Implication is basically the same as ≤, in terms of the truth table.
Streghthening the antecedent: α → γ╞ (α^β) → γ
α≤γ therefore, α^β ≤ γ
(α^β is any number that is smaller than alpha)
Weakening the consequent : α → γ ╞ α → (βvγ)
α≤γ therefore, α ≤ (βvγ)
(Any number that is greater than alpha)
affirming the consequent
α → β, β not entail α
denying the antecedent
α → β, -α not entail -β
-P ╞ P→ Q
eg. Lorenz does not participate in the bauty competiition; therefore, if Lorenz participates in thebeauty competition, he will win it.
Q ╞ P→ Q
eg. Lorenz is smart enough to be a logician, therefore, if Lorenz has an iq of 70, he is smart enough to be a logician.
╞ (P→Q) v (Q→P)
0 is less than or equal to 1, and 1 is less than or equal to 0
Tautology.
P→Q
Tautologous
Pierce’s Law
((P→Q)→P)→P
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Equivalent to the law of excluded middle.
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A logical system in which this fails to be a tautology is also a system wherein (Pv-P) is also not a tautology.
Counterfactional conditions: something that is not truth-functional according to the implication.
Implication can be read as ”if then”, or ”only if” (as in ”we go to the movie only if it rains: R → M”)
Only switches the direction of the arrow however.
Q if: P P→ Q
Q only if: P Q→P
P iff Q: (Q→P)^(P→Q), ≈ P ↔ Qs