Why Conditional Logics Are Needed

This chapter describes a number of different logics that introduce a two-place operator (Ó, ⇒, or >) to help represent conditional expressions – expressions of the form: if A, then B (or of related forms such as the subjunctive conditional: if A were to be the case, then B would be). But why are such logics needed? Why not simply handle conditionals using the symbol → for material implication? The rules for → (namely, Modus Ponens and Conditional Proof) are quite intuitive. Furthermore, we know that the system of propositional logic that employs these rules is sound and complete for a semantics that adopts the material implication truth table embodied in (→).

1. (→) aw(A→B)=T iff aw(A)=F or aw(B)=T.

So it appears that → is all we need to manage ‘if .. then’.

On the other hand, objections to the idea that material implication is an adequate account of conditionals have been with us for almost as long as formal logic has existed. According to (→), A→B is true when A is false, and yet this is hardly the way ‘if A then B’ is understood in natural language. It is false that I am going to live another 1,000 years, but that hardly entails the truth of (1).

1. (1) If I am going to live another 1,000 years, then I will die tomorrow.

When A and B are incompatible with each other as in this case, the normal reaction is to count ‘if A then B’ false, even when the antecedent A is false. This illustrates that in English, the truth of ‘if A then B’ requires some sort of relevant connection between A and B. When A and B are incompatible, as they are in (1), there is no such connection, and so we reject ‘if A then B’.

Chapter 3 introduced the accessibility relation R on the set of worlds W in defining the truth condition for the generic modal operator. In K-models, the frame <W, R> of the model was completely arbitrary. Any nonempty set W and any binary relation R on W counts as a frame for a K-model. However, when we actually apply modal logic to a particular domain and give □ a particular interpretation, the frame <W, R> may take on special properties. Variations in the principles appropriate for a given modal logic will depend on what properties the frame should have. The rest of this chapter explains how various conditions on frames emerge from the different readings we might choose for □.

Conditions Appropriate for Tense Logic

In future-tense logic, □ reads ‘it will always be the case that’. Given (□), we have that □A is true at w iff A is true at all worlds v such that wRv. According to the meaning assigned to □, R must be the relation earlier than defined over a set W of times. There are a number of conditions on the frame <W, R> that follow from this interpretation. One fairly obvious feature of earlier than is transitivity.

1. Transitivity: If wRv and vRu, then wRu.

The Language of Propositional Modal Logic

We will begin our study of modal logic with a basic system called K in honor of the famous logician Saul Kripke. K serves as the foundation for a whole family of systems. Each member of the family results from strengthening K in some way. Each of these logics uses its own symbols for the expressions it governs. For example, modal (or alethic) logics use □ for necessity, tense logics use H for what has always been, and deontic logics use O for obligation. The rules of K characterize each of these symbols and many more. Instead of rewriting K rules for each of the distinct symbols of modal logic, it is better to present K using a generic operator. Since modal logics are the oldest and best known of those in the modal family, we will adopt □ for this purpose. So □ need not mean necessarily in what follows. It stands proxy for many different operators, with different meanings. In case the reading does not matter, you may simply call □A ‘box A’.

First we need to explain what a language for propositional modal logic is. The symbols of the language are ⊥, →, □; the propositional variables: p, q, r, p′, and so forth; and parentheses. The symbol ⊥ represents a contradiction, → represents ‘if . . then’, and □ is the modal operator. A sentence of propositional modal logic is defined as follows:

1. ⊥ and any propositional variable is a sentence.

2. If A is a sentence, then □A is a sentence.

3. If A is a sentence and B is a sentence, then (A→B) is a sentence.

4. No other symbol string is a sentence.

In this book, we will use letters ‘A’, ‘B’, ‘C’ for sentences. So A may be a propositional variable, p, or something more complex like (p→q), or ((p→ ⊥)→q). To avoid eyestrain, we usually drop the outermost set of parentheses. So we abbreviate (p→q) to p→q. (As an aside for those who are concerned about use-mention issues, here are the conventions of this book. We treat ‘⊥’, ‘→’, ‘□’, and so forth as used to refer to symbols with similar shapes. It is also understood that ‘□A’, for example, refers to the result of concatenating □ with the sentence A.)

