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Unformatted text preview: nnective. The graph of ) function ) given above M↔ (P, the
truth table of the negation connective. Similarly, we associate to a connective
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4 Conditional
Perhaps the most important logical connective is the
conditional, also known as implication:
p > q
The statement asserts that q holds on the condition that p
holds. We call p the hypothesis or premise, and q the conclusion
or consequence. Typical usage in proofs:
“If p, then q”; “p implies q”; “q when p”; “q follows from p”
“p is sufficient for q”; “a sufficient condition for q is p”;
“a necessary condition for p is q”; “q is necessary for p” 5 Logical Equivalence
Two statements involving quantifiers and predicates are
logically equivalent if and only if they have the same
truth values no matter which predicates are
substituted into these statements and which domain is
used.
We write A ≡ B for logically equivalent A and B.
You use logical equivalences to derive more convenient
forms of statements.
Example: De Morgan’s laws.
6 De Morgan’s Laws
¬∀xP (x) ≡ ∃x¬P (x)
¬∃xP (x) ≡ ∀x¬P (x)
¬ ( p ∧ q ) ≡ ¬p ∨ ¬ q
¬ ( p ∨ q ) ≡ ¬p ∧ ¬ q 7 Valid Arguments An argument in propositional logic is a sequence of propositions
that end with a proposition called conclusion. The argument is
called valid if the conclusion follows from...
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 Fall '11
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