- Truth-conditions for conditionals are discussed, and the Lukasiewicz conditional '⇒' is adopted; such a conditional is false when the antecedent is true and the consequent false, and true if the truth-value status of the consequent is at least as high as that of the antecedent (counting lack of truth-value as a status intermediate between truth and falsity).
- A conditional 'If A then B' is to be read either with this conditional ('A ⇒ B') or given the "if-true" reading ('!A ⇒ B').
- The Lukasiewicz conditional satisfies modus ponens, modus tollens, hypothetical syllogism, and contraposition, but only a restricted form of conditional proof: taking 'A' as a premiss and deriving 'B' allows you to conditionalize and infer '!A ⇒ B' but not 'A ⇒ B'. (No non-bivalent truth-status-functional conditional satisfies all of modus ponens, modus tollens, and conditional proof, so one can do no better.)
- If one wishes to state Leibniz's Law2 as a conditional, the bare Lukasiewicz conditional form is incorrect, but the "if-true" version is correct: '!a = b ⇒ (φa ⇒ φb)'.
- John Broome gives a different version, which is logically equivalent to this one. Johnson criticizes these versions, but gives a conditionalized-conditional version that is also equivalent.
- A second argument of Williamson is considered which defends bivalent versions of the Tarski biconditionals; it is held that his rationales for the biconditionals are subject to interpretation, and that interpretations that do not make them beg the question rationalize only non-bivalent versions.
Footnote 1: Taken from "Parsons (Terence) - Indeterminate Identity: Analytical Table of Contents".
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