Description: If 𝜑 and 𝜓 are wff's, so is (𝜑 → 𝜓) or "𝜑 implies
𝜓." Part of the recursive
definition of a wff. The resulting wff
is (interpreted as) false when 𝜑 is true and 𝜓 is false; it is
true otherwise. Think of the truth table for an OR gate with input 𝜑
connected through an inverter. After we define the axioms of
propositional calculus (ax-1 6, ax-2 7, ax-3 8, and ax-mp 5), the
biconditional (df-bi 197), the constant true ⊤ (df-tru 1483), and the
constant false ⊥ (df-fal 1486), we will be able to prove these truth
table values: ((⊤ → ⊤) ↔
⊤) (truimtru 1511),
((⊤ → ⊥) ↔ ⊥) (truimfal 1512), ((⊥ → ⊤)
↔ ⊤)
(falimtru 1513), and ((⊥ →
⊥) ↔ ⊤) (falimfal 1514). These
have straightforward meanings, for example, ((⊤
→ ⊤) ↔ ⊤)
just means "the value of (⊤ →
⊤) is ⊤".
The left-hand wff is called the antecedent, and the right-hand wff is
called the consequent. In the case of (𝜑 → (𝜓 → 𝜒)), the
middle 𝜓 may be informally called either an
antecedent or part of the
consequent depending on context. Contrast with ↔ (df-bi 197),
∧ (df-an 386), and ∨ (df-or 385).
This is called "material implication" and the arrow is usually
read as
"implies." However, material implication is not identical to
the meaning
of "implies" in natural language. For example, the word
"implies" may
suggest a causal relationship in natural language. Material implication
does not require any causal relationship. Also, note that in material
implication, if the consequent is true then the wff is always true (even
if the antecedent is false). Thus, if "implies" means material
implication, it is true that "if the moon is made of green cheese
that
implies that 5=5" (because 5=5). Similarly, if the antecedent is
false,
the wff is always true. Thus, it is true that, "if the moon is made
of
green cheese that implies that 5=7" (because the moon is not actually
made
of green cheese). A contradiction implies anything (pm2.21i 116). In
short, material implication has a very specific technical definition, and
misunderstandings of it are sometimes called "paradoxes of logical
implication." |