MathMeanMathLogic042100

In free variable logic it is clear what we mean by a

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Unformatted text preview: c. In free variable logic, it is clear what we mean by a follows from S algebraically. So what does “a follows from S formally” mean here? For this general context, the most a.a.g. friendly way to go is to avoid derivations and use another algebraic notion. 5 We say that a follows from T locally algebraically iff for every algebra D and assignment t to the variables, if all elements of T are true under t then a is true under t. The so called substitution instances of a free variable statement are obtained by replacing identical variables with identical terms. THEOREM 3. (Herbrand’s theorem). Let S be a set of free variable statements and a be a free variable statement. Then a follows from S algebraically iff a follows from a finite set of substitution instances of elements of S locally algebraically. We can think of this finite set of substitution instances as a “Herbrand proof” of a from S, and count the number of occurrences of function symbols as a measure of its size. COROLLARY 4. (Tarski compactness). Let S be a set of free variable statements and a be a free variable statement. Then a follows from S algebraically iff a follows from a finite subset of S algebraically. Let S be a set of free vari-able statements. There is an important process of expanding S through the introduction of new symbols that goes back to Hilbert with his Œ-calculus. It amounts to a relatively a.a.g. friendly treatment of quantifier logic. Let j(x1,...,xn+1) be any free variable statement that uses only function symbols appearing in S. We then introduce a new function symbol F which is n-ary, and add the free variable statement j(x1,...,xn+1) Æ j(x1,...,xn,F(x1,...,xn)) to S. We can repeat this process indefinitely, eventually taking care of all free variable statements in this way involving any of the function symbols that eventually get introduced. Any two ways of doing this are essentially equivalent. We write the result as S*. THEOREM 5. Let S be a set of free variable statements. Any algebra in which S holds universally can be made into an 6 algebra in which S* holds universally without changing the domain and functions of the original algebra. COROLLARY 6. Let S be a set of free variable statements and a be a free variable statement using only function symb...
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