \Delta $ , C is an $\mathcal {L}_{PL}$ -formula; and for all $+\hspace {-0.1em} D \in \Theta $ , $D = \Diamond X \rightarrow X$ for some $\mathcal {L}_{PL}$ -formula X.

2. ii. By Lemma 4.3, any other application of (Weak Inference) can be reduced to (i.).

So, first assume that A and B are $\mathcal {L}_{PL}$ -formulae and $\Gamma ' = \Delta \cup \Theta $ as above. We need to show that for any model $V = \langle W^V,R^V,V^V,w^V \rangle $ of $\Gamma '$ , we have that $V, w^V \Vdash \Diamond B$ , where $\Vdash $ is the usual satisfaction relation for worlds in modal logic. Assume for reductio that there is a counterexample, i.e. a V with $V,w^{V} \Vdash \Box \neg B$ . By the induction hypothesis and in particular (I2), we also have that $V,w^V \Vdash \Diamond A$ . It follows that $V,w^V \Vdash \Diamond (A\land \neg B)$ . Let $v \in W^{V}$ be a witness, i.e. $V,v \Vdash A \land \neg B$ . Note that for all $+\hspace {-0.1em} C \in \Delta $ we have that $V,v^{V} \Vdash C$ , since $V,w^{V} \Vdash \Box C$ .

Now consider the model $V'$ such that: $W^{V'} = \{v\}$ , $V^{V'}(v) = V^{V}(v)$ , $R^{V'} = \{(v,v)\}$ , $w^{V'} = v$ . Note that all C with $+\hspace {-0.1em} C \in \Delta $ are assumed to be $\mathcal {L}_{PL}$ -formulae. That is, the fact that $V,v^V \Vdash C$ is dependent only on the valuation $V(v)$ and not on any other worlds in $W^V$ . Thus it is also the case that $V',w^{V'} \Vdash C$ for all C with $+\hspace {-0.1em} C \in \Delta $ . For the same reason, $V',w^{V'} \Vdash A \land \neg B$ . Also note that, since $V'$ has precisely one world, $V',w^{V'} \Vdash \Diamond X$ iff $V',w^{V'} \Vdash X$ . So $V',w^{V'} \Vdash \Diamond X \rightarrow X$ for any X. Thus $V' \Vdash \Theta $ .

Hence $V',w^{V'} \Vdash \Gamma '$ . By construction, $V',w^{V'} \Vdash \Box A$ and $V', w^{V'} \Vdash \neg \Box B$ . So $V'$ is a countermodel to $\Gamma '\cup \{+\hspace {-0.1em} A\} \models ^{\mathbf {S5}} +\hspace {-0.1em} B$ , but this is true by induction (I1). Contradiction. Thus there is no such V. This shows (i.).

For (ii.), we relax our assumption so that A, B and the formulae in $\Gamma '$ may be arbitrary. By Lemma 4.3, we have a derivation $D''$ such that $\pi ^{D'}[\Gamma '] \cup \Sigma ^{D'}, +\hspace {-0.1em} c_A \vdash ^+_{D''} +\hspace {-0.1em} c_B$ (writing $+\hspace {-0.1em} c_A$ for $\pi ^{D'}(+\hspace {-0.1em} A)$ and same for B).

Note that since $\pi ^{D'}$ maps everything to $\mathcal {L}_{PL}$ -formulae, the elements of $\pi ^{D'}[\Gamma '] \cup \Sigma ^{D'}$ are as described in (i): $\pi ^{D'}[\Gamma '] \cup \Sigma ^{D'} = \Delta \cup \Theta $ with $\Theta $ being exactly all formulae added in clause (e.) in the construction of $\Sigma ^{D'}$ (Definition 1). Thus we obtain $\pi ^{D'}[\Gamma '], \Sigma ^{D'} \models ^{\mathbf {S5}} \Diamond c_B$ by the argument of (i). Note that the argument of (i) rests on the induction hypothesis, but this is still licit here since $D''$ does not contain more applications of (Weak Inference) than $D'$ (by Lemma 4.3).

