forbes_expressive_completeness - 74 Chapter 3: Semantics...

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74 Chapter 3: Semantics for Sentential Logic 7 Expressive completeness At the end of §1 in Chapter 2 we claimed that our five sentential connectives ~ ’, ‘ ’, ‘ & ’, ‘ ’ and ‘ ’ are all we need in sentential logic, since other sentential connectives are either definable in terms of these five or else beyond the scope of sentential logic. When we say that a connective is beyond the scope of clas- sical sentential logic, what we mean is that it is non-truth-functional; in other words, there is no truth-function that it expresses (see §1 of this chapter for a discussion of expressing a truth-function). In the next section we will consider various connectives of this sort. Meanwhile, we will concern ourselves with the definability of other truth-functional connectives. An example of a truth-functional connective which is definable in terms of our five is ‘neither…nor…’, since for any English sentences p and q , [ neither p nor q \ is correctly paraphrased as [ not p and not q \ (see (11) on page 18). But this is just one example. How can we be confident that every truth-functional connective can be defined in terms of ‘ ~ ’, ‘ ’, ‘ & ’, ‘ ’ and ‘ ’? Our confidence is based in the fact that our collection of connectives has a property called expressive completeness , which we now explain. At the end of §1 of this chapter, we listed the function-tables for the one- place, or unary , function expressed by ‘ ~ ’, and the four two-place, or binary , functions expressed by the other connectives. However, there are many more unary and binary truth-functions than are expressed by the five connectives individually. For example, there are three other unary truth-functions: > >> >⊥ ⊥⊥ (a) (b) (c) To show that all unary truth-functional connectives are definable in terms of our five basic connectives, we establish the stronger result that all unary truth- functions are definable, whether or not they are expressed by some English connective. (While (b) is expressed by ‘it is true that’, neither (a) nor (c) has an uncontrived rendering.) Our question is therefore whether we can express all of (a), (b) and (c) in terms of our five chosen connectives. And in this case it is easy to see that (a) is captured by ‘… ~ …’, (b) by ‘ ~~ …’, and (c) by ‘… & ~ …’, where in (a) and (c) the same formula fills both ellipses. What about the other binary truth-functions? We have connectives for four, and we know how to define a fifth, the truth-function >> >⊥ ⊥> ⊥⊥ > which is expressed by ‘neither…nor…’ (to repeat, [ neither p nor q \ is true just in case both p and q are false, so we express it with [ ~ p & ~ q \ ). But there are
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§7: Expressive completeness 75 many more binary truth-functions, and again we are concerned to define all of them, not merely those which correspond to some idiomatic phrase like ‘nei- ther…nor…’. First, how many other binary functions are there? There are as
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This note was uploaded on 08/01/2008 for the course PHIL 140A taught by Professor Fitelson during the Spring '07 term at University of California, Berkeley.

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forbes_expressive_completeness - 74 Chapter 3: Semantics...

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