Report

va ri eties o f n e c e ssit y LECTURE 10 t a uto l ogic al e q ui valen ce t a uto l ogic al c o n s eque nc e LOGICAL TRUTH AND TARSKI’S WORLD NECESSITIES Let’s call a sentence Tarski’s World-necessary (T W -necessary) just in case it is true in all the worlds that can possibly be constructed in Tarski’s World. What’s the relationship between logical truth and T W -necessity? All logical truths are T W -necessar y. Some T W-necessities are not logical truths, e.g., ‘Tet(a) ∨ Cube(a) ∨ Dodec(a).’ So the logical necessities are a strict subset of the T W necessities. EXERCISE 4.8 For each of the following sentences, say where it goes in the following Euler circle diagram 5. Larger(a,b) ∨ ¬Larger(a,b) 6. Larger(a,b) ∨ Smaller (a,b) I>CLICKER QUESTION ‘¬[¬Tet(a) ∧ ¬Cube(a) ∧ ¬Dodec(a)]’ is A. A tautology B. A T W-necessity C. A and B D. None of the above LOGICAL AND TAUTOLOGICAL EQUIVALENCE EQUIVALENCE Our informal definition of equivalence, like our informal definition of logical truth, also referred to possible worlds: P and Q are equivalent if and only if they are true in all and only the same possible worlds. With our dif ferent conceptions of possibility and necessity, we can distinguish dif ferent kinds of equivalence: tautological equivalence logical equivalence, and TW-equivalence. TAUTOLOGICAL EQUIVALENCE To give a more precise definition of tautological equivalence, we will use the notion of sentences’ joint truth tables. The joint truth table for A and B is the truth table with both sentences to the right of the reference columns. A T F A ∨ ¬A T F T T A ∧ ¬A F F F T Sentences A and B are tautologically equivalent if and only if at each row, the truth value under A’s main connective is the same as the truth value under B’s main connective. SOME IMPORTANT LOGICAL EQUIVALENCES DeMorgan’s Laws Double Negation TAUTOLOGICAL EQUIVALENCE AND LOGICAL EQUIVALENCE All tautological equivalences are logical equivalences. But remember how not all logical truths were tautologies? Similarly, not all tautological equivalences are logical equivalences. Example: ‘a = b’ is logically but not tautologically equivalent to ‘¬ FrontOf(a,b).’