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Syntax & Semantics of Our Language for Propositional Logic
NAME | SYNTAX | ENGLISH | E.G. | SEMANTICS | ||||||||||||||||||
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atomic proposition |
p, q, etc. | (any "simple" grammatical declarative sentence) |
Today is Monday. 2+2=4 |
tval(p)=T iff p describes the world correctly |
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molecular propositions: negation |
¬p | It's not the case that p. Not p |
Today isn't Monday. 2+2≠4 |
Use truth tables for the semantics of molecular propositions:
i.e., tval(¬p)=F iff tval(p)=T
|
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conjunction | (p ∧ q) | p and q
p but q p although q etc. |
Today is Monday and 2+2=4.
Today is Monday but there's no school anyway. |
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inclusive disjunction (Latin "vel") |
(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. |
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exclusive disjunction (Latin "aut") |
(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)
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material conditional |
(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
Belgium.
If the sum of the digits of a number is divisible by
3, |
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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)
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((p ∨ q) ∧ ¬(p ∧ q))
input | intermediate | output | |||
---|---|---|---|---|---|
1 | 2 | 3=(1∨2) | 4=(1∧2) | 5=¬4 | 6=(3∧5) |
p | q | (p ∨ q) | (p ∧ q) | ¬(p ∧ q) | ((p ∨ q) ∧ ¬(p ∧ q)) |
T | T | T | T | F | F |
T | F | T | F | T | T |
F | T | T | F | T | T |
F | F | F | F | T | F |