Truth, deduction, computation lecture 8
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Transcript of Truth, deduction, computation lecture 8
![Page 1: Truth, deduction, computation lecture 8](https://reader036.fdocuments.net/reader036/viewer/2022081907/545ae245af7959755d8b5df9/html5/thumbnails/1.jpg)
Truth, Deduction, ComputationLecture 8Conditionals and Other Connectives
Vlad PatryshevSCU2013
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Examples in Plain English
1. It rains because we prayed2. It rains after we prayed3. We go to school unless it rains4. If we go to school, it rains5. We don’t go to school only if it rains
Any logic in these sentences?How about truth tables?
Some of the sentences are not truth-functional
causation,correlation...
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Conditional Symbol →
Material conditional
P Q P→Q
T T T
T F F
F T T
F F T
Looks familiar? How about DNF?
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Conditional Symbol →
Material conditional
P Q P→Q
T T T
T F F
F T T
F F T
¬PvQ
T
F
T
T
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Necessary and Sufficient Conditions
● P only if Q - meaning if P, then Q● Q if P - same thing● Q is necessary● P is sufficient
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Conditions in DeductionP1∧P
2∧...P
i∧...∧P
n→Q is a logical truth
if and only if P
1
…
Pn
Q
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Biconditional Symbol ↔
● A ↔ B● A if and only if B● A iff B● A “just in case” B (in math only)
○ Math: n is even just in case n2 is even○ Real life: We took umbrellas just in case it
rains
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Biconditional Symbol ↔
P Q P↔Q
T T T
T F F
F T F
F F T
Looks familiar? How about DNF?
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Biconditional Symbol ↔
P Q P↔Q
T T T
T F F
F T F
F F T
(P∧Q)v(¬P∧¬Q)
T
F
F
T
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Completeness
Given a truth-valued function, can it be expressed via the connectives we know?E.g. via ∧v¬?
Easy for n=1:
General case? f(P1, P
2, …, P
n)
P f1 f2 f3 f4
T T T F F
F T F T F
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Completeness
∧v¬ is enough.Actually,one of ∧v, and ¬
Other solutions?
Actually...
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Peirce’s Arrow
NOR, aka ↓
A ↓ B ⇔ ¬(AvB)
¬A ⇔ A↓AAvB ⇔ ¬¬(AvB) ⇔ ¬(A↓B) ⇔ (A↓B)↓(A↓B)
“A or B” is “neither (neither A or B) nor (neither A or B)
Other solutions?
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Sheffer Stroke
NAND, aka ↑
A ↑ B ⇔ ¬(A∧B)
¬A ⇔ A∧A
A∧B ⇔ ¬¬(A∧B) ⇔ ¬(A↑B) ⇔ (A↑B)↑(A↑B)
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Exercise
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That’s it for today