Now let $\Sigma = (\pi ^D)^{-1}[\Sigma ^{D'}]$ . Since $\mathbf {S5}$ is closed under uniform substitution, $\Gamma '\cup \Sigma \models ^{\mathbf {S5}} \Diamond B$ . But $\Sigma $ contains only $\mathbf {S5}$ -tautologies (by inspection of Definition 1). Hence $\Gamma ' \models ^{\mathbf {S5}} \Diamond B$ .□

It is worth noting why the argument above does not work for the rules excluded from (Weak Inference), i.e. ( $+\hspace {-0.1em}\Diamond $ E.) and ( $\oplus \hspace {-0.08em}\Diamond $ E.). The reason is that translating these rules in the proof of Lemma 4.3 would require us to add $+\hspace {-0.1em} (c_{\Diamond Z} \rightarrow c_Z)$ to $\Sigma ^D$ in Definition 1. This, however, does not substitute to an $\mathbf {S5}$ -tautology, so the final step in the soundness proof would fail.

4.2 Completeness

We now show that, modulo the translation $\tau $ defined above, $\mathsf {EML}$ is also complete with respect to $\mathbf {S5}$ .

This is shown by a model existence theorem. The construction of a canonical term model has to respect the difference between derivations that use $\Diamond $ -Eliminations and those that do not. To this end, we need some additional definitions. We write $\vdash ^*$ to denote provability in $\mathsf {EML}$ without the rules for $\Diamond $ -Elimination. We say that a set $\Gamma $ is S-consistent if $\Gamma $ contains only strongly asserted formulae and $\Gamma \not \vdash ^* \bot $ . And we say that $\Gamma $ is S-inconsistent if it contains only strongly asserted formulae and $\Gamma \vdash ^* \bot $ . Note that there are inconsistent S-consistent sets, e.g. $\{+\hspace {-0.1em} p, +\hspace {-0.1em}\Diamond \neg p\}$ .

In the typical canonical construction, one takes maximally consistent sets of formulae to be the worlds. We will instead take maximally S-consistent sets of formulae. In the construction, we shall use the following technical lemmas.

Now we are ready to demonstrate a model existence result.

Intuitively, the reason why the worlds of the term model may denote inconsistent sets of formulae is as follows. The inference $+\hspace {-0.1em} A \vdash +\hspace {-0.1em}\Box A$ must be excluded when computing maximal consistent sets. That is, if we were taking consistent sets (instead of S-consistent sets) as the worlds of the term model, then whenever $+\hspace {-0.1em} A \in w$ for some consistent w, it would also be the case that $+\hspace {-0.1em}\Box A \in w$ . But such sets are not useful as worlds in a canonical model, since they can only see themselves according to the definition of R in the canonical model construction. The notion of S-consistency takes care of this issue.

One may also wonder why the same argument does not work when we weaken the demand that $\Gamma $ be consistent in the statement of Theorem 4.7 to $\Gamma $ being merely S-consistent. The stronger assumption of consistency is used in the step for $+\hspace {-0.1em}\Diamond A$ in the induction on the complexity of sentences in the proof of the theorem. For this step of the proof relies on the fact that $Cl^+\hspace {-0.1em}(\Gamma )$ is closed under the derivability relation in the full $\mathsf {EML}$ calculus and in particular under ( $+\hspace {-0.1em}\Box $ I.). But if we were to allow inconsistent S-consistent $\Gamma $ , then closure under the full $\mathsf {EML}$ calculus would result in the trivial theory. Thus, only consistent $\Gamma $ have a canonical model (but the worlds in this canonical model may be inconsistent sets).

4.3 Some corollaries

The following is a noteworthy corollary of the completeness theorem.

That is, the $\mathsf {EML}$ logic of strong assertion is the logic that preserves $\mathbf {S5}$ -validity. Schulz (Reference Schulz2010, p. 389) noted the same result about Yalcin’s (Reference Yalcin2007) informational consequence (IC). That is, $\Gamma \models ^{\text {IC}} A$ iff $\{\Box B \mid B\in \Gamma \} \models ^{\mathbf {S5}} \Box A$ . Thus, we may conclude, the logic of strong assertion coincides with informational consequence.

