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Syntax & Semantics of Our Language for Propositional Logic
|p, q, etc.||(any "simple" grammatical
|Today is Monday.
p describes the world correctly
|¬p||It's not the case that p.
|Today isn't Monday.
|Use truth tables
for the semantics of
i.e., tval(¬p)=F iff tval(p)=T
|conjunction||(p ∧ q)||p and q
p but q
p although q
|Today is Monday and 2+2=4.
Today is Monday but there's no school anyway.
|(p ∨ q)||(Either) p or q (or both)
p and/or q
|Today is Monday or 2+2=4.
You'll pass if you study or if you already know the material.
|(p ⊕ q)||(Either) p or (else) q (but not both)||Today is a weekday or a weekend.
You'll pass 191 or else you'll fail 191.
Note: tval(p ⊕ q)=T iff tval(p)≠tval(q)
|(p → q)
p is the "antecedent"
q is the "consequent"
If p, then q
q if p
p only if q
p is a sufficient condition for q
q is a necessary condition for p
|If today is Tuesday, this must be
If the sum of the digits of a number is divisible by
|biconditional||(p ↔ q)||p if & only if q
p iff q
p is necessary & sufficient for q
q is necessary & sufficient for p
|A plane figure is a triangle iff it is a 3-sided polygon.|
Note: tval(p ↔ q)=T iff tval(p)=tval(q)
((p ∨ q) ∧ ¬(p ∧ q))
|p||q||(p ∨ q)||(p ∧ q)||¬(p ∧ q)||((p ∨ q) ∧ ¬(p ∧ q))|