Unit II LOGICSection 3.7: Argument and Truth Tables

What is an argument???

An argument consists of TWO parts!1. Premises given statements2. Conclusion decision reached

True or not…that is the QUESTION!• Valid Argument : If premises are true,

the conclusion is true

• Invalid Argument (fallacy): argument that isn’t true

Testing the Validity of a Statement• Use TRUTH TABLES!!!

• An argument is valid if the statement is a tautology (basically the last column of the truth table is true)

Types of Valid ArgumentsReasoning Symbolic Representation

Direct 𝑝⟶𝑞 𝑝 ∴ 𝑞

Contrapositive 𝑝⟶𝑞 ~𝑞 ∴ ~𝑝

Disjunctive 𝑝∨𝑞 𝑝∨𝑞 ~𝑝 ∴ 𝑞 ~𝑞 ∴ 𝑝

Transitive

𝑝⟶𝑞 𝑞 ⟶𝑟 ∴ 𝑝⟶𝑟 ∴ ~𝑟⟶~𝑝

Types of Invalid ArgumentsReasoning Symbolic Representation

Fallacy of the Converse 𝑝⟶𝑞 𝑞 ∴ 𝑝

Fallacy of the Inverse 𝑝⟶𝑞 ~𝑝 ∴ ~𝑞

Misuse of Disjunctive Reasoning

𝑝∨𝑞 𝑝∨𝑞 𝑝 ∴ ~𝑞 𝑞 ∴ ~𝑝 Misuse of Transitive Reasoning

𝑝⟶𝑞 𝑞 ⟶𝑟 ∴ 𝑟⟶𝑝 ∴ ~𝑝⟶~𝑟

Example :

𝒑 𝒒 ~𝒑 ~𝒒 𝒑⟶𝒒 ሺ𝒑⟶𝒒ሻ∧~𝒑 ሾሺ𝒑⟶𝒒ሻ∧~𝒑ሿ⟶~𝒒

1.

𝑝⟶𝑞 ~𝑝 ∴ ~𝑞 ሾሺ𝒑⟶𝒒ሻ∧~𝒑ሿ⟶~𝒒

SOLUTION Example :

𝒑 𝒒 ~𝒑 ~𝒒 𝒑⟶𝒒 ሺ𝒑⟶𝒒ሻ∧~𝒑 ሾሺ𝒑⟶𝒒ሻ∧~𝒑ሿ⟶~𝒒

T T

T F

F T

T F

1. Fallacy of inverse invalid 𝑝⟶𝑞 ~𝑝 ∴ ~𝑞

Solution using TRUTH TABLE:

SOLUTION Example :

𝒑 𝒒 ~𝒑 ~𝒒 𝒑⟶𝒒 ሺ𝒑⟶𝒒ሻ∧~𝒑 ሾሺ𝒑⟶𝒒ሻ∧~𝒑ሿ⟶~𝒒

T T F F

T F F T

F T T F

F F T T

1. Fallacy of inverse invalid 𝑝⟶𝑞 ~𝑝 ∴ ~𝑞

Solution using TRUTH TABLE:

SOLUTION Example :

𝒑 𝒒 ~𝒑 ~𝒒 𝒑⟶𝒒 ሺ𝒑⟶𝒒ሻ∧~𝒑 ሾሺ𝒑⟶𝒒ሻ∧~𝒑ሿ⟶~𝒒

T T F F T

T F F T F

F T T F T

F F T T T

1. Fallacy of inverse invalid 𝑝⟶𝑞 ~𝑝 ∴ ~𝑞

Solution using TRUTH TABLE:

SOLUTION Example :

𝒑 𝒒 ~𝒑 ~𝒒 𝒑⟶𝒒 ሺ𝒑⟶𝒒ሻ∧~𝒑 ሾሺ𝒑⟶𝒒ሻ∧~𝒑ሿ⟶~𝒒

T T F F T F

T F F T F F

F T T F T T

F F T T T T

1. Fallacy of inverse invalid 𝑝⟶𝑞 ~𝑝 ∴ ~𝑞

Solution using TRUTH TABLE:

SOLUTION Example :

𝒑 𝒒 ~𝒑 ~𝒒 𝒑⟶𝒒 ሺ𝒑⟶𝒒ሻ∧~𝒑 ሾሺ𝒑⟶𝒒ሻ∧~𝒑ሿ⟶~𝒒

T T F F T F T

T F F T F F F

F T T F T T T

F F T T T T T

1. Fallacy of inverse invalid 𝑝⟶𝑞 ~𝑝 ∴ ~𝑞

Solution using TRUTH TABLE:

Since the second row of the last column is FALSE, the statement is a fallacy, or INVALID