truth, deduction, computation lecture 8
DESCRIPTION
My logic lectures at SCU Conditionals and Other ConnectivesTRANSCRIPT
Truth, Deduction, ComputationLecture 8Conditionals and Other Connectives
Vlad PatryshevSCU2013
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...
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?
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
Necessary and Sufficient Conditions
● P only if Q - meaning if P, then Q● Q if P - same thing● Q is necessary● P is sufficient
Conditions in DeductionP1∧P
2∧...P
i∧...∧P
n→Q is a logical truth
if and only if P
1
…
Pn
Q
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
Biconditional Symbol ↔
P Q P↔Q
T T T
T F F
F T F
F F T
Looks familiar? How about DNF?
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
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
Completeness
∧v¬ is enough.Actually,one of ∧v, and ¬
Other solutions?
Actually...
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?
Sheffer Stroke
NAND, aka ↑
A ↑ B ⇔ ¬(A∧B)
¬A ⇔ A∧A
A∧B ⇔ ¬¬(A∧B) ⇔ ¬(A↑B) ⇔ (A↑B)↑(A↑B)
Exercise
That’s it for today