Truth-functional schemata are equivalent if they mutually imply each other. They will therefore agree case by case under truth-value analysis. You can say that no interpretation of literals will make the truth-values of one sentence unlike those of a sentence with which it is equivalent.
As implication is validity of the conditional, equivalence is validity of the biconditional.
Though logic is said to have the primary purpose of justifying inference, another perhaps equally important task of logic is to manipulate statements.
(1) P is equivalent to 1. ~(~p), 2. pp, 3. p ∨ p, 4. p ∨ pq, 5. p. p ∨ q, 6. pq ∨ p~q, 7. p ∨ q.p ∨ ~q.
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There are several laws of interchange. The first one is to:
The third law of interchange is that interchange does not preserves validity, implication, equivalence, and inconsistency. Unlike substitution it also preserves consistency, non-validity, nonimplication, and nonequivalence.
Because of this, just like select obvious truth-functional reductions, Quine recommends we immediately supplant any schema of any of the seven forms found in (1) with p (or q or r, etc. as the case may be).
"Pure biconditionals are valid if and only no letter occurs in it an odd number of times" (p. 66)
As implication is validity of the conditional, equivalence is validity of the biconditional.
Though logic is said to have the primary purpose of justifying inference, another perhaps equally important task of logic is to manipulate statements.
(1) P is equivalent to 1. ~(~p), 2. pp, 3. p ∨ p, 4. p ∨ pq, 5. p. p ∨ q, 6. pq ∨ p~q, 7. p ∨ q.p ∨ ~q.
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i. Equivalence is mutual implicationPutting a schemata for a literal is called substitution, but putting a schemata for another schemata is called interchange.
ii. Any schema is equivalent to itself
iii. If one schema is equivalent to a second, and the second is equivalent to a third then the first is equivalent to the third.
iv. If one schema is equivalent to a second then the second is equivalent to the first...
v. Valid schemata are equivalent to one another and to no others, and similarly for inconsistent schemata.
vi S1 implies S2 if and only if S1 is equivalent to the conjunction of S1 and S2.
vii. S1 implies S2 if and only if S2 is equivalent to the alternation of S1 and S2.
viii. S implies each of S1 and S2 if and only if S implies the conjunction of S1 and S2.
ix. S1 and S2 each imply S if and only if the alternation of S1 and S2 implies S.
-p. 62
There are several laws of interchange. The first one is to:
Think of '...p...' as any schema containing 'p', and of '...q...' as formed from '...p...' by putting 'q' for one or more occurences of 'p'; thenThe second law of interchange is:'p↔q' implies '...p....↔....q...'-p. 63-64, underline mine
If S1 annd S2 are equivalent, and S2' is formed from S1' by putting S2 for one or more occurences of S1, then S1' and S2 are equivalent.so for example if 'p→q' and '~p(p~q)' are equivalent, then we can interchange 'p→q. ∨ r' and '~(p~q) ∨ r'. Once you decipher each matter its truth is evident, but the process of teaching yourself to think in certain patterns yields fruit when coming across complex expressions.
-p. 64
The third law of interchange is that interchange does not preserves validity, implication, equivalence, and inconsistency. Unlike substitution it also preserves consistency, non-validity, nonimplication, and nonequivalence.
Because of this, just like select obvious truth-functional reductions, Quine recommends we immediately supplant any schema of any of the seven forms found in (1) with p (or q or r, etc. as the case may be).
"Pure biconditionals are valid if and only no letter occurs in it an odd number of times" (p. 66)
1 comment:
Where is the rest of the book? Did you really stop with 9. Equivalence? I was hoping to see your commentary on Quine's rendition of Venn Diagrams and Quantification Theory
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