Since there are so many different possible systems for modal logic, it is important to determine which system are equivalent, and which ones distinct from others. Figure 11.1 (on the next page) lays out these relationships for some of the best-known modal logics. It names systems by listing their axioms. So, for example, M4B is the system that results from adding (M), (4), and (B) to K. In boldface, we have also indicated traditional names of some systems, namely, S4, B, and S5. When system S appears below and/or to the left of S′ connected by a line, then S′ is an extension of S. This means that every argument provable in S is provable in S′, but S is weaker than S′, that is, not all arguments provable in S′ are provable in S.

Showing Systems Are Equivalent

One striking fact shown in Figure 11.1 is the large number of alternative ways of formulating S5. It is possible to prove these formulations are equivalent by proving the derivability of the official axioms of S5 (namely, (M) and (5)) in each of these systems and vice versa. However, there is an easier way. By the adequacy results given in Chapter 8 (or Chapter 9), we know that for each collection of axioms, there is a corresponding concept of validity. Adequacy guarantees that these notions of provability and validity correspond. So if we can show that two forms of validity are equivalent, then it will follow that the corresponding systems are equivalent. Let us illustrate with an example.

There are a number of different approaches one can take to giving the semantics for the quantifiers. The simplest method uses truth value semantics with the substitution interpretation of the quantifiers (Leblanc, 1976). Although the substitution interpretation can be criticized, it provides an excellent starting point for understanding the alternatives, since it avoids a number of annoying technical complications. For students who prefer to learn the adequacy proofs in easy stages, it is best to master the reasoning for the substitution interpretation first. This will provide a core understanding of the basic strategies, which may be embellished (if one wishes) to accommodate more complex treatments of quantification.

Truth Value Semantics with the Substitution Interpretation

The substitution interpretation is based on the idea that a universal sentence ∀xAx is true exactly when each of its instances Aa, Ab, Ac, . . , is true. For classical logic, ∀xAx is T if and only if Ac is T for each constant c of the language. In the case of free logic, the truth condition states that ∀xAx is T if and only if Ac is T for all constants that refer to a real object. Since the sentence Ec indicates that c refers to a real object, the free logic truth condition should say that Ac is T for all those constants c such that Ec is also true.

Worlds and Intensions

A pervasive feature of natural languages is that sentences depend for their truth value on the context or situation in which they are evaluated. For example, sentences like ‘It is raining’ and ‘I am glad’ cannot be assigned truth values unless the time, place of utterance, and the identity of the speaker are known. The same sentence may be true in one situation and false in another. In modal language, where we consider how things might have been, sentences may be evaluated in different possible worlds.

In the standard extensional semantics, truth values are assigned directly to sentences, as if the context had no role to play in their determination. This conflicts with what we know about ordinary language. There are two ways to solve the problem. The first is to translate the content of a sentence uttered in a given context into a corresponding sentence whose truth value does not depend on the context. For example, ‘It is raining’ might be converted into ‘It is raining in Houston at 12:00 EST on Dec. 9, 1997..’. The dots here indicate that the attempt to eliminate all context sensitivity may be a never-ending story. For instance, we forgot to say that we are using the Gregorian calendar, or that the sentence is to be evaluated in the real world.

Russell’s Theory of Descriptions

English phrases that begin with ‘the’, such as ‘the man’ and ‘the present king of France’, are called definite descriptions (or descriptions, for short). So far, we have no adequate logical notation for descriptions. It is possible to translate ‘the man is bald’ by choosing a constant c for ‘the man’, a predicate letter P for ‘is bald’, and writing: Pc. However, treating the description as if it were a constant will cause us to classify some valid arguments as invalid.

For example, it should be clear that (1) entails (2).

1. (1) Aristotle is the philosopher who taught Alexander the Great.

2. (2) Aristotle taught Alexander the Great.

If we choose the constants: a for Aristotle, and g for Alexander the Great, we might notate (2) as (2′).

