Logic for Computer Science: Foundations of Automatic Theorem by Jean H. Gallier

By Jean H. Gallier

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Additional info for Logic for Computer Science: Foundations of Automatic Theorem Proving (REVISED ON-LINE VERSION (2003))

Example text

Justify your answer. 3. 1. 4. The function sub : P ROP → 2P ROP which assigns to any proposition A the set sub(A) of all its subpropositions is defined recursively as follows: sub(⊥) = {⊥}, sub(Pi ) = {Pi }, for a propositional symbol Pi , sub(¬A) = sub(A) ∪ {¬A}, sub((A ∨ B)) = sub(A) ∪ sub(B) ∪ {(A ∨ B)}, sub((A ∧ B)) = sub(A) ∪ sub(B) ∪ {(A ∧ B)}, sub((A ⊃ B)) = sub(A) ∪ sub(B) ∪ {(A ⊃ B)}, sub((A ≡ B)) = sub(A) ∪ sub(B) ∪ {(A ≡ B)}. Prove that if a proposition A has n connectives, then sub(A) contains at most 2n + 1 propositions.

In case (1), the subsequence consisting of these distinct elements forms a decreasing sequence in A, contradicting the fact that ≤ is well founded. In case (2), there is some k such that for all i ≥ k, xi = xi+1 . By definition of <<, the sequence (yi )i≥k is a decreasing sequence in A, contradicting the fact that ≤ is well founded. Hence, << is well founded on A × A. As an illustration of the principle of complete induction, consider the following example in which it is shown that a function defined recursively is a total function.

Hence, tautologies play an important role. For example, let “John is a teacher,” “John is rich,” and “John is a rock singer” be three atomic propositions. Let us abbreviate them as A,B,C. ” This amounts to showing that the (formal) proposition (∗) (A and not(A and B) and (C implies B)) implies (not C) is a tautology. Informally, this can be shown by contradiction. The statement (∗) is false if the premise (A and not(A and B) and (C implies B)) is true and the conclusion (not C) is false. This implies that C is true.

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