There is a caveat, however. Yalcin’s semantics includes a conditional $\rightarrow $ that is not a material conditional, but a version of a restricted strict conditional (Yalcin, Reference Yalcin2007, p. 998). Clearly, then, Yalcin’s informational consequence only preserves $\mathbf {S5}$ -validity on the signature $\{\neg ,\land ,\Diamond \}$ . Hence the $\mathsf {EML}$ logic of strong assertion only corresponds to the nonimplicative fragment of informational consequence. In either logic, however, the material conditional is definable from $\neg $ and $\land $ .

Now, since in $\mathbf {S5}$ tautological truths are validities, Proposition 4.8 entails that the strongly asserted theorems of $\mathsf {EML}$ are exactly the $\mathbf {S5}$ -tautologies. This shows the converse direction of Theorem 3.5 to establish Theorem 4.9).

From these results, we also obtain two corollaries about the logic of rejection in $\mathsf {EML}$ .

Proposition 4.11 mentions $\Box A$ , whereas Theorem 4.9 does not. This difference is due to the fact that $\models ^{\mathbf {S5}} A$ iff $\models ^{\mathbf {S5}} \Box A$ , whereas, in general, it is not the case that $A \models ^{\mathbf {S5}} \bot $ iff $\Box A\models ^{\mathbf {S5}} \bot $ . A counterexample to the latter is obtained by letting A be $p \land \Diamond \neg p$ .

But what is the relation of $\mathsf {EML}$ to the consequence relation $\models ^{\mathbf {S5}}$ with premisses? We can approximate this relation from below by considering $\vdash ^*$ , the derivability relation of $\mathsf {EML}$ without rules for $\Diamond $ -Elimination.

However, we used $\Diamond $ -Elimination rules to derive Axioms K and 5. Nonetheless, one can recapture S5 using the derivability relation $\vdash ^{**}$ that results from adding to $\vdash ^*$ the following restricted rule of $\Diamond $ -Elimination.

The restriction serves to recover the Necessitation rule in the proof of the following theorem.

Proof. We first prove the right-to-left direction. Inspection of the proofs in §3.4 reveals that $\vdash ^{**}$ derives all S5 axioms. Furthermore, the following derivation shows that $\vdash ^{**}$ satisfies a version of the Necessitation rule, i.e. that if $\emptyset \vdash ^{**} +\hspace {-0.1em} A$ , then $\emptyset \vdash ^{**} +\hspace {-0.1em}\Box A$ .

The application of ( $\oplus \hspace {-0.08em}\Diamond $ E.*) is legitimate here, since the (SR $_2$ ) derivation uses no premisses or assumptions other than the one being discharged and $+\hspace {-0.1em} A$ (which is a theorem by assumption.)

Now, since $\Gamma \models ^{\mathbf {S5}} A$ and S5 is complete with respect to its model theory, there is a natural deduction proof of A that requires only the premisses $\Gamma $ , the S5 Axioms, the Necessitation rule and modus ponens. By the above, this proof can be performed in $\vdash ^{**}$ to derive $+\hspace {-0.1em} A$ from $\{+\hspace {-0.1em} B \mid B \in \Gamma \}$ . This concludes the right-to-left direction.

For left-to-right direction, the proof of Proposition 4.12 works, but the following step is nontrivial:

1. (*) Assume that $\{+\hspace {-0.1em} B \mid B \in \Gamma \} \vdash ^{**} +\hspace {-0.1em} A$ . By ( $+\hspace {-0.1em}\rightarrow $ I.), this means that $\vdash ^{**} +\hspace {-0.1em} \left(\left(\bigwedge _{B\in \Gamma } B\right) \rightarrow A\right)$ .

It is not clear that ( $+\hspace {-0.1em}\rightarrow $ I.) can be applied here, since under $\vdash ^{**}$ one may apply the rule ( $\oplus \hspace {-0.08em}\Diamond $ E.*), and we have not established that this rule is permitted in ( $+\hspace {-0.1em}\rightarrow $ I.). To see that the step (*) is nonetheless correct here, note that any occurrence of ( $\oplus \hspace {-0.08em}\Diamond $ E.*) can only be in an (SR $_2$ ) subderivation that establishes $+\hspace {-0.1em} X$ for some sentence X. Let R be the set of all $+\hspace {-0.1em} X$ that are established this way anywhere in the proof. Then the following version of (*) is obviously correct, as all subderivations using ( $\oplus \hspace {-0.08em}\Diamond $ E.*) can be replaced by a premiss from R.

