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. Aristotle is the philosopher who taught Alexander the Great.

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.

In order to do that, let us introduce the symbol! for ‘the’. We can use! with variables much as we do with quantifiers, and notate (3) by (3′).

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 chose 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.

When wRv (w is earlier than v) and vRu (v is earlier than u), it follows that wRu (w is earlier than u). So let us define a new kind of satisfiability that corresponds to this condition on R.

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 is 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, for example, ‘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.

Since there are so many different possible systems for modal logic, it is important to determine which systems are equivalent, and which ones distinct from others. Figure 11.1 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.

Many different systems of quantified modal logic have been presented in this book, each one based on the minimal system fK. In the next few chapters, we will show the adequacy of many of these logics by showing both their soundness and completeness. When S is one of the quantified modal logics discussed, and the corresponding notion of an S-model has been defined, soundness and completeness together amount to the claim that provability-in-S and S-validity match.

1. (Soundness) If H ⊢S C then H ⊧S C.

2. (Completeness) If H ⊧S C then H ⊢S C.

This chapter will be devoted to soundness and to some theorems that will be useful for the completeness proofs to come. Some of these results are interesting in their own right since they show how the various treatments of the quantifier are interrelated. Sections 15.4–15.8 will explain how notions of validity for the substitution, intensional, and objectual interpretations are shown equivalent to corresponding brands of validity on truth value models – the simplest kind of models. This will mean that the relatively easy completeness results for truth value models can be quickly transferred to substitution, intensional, and objectual forms of validity. Readers who wish to study only truth value models may omit those sections.

Two different strategies will be presented to demonstrate completeness for truth value models. Chapter 16 covers completeness using a variation on the tree method found in Chapter 8. The modifications needed to extend the completeness result to systems with quantifiers are fairly easy to supply.

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.

The main purpose of this book is to help bridge a gap in the landscape of modal logic. A great deal is known about modal systems based on propositional logic. However, these logics do not have the expressive resources to handle the structure of most philosophical argumentation. If modal logics are to be useful to philosophy, it is crucial that they include quantifiers and identity. The problem is that quantified modal logic is not as well developed, and it is difficult for the student of philosophy who may lack mathematical training to develop mastery of what is known. Philosophical worries about whether quantification is coherent or advisable in certain modal settings partly explains this lack of attention. If one takes such objections seriously, they exert pressure on the logician to either eliminate modality altogether or eliminate the allegedly undesirable forms of quantification.

Even if one lays those philosophical worries aside, serious technical problems must still be faced. There is a rich menu of choices for formulating the semantics of quantified modal languages, and the completeness problem for some of these systems is difficult or unresolved. The philosophy of this book is that this variety is to be explored rather than shunned. We hope to demonstrate that modal logic with quantifiers can be simplified so that it is manageable, even teachable. Some of the simplifications depend on the foundations – in the way the systems for propositional modal logic are developed.

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.

Semantics for quantified modal logic can be defined by incorporating these ideas into the definition of a model for propositional modal logic.

Here we give completeness proofs for many quantified modal logics, using a variant of the method of maximally consistent sets. Although the previous chapter already established completeness for many quantified modal logics using the tree method, there are good reasons for covering the method of maximally consistent sets as well. First, this is the standard approach to obtaining completeness results, so most students of modal logic will want some understanding of the method. Second, the tree method applied only to those systems for which it was shown how to convert a tree into a proof. The method of maximally consistent sets applies to more systems, though it has limitations described below in Section 17.2.

How Quantifiers Complicate Completeness Proofs

One might expect that proving completeness of quantified modal logic could be accomplished by simply “pasting together” standard results for quantifiers with those for propositional modal logic. Unfortunately, it is not so easy. In order to appreciate the problems that arise, and how they may be overcome, let us first review the strategies used to show completeness for propositional modal logic with maximally consistent sets. Then it will be possible to outline the difficulties that arise when quantifiers are added.

The basic idea behind completeness proofs that use maximally consistent sets is to show that any argument H / C that is not provable has a counterexample.

Converting Trees to Proofs in K

Not only is the tree method useful for checking validity in modal logics, but it may also be used to help construct proofs. Trees provide a mechanical method for finding proofs that might otherwise require a lot of ingenuity. If an argument has a proof at all in a system S, the tree method can be used to provide one. The process is easiest to understand for system K, so we will explain that first, leaving the stronger systems for Sections 7.3–7.9. The fundamental idea is to show that every step in the construction of a closed tree corresponds to a derivable rule of K.

It is easiest to explain how this is done with an example, where we work out the steps of the tree and the corresponding steps of the proof in parallel. We will begin by constructing a proof of □(p→q) / □p→□q using the steps of the tree as our guidepost. The tree begins with □(p→q) and the negation of the conclusion: ∼(□p→□q). The first step in the construction of the proof is to enter □(p→q) as a hypothesis. In order to prove □p→□q, enter ∼(□p→□q) as a new hypotheses for Indirect Proof. If we can derive ⊥ in that subproof, the proof will be finished.