Lecture Notes: Semantics for S5

April 8, 1998

As stated in the last lecture, Lewis's goal in creating modal logic was to give a rigorous treatment of the notion of implication, a concept which he understood in terms of the dual notions of necessity and possibility. But Lewis did not have a clear interpretation of these fundamental modalities. He lacked a semantics, i.e., a systematic way to determine the meanings of the formulas in the formal language.

Lewis provided a partial analysis of the notions of impossibility and necessity.

No proposition is "impossible" in the sense of p except as imply their own negation; and no proposition is "necessary" in the sense of p unless its negation is self-contradictory. (SSL, p. 335)

Taking "imply" to indicate strict implication, it seems that Lewis is asserting that if it is true that

p then p

p. This in fact holds in his systems but it is hardly enlightening, since strict implication is itself defined in terms of possibility. Thus p

p is equivalent to

(p &

p), which in turn is equivalent to

(p & p) and finally to

p, which is where we began. (In fact, although in Bonevac's text both possibility and strict implication are defined in terms of necessity, one could as well begin with strict implication as undefined and define possibility and necessity in terms of it.)

It was not until the 1940s that Rudolf Carnap hit upon a semantics for S5, which was as well the first semantics for a modal system. S5 was one of the five systems of modal logic introduced by Lewis in his 1932 book (co-authored by C. H. Langford) Symbolic Logic. It is the strongest of the systems, in the sense that it contains all the theorems of all the other systems (S1-S4) as well as some which they do not contain. Here is how Carnap described what he did.

The logic of modalities had been constructed for many years in the framework of symbolic logic, beginning mainly with the work of C. I. Lewis (1918). However, so far no clear interpretation of the modal terms had been given. After defining semantical concepts like logical truth and related ones, I proposed to interpret the mdoalities as those properties of propositions which correspond to certain semantical properties of the sentences expressing the proposition. For example, a proposition is logically necessary if and only if a sentence expressing it is logically true. ("Rudolf Carnap, Intellectual Autobiography," p. 62, in Paul Schilpp, ed., The Philosophy of Rudolf Carnap (LaSalle, Illinois, 1963).

A sentence (or formula) of a symbolic logic is logically true if and only if it is true on all interpretations. For sentential logic, this amounts to claiming that it has the value true on every line of its truth-table, no matter what the value of its components. Thus, on Carnap's interpretation, every logically true formula of sentential logic expresses a proposition which is necessary. This is part of what Bonevac asserts in his Thesis 1, on page 231.

But what are we to say about the logical truth of modal formulas, i.e., formulas containing modal operators? We know that p p is a logical truth of sentential logic, which will make (p p) true on Carnap's interpretation. But suppose that we want to know whether it is logically true and therefore necessarily true. We do not have truth-tables for modal formulas, so we cannot apply the definition of logical truth of sentential logic to modal formulas. Something more is needed. For Carnap, it was an adaptation of Leibniz's notion of a possible world.

In my search for an explication [of logical truth] I was guided, on the one hand, by Leibniz' view that a necessary truth is one which holds in all possible worlds, and on the other hand, by Wittgenstein's view that a logical truth or tautology is characterized by holding for all possible distributions of truth values. Therefore the various forms of my definition of logical truth are based either on the definition of logically possible states or on the definition of sentences describing those states (state descriptions). ("Autobiography, p. 63)

The we can look at a possible world as a possible distribution of truth-values, i.e., as a line on a truth table. A possible world, then, matches sentence letters with truth values. A simple interpretation will illustrate the transition from truth-tables to possible worlds. Let one possible world w₁ match the single atomic formula p with the value true, and another one w₂ match p with the value False. We determine the value of a non-modal formula at a world by the values of its components at that same world. Thus the value of p

p at w₁ is true given that p is true at the first world, and it is also true at w₂ given that the value of p is false at the second world.

	p	p p
w₁	True	True
w₂	False	True

Now we can describe how to assign a value to (p p). This formula is true at a world just in case p p is true at all possible worlds. The two worlds in our small interpretation both make the formula p p true, so the modal formula (p p) is true at both worlds, just as p p is true at both worlds.

	p	p p	(p p)
w₁	True	True	True
w₂	False	True	True

The subtle difference between the evaluation of the modal and the non-modal formula in the example is this: the non-modal formula is evaluated on the basis of the value of its components in a single world, while the modal formula is evaluated on the basis of the value of its one component at two different worlds. In non-modal sentential logic, one never uses information from another line on the truth-table to determine the value of a formula on a given line.

This example shows why Thesis 1 ("Every logical truth is necessary") holds for the S5 semantics. Recall that a logical truth is one that is true on every interpretation, and hence true at every possible world, since an interpretation is equated to a possible world. A non-modal logical truth is true on every line of a truth-table, and hence at any possible world that can be constructed. A modal logical truth of S5 such as (p p) is also true at every possible world, since by virtue of being a logical truth it is true on every interpretation. Hence the modal formula is necessary as well; in the present example, (p p) is true at every possible world, and so on.

It is easy to confuse this last result with what Bonevac calls Thesis 4 ("What is necessary is necessarily necessary"), p. 232. An instance of the thesis is that if it is true that p then it is true that p. The sentence letter p is not a logical truth, nor is it in any way a necessary truth, but we can see what happens if it is necessarily true. To see how this thesis holds, let us suppose that p is true at a world w₁. Then at every world, p is true. As a consequence, p is true at all possible worlds, in which case p is true at all possible worlds, including w₁.

Perhaps now the difference between Thesis 1 and Thesis 4 can be more clearly understood. A logical truth is true at all possible worlds by virtue of the semantics itself, so it is always necessary. But the truth of p at all possible worlds in the demonstration just given is the result of the assumption that p is true at a world. On the semantics for many other modal systems other than S5, this assumption will not require that p be true at all possible worlds, but only at those which are "accessible" from the world in which p is true. We will consider these more restricted semantical systems later in the course.

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