Consistency
A truth-functional schema is "consistent" when at least one of its possible outcomes is true, otherwise "inconsistent".Validity
A truth-functional schema is "valid" if each of its possible outcomes is true, otherwise "invalid".Patently valid schemata may be reduced to ⊤, and patently invalid schemata to ⊥. "Patent" in this case is arbitrary, but classroom rules may be applied for consistency (get it?).
Quine names these two categories of patently inconsistent schemata: (a) conjunctions in which some part appears both plain and negated as component of the conjunction, and (b) biconditionals in which some part appears both plain and negated on either side of the biconditional.
Also note that p→~p is actually counted as consistent.
Quine names these two categories of patently valid schemata: (a) alternations in which some part appears both plain and negated as component of the alternation, and (b) conditionals or biconditionals whose two sides are alike.
Substitution of schemata for letters preserves validity!
Substitution of schemata for letters preserves inconsistency!
Substitution of schemata for letters DOES NOT preserve consistency!
-Jonathan Charles Wright and W. V. Quine
2 comments:
How can p→~p be counted as consistent? Is it consistent but invalid?
Good question.
p→~p
is considered consistent, because at least one of its possible outcomes is true.
Specifically, for a false interpretation of p, the schema is interpreted thusly:
⊥→~⊥
While such interpretation would resolve to
⊥→⊤
as the negation of a false proposition is true, such fact is irrelevant.
The relevant fact is that as strange as it is, all truth functions with false antecedents are treated as true (cf. §3. The Conditional).
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