1. Assume that $\{+\hspace {-0.1em} B \mid B \in \Gamma \} \vdash ^{**} +\hspace {-0.1em} A$ . By ( $+\hspace {-0.1em}\rightarrow $ I.), this means that $R \vdash ^{**} +\hspace {-0.1em} \left(\left(\bigwedge _{B\in \Gamma } B\right) \rightarrow A\right)$ .

But all members of R are theorems under $\vdash ^{**}$ , since they can be established by a Smileian reductio that does not require any side premisses. Thus the step (*) is correct. □

5.1 Yalcin sentences

Yalcin (Reference Yalcin2007) famously observed that sentences of the form p and it might not be that p sound bad even when occurring in certain embedded environments, such as suppositions, in which Moore-paradoxical sentences (p and I don’t believe that p) sound fine. The following proof in EML shows $+\hspace {-0.1em} p \land \Diamond \neg p$ to be absurd.

This proof shows that uttering the sentence p and it might not be that p immediately commits one to having incompatible attitudes (namely, assent to p and dissent from p), which is absurd. Thus it also explains why suppose that p and might not p sounds odd, as to suppose p and it might not be that p is to suppose something manifestly absurd. This is the argument we give in Incurvati & Schlöder (Reference Incurvati and Schlöder2019), but we can clarify why precisely it is absurd to suppose p and it might not be that p.

To formalize supposition, we add a new primitive force-marker $\mathcal {S}$ to our language. In English, supposition refers both to an attitude and to its expression (Green, Reference Green2000, pp. 377–378). That is, the speech act of supposing that A expresses the attitude of supposing that A. Accordingly, $\mathcal {S}A$ stands for the speech act of supposing A, performed through locutions such as suppose that A, and expresses the attitude of supposing A.

Following Stalnaker (Reference Stalnaker2014), pp. 150–151, we take the supposition of A to consists in a proposal to add A to the common ground, but to do so temporarily.Footnote ⁶ In supposing that A, one is not committing to A, but is probing what happens if one were to commit to A. That is, one is checking what the consequences would be of adding A to the common ground.Footnote ⁷ For this process to work as desired, the internal logic of supposition must be the same as the logic of strong assertion. This sanctions the coordination principle ( $\mathcal {S}$ -Inference), which states that the suppositional consequences of a suppositional context mirror the strongly assertoric consequences of the corresponding strongly assertoric context.Footnote ⁸

This coordination principle immediately implies that suppose p and might not p is absurd. For as shown above, $+\hspace {-0.1em} p\land \Diamond \neg p$ entails incompatible attitudes, which is absurd. By ( $\mathcal {S}$ -Inference), this means that supposing p and might not p is absurd as well, i.e. one derives $\bot $ from $\mathcal {S}(p\land \Diamond \neg p)$ .

The absurdity of suppose p and might not p does not quite explain its infelicity, since not all absurd suppositions sound bad. One may felicitously suppose certain logical contradictions. For instance, someone not familiar with the derivability of Peirce’s Law in classical logic may felicitously suppose its negation. However, to see that the negation of Peirce’s Law is absurd requires a complex argument. By contrast, anyone grasping the meaning of and and not will immediately recognize the absurdity of, say, p and not p. This explains why p and not p sounds bad and continues to do so in embedded contexts. The same holds for p and might not p: its absurdity can be immediately inferred by applying the meaning-conferring rules of and, not and might. This absurdity is therefore manifest to anyone who grasps the meaning of these expressions, which explains why suppose p and might not p is infelicitous.

In addition to explaining the infelicity of Yalcin sentences under suppose, our account has the resources to explain why Moore sentences sound bad in ordinary contexts but cease to do so in suppositional ones. For while strongly asserting p and I do not believe that p is improper, it is not absurd in the sense of $+\hspace {-0.1em} p \land \neg Bp$ entailing $\perp $ . In our view, uttering a Moore sentence is infelicitous because it violates one of the preparatory conditions of strong assertion, e.g. that one should know or believe what one strongly asserts. But such preparatory conditions do not factor into the commitments undertaken by a strong assertion. Thus, the violation of these preparatory conditions does not proof-theoretically reduce to the speaker having both strongly asserted and rejected the same proposition. Hence, suppose that p and I do not believe that p is not predicted to be absurd. Since supposition and strong assertion have different preparatory conditions—in particular, one need not believe what one supposes—suppose that p and I do not believe that p is not predicted to be improper either.