1. (2′) Tag

If we treat ‘the philosopher who taught Alexander the Great’ as a constant g, then (1) is notated by (1′).

1. (1′) a≈g

However, there is no logical relationship between the atomic sentences (1′) and (2′) that would cause us to recognize that the argument from (1′) to (2′) is valid. Clearly we need a way to notate the internal structure of ‘the philosopher who taught Alexander the Great’ if we are ever to show that (1) entails (2) in logic.

De Re and De Dicto

We have already encountered the de re – de dicto distinction at a number of points in this book. In this section, we will investigate the distinction more carefully, explain methods used to notate it, and develop quantified modal logics that are adequate for arguments involving the new notation.

Some of the best illustrations of the de re – de dicto distinction can be found among sentences of tense logic. For example, consider (S).

1. (S) The president was a crook.

This sentence is ambiguous. It might be taken to claim of the present president that he (Obama at the time this was written) used to be a crook. On the other hand, it might be read ‘At some time in the past the president (at that time) was a crook’. On this last reading, we are saying that the whole sentence (or dictum, in Latin) ‘the president is a crook’ was true at some past time. This is the de dicto reading of (S). Here both ‘the president’ and ‘is a crook’ are read in the past tense. We can represent this interpretation of (S) by applying the past tense operator P to the sentence ‘the president is a crook’, so that both ‘the president’ and ‘is a crook’ lie in its scope.

1. P(the president is a crook) de dicto reading of (S)

On the first reading of (S), we are saying a certain thing (in Latin, res) has a past tense property: of having been a crook. This is the de re reading of (S). Here we read ‘the president’ in the present tense, and ‘is a crook’ in the past tense. We can represent this reading by restricting the scope of the past tense operator P to the predicate ‘is a crook’.

1. the president P(is a crook) de re reading of (S)

The distinction between these two readings of (S) is a crucial one, for given that Obama never was a crook, and that Nixon was, the de dicto version of (S) is true, while the de re version is false.

We have shown that systems based on natural deduction rules do not (globally) express classical semantics; they select intensional rather than standard interpretations for the connectives. Although S& forces the classical reading, the rules for →, ↔, ∨, and ~ (whether classical or intuitionistic) pick out intuitionistic interpretations that are weaker than their classical ones. (See Chapters 6, 7, and 8.) It is well known that when local expression is the criterion at issue, the same “failing” does not apply to Gentzen sequent calculi with multiple conclusions (or sequent systems for short), for they are strong enough to force classical readings of the connectives. (See Shoesmith and Smiley (1978, p. 3) and Hacking (1979, p. 312).) That result is far from surprising, as we learned in Chapter 3 that even the natural deduction rules locally express the classical interpretation.

However, this leaves open the question as to whether the sequent rules globally express the classical truth conditions. At the end of Section 1.5, we gave a simple argument that the system G→~ deductively expresses the classical readings. Since every global model of a system is a deductive model of it, it follows that G→~ forces the classical conditions and so expresses them. In Section 11.1, that result will shown for a full range of sequent systems using a more powerful line of proof. This result has interesting consequences for our understanding of the relationships between natural deduction and sequent systems. Section 11.2 shows that it explains the “magical fact” (Hacking, 1979, p. 293) that when one restricts sequent rules to a single formula on the right-hand side, one obtains exactly intuitionistic (and not classical) propositional logic. In Section 11.3 it is shown that the distinction between global and local expression collapses for sequent systems, so that classical conditions forced by local models are exactly the ones forced by global models. Since we have argued in Section 3.5 that local expression is incompatible with an inferentialist account of connective meaning, sequent calculi forfeit a role in the project of this book. Another untoward feature of sequent systems was shown in Theorem 3.5. The only truth conditions that sequent rules can express are extensional. The collapse of global to local validity corresponds to a prejudice for extensional over intensional truth conditions. Section 11.4 will deploy these results to help adjudicate claims that classical systems are the only logics.

