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Elements of Logic (3) Connectives – Material (Philoan1) Implication 2 (p ⊃ q), (p :T
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aS V q), (~(p Λ ~q)) Known also as the “material conditional”, material implication is a binary “truth-function” according to which the argument (p :T) is true whenever (a), the consequent (q) is true; or (b), the antecedent/s (p) is false (or irrelevant). Thus, the material conditional asserts that: If (p is the case) then (q is the case) In other words, the expression “if p then not q” (p :aT
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is false – it cannot be the case that both (p is the case) and (q is not the case). But that’s the only one that’s false – the other three combinations of p and q are valid implications, and evaluate as true. Here’s the whole “truth table”: MATERIAL IMPLICATION p q (p :T
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Elements of Logic (8) Connectives – Material Implication (cont.) And things just keep getting worse! Given that all contradictions (t Λ ~t) are necessarily false, it follows that according to any rule of material implication, any contradiction necessarily (materially) implies any consequent at all! In other words, for any rule (p :T), any conditional of the form ((t Λ ~t) :q), however absurd, will always evaluate as true! For example, according to our earlier material implication rule MC1, even If ((2+2=4) Λ (2+2
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“Kinds” of Logic (1)Modal Logic An extension of classical logic that deals with the various “modes” of truth; viz.: Necessity and non-necessity (contingency) Possibility and non-possibility (impossibility) To achieve this, modal logic introduces two additional unary operators: necessarily: (
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³ER[´ possibly: (◊) “diamond” ◊ and hence: contingently: (~
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