The rule ( $\mathcal {S}$ -Inference) is sufficient to explain the relevant data about Yalcin sentences, but to give a complete account of supposition further rules may need to be added. For our present project, there are more immediate concerns.

5.2 Generalized Yalcin sentences

Paolo Santorio (Reference Santorio2017) observed that sentences like (3) seem to sound as bad as the original Yalcin sentences, and continue to do so when embedded under suppose.

1. (3) (If p, then it might be that q) and (if p, then $\neg $ q).

We have seen that there are good reasons to treat Yalcin sentences as semantically contradictory. But then, Santorio argues, one should also treat sentences like (3) as semantically contradictory. However, it is difficult to find a conditional $>$ and a consequence relation $\models $ such that $p>\Diamond q \land p>\neg q \models \bot $ . This is because if $>$ can be vacuous, then the incompatibility of $\Diamond q$ and $\neg q$ does not force us to conclude $\perp $ , but only that p satisfies the vacuity-condition for $>$ .

Santorio (Reference Santorio2017) succeeds in defining such a $>$ and $\models $ , but we want to offer a different diagnosis of the problem posed by (3), one which does not require adopting a revised notion of consequence. Observe that an utterance of (4), which does not contain an epistemic modal, already sounds odd.

1. (4) (If p, then q) and (if p, then $\neg $ q).

The reason why (3) and (4) both sound odd seems to be similar. To wit, both utterances appear to prompt one to suppose that p, i.e. to consider what follows should p be the case—but to do so is absurd, given the information the utterances contain. That is, we will argue that (3) and (4) sound bad for the same reason that (5a) and (5b) sound bad—their antecedents cannot be supposed.

1. (5) a. If (p and not p), then q.

   b. If (p and it might not be p), then q.

We propose to explain the infelicitousness of (3), (4) and (5) by using a notion of supposability based on our characterization of supposition $\mathcal {S}$ . We have followed Stalnaker in taking the supposition of A to be a proposal to temporarily update the common ground with A. The supposability of A in a given context is the possibility of supposing A in that context.

As we explain below, a strong assertion of an indicative conditional pragmatically presupposes the supposability of the conditional’s antecedent in a context C that corresponds to the current common ground updated with the strong assertion’s content. So, in the case of (3), the supposability of p is to be checked with respect to a context C that contains at least $(p> \neg q) \land (p> \Diamond q)$ . But we have that $\mathcal {S} (p \land (p> \neg q) \land (p> \Diamond q)) \vdash \bot $ (assuming that $>$ satisfies modus ponens), so the presupposition fails.

Why should the indicative conditional have such a presupposition? Following Stalnaker (Reference Stalnaker1978), the pragmatic presuppositions of a strong assertion include all the information that can be inferred from the performance of the strong assertion itself. In particular, a strong assertion presupposes that the context is such that its essential effect (i) changes the context in a nontrivial and well-defined way and (ii) everyone in the conversation can compute this change. Now, when one proposes to update the common ground with a conditional, one proposes to change it in such a way that if the antecedent is added to the common ground, its consequent should be added too. Everyone must be able to compute what this change amounts to, as this is what they base their decision to accept or reject the update proposal on. Thus, everyone must be able to consider the common ground updated with the conditional and then temporarily (for the purpose of deliberation) add the conditional’s antecedent and arrive at a well-defined result. This may be seen as a discoursive analogue of the Ramsey test. Hence, strong assertions of conditional content pragmatically presuppose that the conditional’s antecedent is supposable in the context of the current common ground updated with the assertion’s content. But this presupposition cannot be met for the conditionals (3), (4) and (5). Many have claimed that conditionals presuppose, in some sense, that their antecedents are possible (Gillies, Reference Gillies2010; Mandelkern & Romoli, Reference Mandelkern and Romoli2017; Crespo, Karawani, & Veltman, Reference Crespo, Karawani, Veltman, Ball and Rabern2018). Our argument shows that supposability is the right way to specify what kind of possibility should be meant here, at least within the Stalnakerian framework.

