Formal Logic - Lesson 4 - Tautology, Contradiction and Contingency

Lecture / Presentation in Discrete Structure I

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FORMAL LOGIC
Discrete Structures I
FOR-IAN V. SANDOVAL

2. z
Lesson 4
TAUTOLOGY, CONTRADICTION
AND CONTINGENCY
Source: Google Images

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LEARNING OBJECTIVES
❑Distinguish classes of compound statement

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TAUTOLOGY
❑acompoundstatementthatistrueforallpossible
combinationofthetruthvaluesofthepropositional
variablesalsocalledlogicallytrue.
❑i.e.(~p^q)→q
p q ~p ~p ^ q (~p ^ q ) →q
T T F F T
T F F F T
F T T T T
F F T F T

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CONTRADICTION
❑acompoundstatementthatisfalseforallpossible
combinationofthetruthvaluesofthepropositional
variablesalsocalledlogicallyfalseorabsurdity.
❑i.e.(~pvq)⊕(p→q)
p q ~p~ p v q p →q (~ p v q ) ⊕(p →q )
TTF T T F
TFF F F F
FTT T T F
FFT T T F

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CONTINGENCY
❑acompoundstatementthatcanbeeithertrueoffalse,
dependingonthetruthvaluesofthepropositionalvariables
areneitheratautologynoracontradiction..
❑i.e.(p→q)^(p→~q)
p q p →q ~q p →~q (p →q ) ^ (p →~q )
TT T F F F
TF F T T F
FT T F T T
FF T T T T

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
❑EnrichmentExercise
Constructthetruthtableofthefollowingand
determinewhetherthecompoundstatementisatautology,
contradictionandcontingency.
1.p⊕(~p↔q)
2.[r^(p→q)]→q
3.p→(q→r)

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
1.p⊕(~p↔q)
p q
T T
T F
F T
F F

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
1.p⊕(~p↔q)
p q ~p
T T F
T F F
F T T
F F T

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
1.p⊕(~p↔q)
p q ~p (~p ↔ q)
T T F F
T F F T
F T T T
F F T F

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
1.p⊕(~p↔q)
p q ~p (~p ↔ q)p ⊕(~p ↔q)
T T F F T
T F F T F
F T T T T
F F T F F
Therefore,p⊕(~p↔q)iscontingency.

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
2.[r^(p→q)]→q
p q r
TTT
TTF
TFT
TFF
FTT
FTF
FFT
FFF

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
2.[r^(p→q)]→q
p q rp →q
TTT T
TTF T
TFT F
TFF F
FTT T
FTF T
FFT T
FFF T

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
2.[r^(p→q)]→q
p q rp →qr ^ (p →q)
TTT T T
TTF T F
TFT F F
TFF F F
FTT T T
FTF T F
FFT T T
FFF T F

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
2.[r^(p→q)]→q
p q rp →qr ^ (p →q) [r ^ (p →q)] →q
TTT T T T
TTF T F T
TFT F F T
TFF F F T
FTT T T T
FTF T F T
FFT T T F
FFF T F T
Therefore,[r^(p→q)]→qiscontingency.

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
3.p→(q→r)
p q r
TTT
TTF
TFT
TFF
FTT
FTF
FFT
FFF

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
3.p→(q→r)
p q rq → r
TTT T
TTF F
TFT T
TFF T
FTT T
FTF F
FFT T
FFF T

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
3.p→(q→r)
p q rq → r p → (q → r )
TTT T T
TTF F F
TFT T T
TFF T T
FTT T T
FTF F T
FFT T T
FFF T T
Therefore,[p→(q→r)iscontingency.

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
4.[p^(p→q)]→q
p q r
TTT
TTF
TFT
TFF
FTT
FTF
FFT
FFF

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
4.[p^(p→q)]→q
p q rp →q
TTT T
TTF T
TFT F
TFF F
FTT T
FTF T
FFT T
FFF T

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
4.[p^(p→q)]→q
p q rp →qp ^ (p →q)
TTT T T
TTF T T
TFT F F
TFF F F
FTT T F
FTF T F
FFT T F
FFF T f

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
4.[p^(p→q)]→q
p q rp →qp ^ (p →q)[p^ (p →q)] →q
TTT T T T
TTF T T T
TFT F F T
TFF F F T
FTT T F T
FTF T F T
FFT T F T
FFF T f T
Therefore,[p^(p→q)]→qistautology.

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
5.p→(p↔r)
p q r
TTT
TTF
TFT
TFF
FTT
FTF
FFT
FFF

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
5.p→(p↔r)
p q rp ↔ r
TTT T
TTF F
TFT T
TFF F
FTT F
FTF T
FFT F
FFF T

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TAUTOLOGY, CONTRADICTION & CONTINGENCY
5.p→(p↔r)
p q rp ↔ r p → (p ↔ r )
TTT T T
TTF F F
TFT T T
TFF F F
FTT F T
FTF T T
FFT F T
FFF T T
Therefore,p→(p↔r)iscontingency.

26. z
•Levin, O. (2019). Discrete Mathematics: An Open Introduction 3
rd
Edition. Colorado: School of Mathematics Science
University of Colorado.
•Aslam, A. (2016). Proposition in Discrete Mathematics retrieved from https://www.slideshare.net/AdilAslam4/chapter-1-
propositions-in-discrete-mathematics
•Operator Precedence retrieved from http://intrologic.stanford.edu/glossary/operator_precedence.html
REFERENCES