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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