One might reply that our pragmatic explanation is not general enough. For Santorio’s generalized Yalcin sentence (3) also sounds bad when the conditionals it contains are read as subjunctives. And, the reply goes, the assertion of a subjunctive conditional does not have the supposability presupposition associated with the assertion of its indicative counterpart. While indicative conditionals change the common ground in a way that can be evaluated by provisionally updating the common ground with their antecedents, subjunctive conditionals change the common ground in a way that can be evaluated by provisionally revising the common ground with their antecedent (Stalnaker, Reference Stalnaker, Harper, Stalnaker and Pearce1968). Thus, the strong assertion of a subjunctive conditional does not presuppose that its antecedent be supposable in the common ground updated with the strong assertion’s content, since some of this content might be revised in order to suppose the antecedent.

The reply is unsuccessful. Although the assertion of a subjunctive does not have the same supposability presupposition as the assertion of the corresponding indicative, it does have a supposability presupposition. And this presupposition cannot be met when Santorio’s (3) is read as a subjunctive conditional. In particular, since not all information in the common ground is up for revision when considering a subjunctive antecedent, the assertion of a subjunctive conditional presupposes that its antecedent be supposable in C, where C is the nonrevisable part of the common ground updated with the conditional. Now consider the strong assertion (if it were p, then q) and (if it were p, then not q). Clearly, p is not supposable in the context $C'$ that results from updating C with the strong assertion, since $\mathcal {S}(C' \land p) \vdash \mathcal {S}(p\land \neg p)$ , which entails $\bot $ . The same goes for (if it were p, then q) and (if it were p, then it might be that not q): p is not supposable in the context $C'$ that results from updating C with this strong assertion, since $\mathcal {S}( p\land C') \vdash \mathcal {S}(p\land \Diamond \neg p)$ , which entails $\bot $ .

One may wonder how this pragmatic explanation of the infelicity of generalized Yalcin sentences can account for their infelicity when embedded under suppose. After all, the special problem raised by Yalcin sentences is that, unlike Moore sentences, they continue to sound bad under suppose and similar environments. This, the usual story goes, precludes a pragmatic explanation of their infelicity similar to the one given for Moore sentences, since pragmatic inferences are suspended under suppose.

This story overplays the power of suppose to suspend pragmatic inferences: although the pragmatic inferences used to explain the infelicity of Moore sentences are suspended under suppose, it does not follow that all such inferences are. The pragmatic inference used to explain the infelicity of Moore sentences is suspended under supposition because while it seems a preparatory condition for strong assertion that one ought to believe what one strongly asserts, there is no analogue preparatory condition for supposition. By contrast, the pragmatic inference we outlined in the previous paragraphs clearly goes through under supposition. In particular, just as the actual update of the common ground with a conditional is only well-defined if its antecedent is supposable in the right context C, so is the temporary update with the same conditional only well-defined if its antecedent is supposable in C. That is, this supposability requirement—unlike the requirement that one ought to believe the content of the speech act—is shared by strong assertions and suppositions of conditionals alike.

There are some further generalized Yalcin sentences that we can explain using this suppositional strategy. Matthew Mandelkern (Reference Mandelkern2019) observed that sentences like (6a) and (6b) seem to sound as bad as the original Yalcin sentences.

1. (6) #a. (p and it might not be that p) or (q and it might not be that q)

   #b. might (p and it might not be that p)

However, on our account such sentences are not absurd: there are models of $\mathsf {EML}$ in which (7a) and (7b) hold and hence neither sentence entails $\bot $ (on the assumption that neither A nor B are classical contradictions).

1. (7) a. +((A $\land \Diamond \neg $ A) $\lor $ (B $\land \Diamond \neg $ B)).

   b. + $\Diamond $ (A $\land \Diamond \neg $ A).

Similarly to Santorio’s cases, Mandelkern’s cases can only be accounted for semantically by making substantial revisions to classical logic. Any attempt to add further rules to $\mathsf {EML}$ so that (7a) and (7b) entail $\bot $ would trivialize the epistemic possibility modal.

