Statement Forms and Material Equivalence

Kareem Khalifa

Department of Philosophy

Middlebury College

Overview

• Why this matters

• Material equivalence

• Tautologies, contradictions, and contingencies

• Sample exercises

Why this matters

• Identifying materially equivalent statements facilitates the understanding of otherwise-torturous prose.– Ex. Being neither honest nor worthy does not entail

being either distrusted or disrespected.– Trusted and respected people can be both dishonest

and unworthy.

• Good writing avoids tautologies and contradictions.– Say nothing false, but say something.

IMPORTANT!!• Last class, we covered:

1. Testing for validity

• Today, we’ll be covering two new topics:2. Material Equivalence

3. Tautologies, Contradictions, and Contingencies

• For EACH of these topics, you must look for DIFFERENT things on the truth-table.

• Make sure you understand these differences!

Today’s First Topic:Material Equivalence (, )

• General idea: “p q” says that wherever p is true, q is true, and wherever p is false, q is false.

• In English, we represent “” as “if and only if” or “is necessary and sufficient for.”– Ex. A creature is a human if and only if it is a

member of the species Homo sapiens.– Being human is necessary and sufficient for

being a member of Homo Sapiens.

Truth table for

p q p q

T T T

T F F

F T F

F F T

Relationship between and

• Recall: – “p only if q” = p q– “p if q” = q p

• So, since “p if and only if q” = p q, then p q = [(p q) & (q p)].

• Similarly,– “p is sufficient for q” = p q– “p is necessary for q” = q p

• So, since “p is necessary and sufficient for q” = p q, then p q = [(p q) & (q p)].

Example• (~q p) (p v q)

p q ~q ~q p p v q (~q p) (p v q)

T T F

T F T

F T F

F F T

T

T

T

F

T

T

T

F

T

T

T

T

ATTENTION!!

• We have completed the first topic for today: material equivalence.

• We are moving on to today’s 2nd topic: tautologies, contradictions, and contingencies.

• These two ideas require different uses of the truth-table.

Tautologies

• Tautologies are statements such that there is no way that they can be false.

• Where there’s a row, there’s a way.

• So, tautologies are statements such that there is no row on a truth table in which they are false.

Example: p p

• “It ain’t over ‘til it’s over”

• If it’s over, then it’s over.

p p p

T T

F T

What’s wrong with tautologies?

• They are completely uninformative.– A statement’s “informativeness” is

proportional to its probability of being false.• Ex. Someone is in Twilight 302.• Ex. Heidi Grasswick is in Twilight 302.

– A tautology has no way of being false.– So the probability of a tautology being false is

zero.– So tautologies are completely uninformative.

Contradictions

• Contradictions are statements such that there is no way that they can be true.

• Where there’s a row, there’s a way.

• So, contradictions are statements such that there is no row on a truth table in which they are true.

Example

P ~P P & ~P

T F F

F T F

Contingent statements

• Contingencies are statements such that there is some way that they can be true and some way that they can be false.

• Where there’s a row, there’s a way.

• So, contingencies are statements such that there is some row on a truth table in which they are true, and some row on a truth table in which they are false.

Example

• You will receive either an A or a B.

A B A v B

T T T

T F T

F T T

F F F

Exercise B2• p [(p q) q]

p q

T T

T F

F T

F F

p q (p q) q p [(p q) q]

T

F

T

T

T

T

T

F

T

T

T

T

TAUTOLOGY

Exercise B6• (p p) (q & ~q)

p q ~q p p q & ~q (p p) (q & ~q)

T T F

T F T

F T F

F F T

T

T

T

T

F

F

F

F

F

F

F

F

CONTRADICTION

Exercise C3• [(p q) r] [(q p) r]

p q r p q q p (p q) r

(q p) r

[(p q) r]

[(q p) r]

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

F

F

T

T

T

T

T

T

F

F

T

T

T

T

T

T

F

F

F

T

T

T

T

T

F

F

F

T

T

T

T

T

T

T

T

F

CONTINGENT

T

F

T

T