14. Elements of Logic (3)
Connectives – Material (Philoan1) Implication 2 (p ⊃ q), (p :T
15. 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
16. 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
22. 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
42. “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: (‘
43. ³ER[´
possibly: (◊) “diamond”
◊
and hence:
contingently: (~‘