Proof. Suppose that (a) is the case, letting A be p and B be $\neg p$ . That is, we have that $+\hspace {-0.1em} ((p \land \Diamond \neg p) \lor (\neg p \land \Diamond p)) \vdash \bot $ . By Smileian reductio, this means that $\vdash \ominus \hspace {-0.08em} ((p \land \Diamond \neg p) \lor (\neg p \land \Diamond p))$ , which by ( $\oplus \hspace {-0.08em}\neg $ I.) means that $\vdash \oplus \hspace {-0.08em} \neg ((p \land \Diamond \neg p) \lor (\neg p \land \Diamond p))$ .

By De Morgan, $\neg ((p \land \Diamond \neg p) \lor (\neg p \land \Diamond p))$ can be written as $(\neg p \lor \neg \Diamond \neg p) \land (p \lor \neg \Diamond p)$ , which is classically equivalent to $\neg \Diamond \neg p \lor \neg \Diamond p$ . This sentence can be rewritten as $\Diamond p \rightarrow \Box p$ . Hence, (a) implies that $\vdash \oplus \hspace {-0.08em} (\Diamond p \rightarrow \Box p)$ . But then, we can derive $+\hspace {-0.1em} p$ from $+\hspace {-0.1em}\Diamond p$ as follows.

For the second part, suppose that (b) is the case, letting A be $\neg p$ . That is, we have that $+\hspace {-0.1em} \Diamond (\neg p \land \Diamond p) \vdash \bot $ . By Smileian reductio, this means that $\vdash \ominus \hspace {-0.08em} \Diamond (\neg p \land \Diamond p)$ . By ( $\oplus \hspace {-0.08em}\neg $ I.), this is equivalent to $\vdash \oplus \hspace {-0.08em}\neg \Diamond (\neg p \land \Diamond p)$ . By Lemma 3.7, it follows that $\vdash +\hspace {-0.1em} \neg (\neg p \land \neg \Diamond p)$ , which by Classicality is equivalent to $\vdash +\hspace {-0.1em} \Diamond p\rightarrow p$ .□

One might take issue with Lemma 3.7 here, but inspection of the proof of the lemma shows that it rests on well-motivated assumptions. Challenging the application of (Weak Inference) in the proof of (a) would not help, since the proof of (b) does not require this principle. Thus, we cannot semantically account for these generalized cases without giving up on classical logic. Mandelkern’s (Reference Mandelkern2019) account avoids these results by denying the universal validity of the relevant classically valid transformations (e.g. the application of De Morgan in the above proof).

Instead of making deep revisions to our semantics, we again provide a pragmatic explanation of the relevant data. We explain the infelicitousness of (6a) by taking the strong assertion of A or B to presuppose the supposability of A and the supposability of B. And we explain the infelicitousness of (6b) by taking the strong assertion of might A to pragmatically presuppose the supposability of A. The pragmatic inferences from A or B and might A to supposability are evinced by the felicitousness of sequences like the following.

1. (8) a. It is either p or q. So suppose that it in fact the case that p. b. It might be that p. So suppose that it in fact the case that p.

At this point, one might suggest taking the inferences from A or B and might A to supposability to be not pragmatic, but semantic. In particular, one might identify the meaning of might with supposability and the meaning of A or B with might A and might B.Footnote ⁹ However, supposition can be counterfactual, so having asserted not A one may go on to suppose that A, possibly just for the sake of argument. By contrast, having asserted not A, it is a mistake to also assert might A. Thus might cannot be identified with supposability.

Pragmatically explaining the data in (6) has also an empirical advantage over Mandelkern’s (Reference Mandelkern2019) own semantic approach. He claims that the infelicitousness of (6a) is explained by the fact that Yalcin sentences are classical contradictions and that disjunctions of classical contradictions are themselves classical contradictions. But now consider (9).

1. (9) (p and it might not be that p) or q.

If q is true, then according to the usual truth-functional meaning of disjunction (which Mandelkern does not dispute), (9) is true, since it has a true disjunct. However, (9) sounds odd. Thus Mandelkern would seem to need some further mechanism to explain the oddity of (9), e.g. our pragmatic presupposition or another principle entailing that disjunctions one of whose disjuncts is a classical contradiction sound generally bad. But then, any such mechanism would also account for the infelicitousness of (6). Hence, the more parsimonious approach is to stick with classical consequence and explain both (6) and (9) pragmatically, as we have done.