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