Of the accounts of expressive power examined so far, one definition is too weak and the other too strong. The Deductive Expression Theorem of Section 2.2 showed that indeterminacy of connective meaning affects all symbols other than & and ⊥ when deductive expression is chosen as the criterion for what rules express. However, when local expression is chosen in its place, the Local Expression Theorem of Section 3.3 applies, and so natural deduction rules determine classical interpretations of all the connectives. Unfortunately, this has the bizarre consequence that systems of rules locally express semantical conditions for which they are incomplete. Furthermore, the use of local expression is simply incompatible with predicate logic or modal logic, and it flies in the face of the fact that it is validity, not satisfaction, that serves as our criterion for the correctness of arguments. (See the objections discussed in Section 3.5.)

In this chapter, we will define global expression, the notion of expression that gets it just right. It is neither too strong nor too weak, and it fits nicely with an interest in systems stronger than propositional logic. Here preservation of validity is taken as the appropriate benchmark for what counts as a model of a rule. Therefore system S globally expresses property P of models iff for every model V, the rules of S preserve V-validity exactly when V obeys P. The interesting thing about global expression is that natural deduction (ND) rules express intuitionistic truth conditions. The present chapter will set the stage for discussing these results by defining global expression and exploring its features. We will first show (Section 4.1) that the natural deduction rules for & and ⊥ globally express the standard interpretations for & and ⊥. In Section 4.2, we will define and explore the notion of a natural semantics for a system of rules S, that is, a condition on models that is globally expressed by S, and that provides a recursive account of the truth behavior of the connectives that S regulates. After that, it will be shown (Section 4.4) that the natural semantics for the natural deduction systems regulating →, ↔, and ∨ is not classical, by exploiting results in Section 4.3 concerning what is called the canonical model.

It has been relatively easy to locate a natural semantics for propositional logics that involve connectives other than disjunction. Discussion of disjunction has been postponed because the situation here is more complicated. It is possible to locate a condition ‖∨‖ on models that the rules for disjunction express (Section 7.2). However, it is unfamiliar; ‖∨‖ is neither the classical condition nor the intuitionistic reading of ∨ that was introduced by Beth (Section 7.1). Furthermore, there is reason to worry whether ‖∨‖ qualifies as a recursive characterization of truth conditions (Section 7.3). So it will be necessary to try to rescue ‖∨‖ (if we can) with a new isomorphism result. In Section 7.4, a variant of the Kripke semantics will be introduced (called path semantics) that includes an additional structure used in the disjunction truth condition. An isomorphism is shown to exist between the models in path semantics and models that obey ‖∨‖ (Section 7.5). This goes part of the way towards legitimizing the condition expressed by the disjunction rules as qualifying as a semantics. However Section 7.6 demonstrates that neither functionality nor a desirable form of compositionality holds for ‖∨‖. So whether the isomorphism result goes far enough to offset this pathology is in doubt. Whether we should accept ‖∨‖ as a legitimate reading for ∨ will be something left for the reader to judge.

The last chapter has paved the way for showing that natural deduction systems for propositional logics express intuitionistic semantics. It is now time to lay out the results connective by connective. We already proved that the natural semantics for S& and S⊥ is classical (Sections 4.1 and 4.2); but intuitionistic semantics agrees with these interpretations of & and ⊥. So it is time to turn to cases where classical and intuitionistic readings differ. We begin with the rules for the conditional → and the biconditional ↔. It is shown that the natural deduction rules S→ for → express the corresponding intuitionistic truth condition ‖→‖ (Section 6.1). So alien anthropologists who learn that S→ describes our deductive behavior must conclude that we assign → the intuitionistic reading (whether we know it or not). In fairness to those who claim that our interpretation of → must be classical, systems that strengthen S→ with Peirce’s Law are then considered (Section 6.2). However, the result of that investigation will be to show that those stronger systems express conditions that are neither classical nor appropriate for defining connective truth conditions. Section 6.3 deals with ↔, and 6.4 provides a summary. These results will supply the basic understanding necessary for handling disjunction (Chapter 7) and negation (Chapter 8) where the situation is more complicated.