This document discusses connectives and conditionals in logic. It introduces the material conditional operator "→" and shows its truth table. The conditional expresses "P only if Q" or "Q if P", where Q is necessary and P is sufficient. A biconditional "↔" is also discussed, which means "A if and only if B". The document explores expressing any truth-valued function using connectives like "∧" and "¬". It introduces the Peirce arrow "↓" and Sheffer stroke "↑" as alternatives to the standard connectives for representing logical operations.

1. Truth, Deduction,
Computation
Lecture 8
Conditionals and Other Connectives
Vlad Patryshev
SCU
2013

2. Examples in Plain English
1.
2.
3.
4.
5.
It rains because we prayed
causation,
correlation...
It rains after we prayed
We go to school unless it rains
If we go to school, it rains
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

3. 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?

4. Conditional Symbol →
Material conditional
P
Q
P→Q
¬PvQ
T
T
T
T
T
F
F
F
F
T
T
T
F
F
T
T

5. Necessary and Sufficient Conditions
●
●
●
●
P only if Q - meaning if P, then Q
Q if P - same thing
Q is necessary
P is sufficient

6. Conditions in Deduction
P1∧P2∧...Pi∧...∧Pn→Q is a logical truth
if and only if
P1
…
Pn
Q

7. Biconditional Symbol ↔
●
●
●
●
A
A
A
A
↔B
if and only if B
iff B
“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

8. Biconditional Symbol ↔
P
Q
P↔Q
T
T
T
T
F
F
F
T
F
F
F
T
Looks familiar? How about DNF?

9. 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

10. Completeness
Given a truth-valued function, can it be
expressed via the connectives we know?
E.g. via ∧v¬?
Easy for n=1:
P
f1
f2
f3
f4
T
T
T
F
F
F
T
F
T
F
General case? f(P1, P2, …, Pn)

11. Completeness
∧v¬ is enough.Actually,one of ∧v, and
¬
Other solutions?
Actually...

12. Peirce’s Arrow
NOR, aka ↓
A ↓ B ⇔ ¬(AvB)
¬A ⇔ A↓A
AvB ⇔ ¬¬(AvB) ⇔ ¬(A↓B) ⇔ (A↓B)↓(A↓B)
“A or B” is “neither (neither A or B) nor (neither A or B)
Other solutions?

13. Sheffer Stroke
NAND, aka ↑
A ↑ B ⇔ ¬(A∧B)
¬A ⇔ A∧A
A∧B ⇔ ¬¬(A∧B) ⇔ ¬(A↑B) ⇔ (A↑B)↑
(A↑B)

14. Exercise

15. That’s it for today