Chapter 1: Formal Logic

Section 1.2 Propositional Logic

Verbal Arguments

An argument in English (an attorney's trial summary, an advertisement, or a political speech) that consists of simple statements can be tested for validity by a two-step process:

1. Symbolize the argument using propositional wffs.
2. Prove that the argument is valid by constructing a proof sequence for it using the derivation rules for propositional logic.

The rule of hypothetical syllogism (hs)

From \(P → Q\) and \(Q → R\), one can derive \(P → R\).

This rule makes the claim that:

\( (P → Q) ∧ (Q → R) → (P → R) \) is a valid statement.

Example 1

Consider the argument, "If interest rates drop, the housing market will improve. Either the federal discount rate will drop or the housing market will not improve. Interest rates will drop. Therefore the federal discount rate will drop." Using

I: Interest rates drop.
H: The housing market will improve.
F: The federal discount rate will drop.

the argument is

\( (I → H) ∧ (F ∨ H') ∧ I → F \)
A proof sequence to establish validity:
1. \(I → H\) (hyp)
2. \(F ∨ H'\) (hyp)
3. \(I\) (hyp)

4. \( H' ∨ F \) (2 comm)
5. \( H → F\) (4 imp)
6. \( I → F\) (1,5,hs)
7. \( F\) (3,6, mp)

Example 2

Use propositional logic to prove that the following argument is valid. Use statement letters S, R, and B.

"If security is a problem, then regulation will be increased. If security is not a problem, then business on the Web will grow. Therefore if regulation is not increased, then business on the Web will grow."

S: Security is a problem
R: Regulation will be increased.
B: Business on the Web will grow.

the argument is:
\( (S → R) ∧ (S' → B) → (R' → B)\)
1. \(S → R\) (hyp)
2. \(S' → B\) (hyp)
3. \(R'\) (hyp)

4. \( S'\) (1,3 mt)
5. \(B\) (2,4 mp)

Example 3

Use propositional logic to prove that the following argument is valid. Use statement letters L and D.

"My client is left-handed, but if the diary is not missing, then my client is not left-handed; therefore, the diary is missing."

L: My client is left-handed.
D: The diary is missing.

the argument is:
\( L ∧ (D' → L') → D \)
1. \( L \) (hyp)
2. \(D' → L' \) (hyp)

3. \( (D')' ∨ L' \) (2, imp)
4. \( D ∨ L' \) (3, dn)
5. \( L' ∨ D \) (4, comm)
6. \( L → D \) (5, imp)
7. \( D \) (1,6, mp)

Practice

1. Use propositional logic to prove that the following argument is valid. Use statement letters E, Q and B.

"If the program is efficient, it executes quickly. Either the program is efficient, or it has a bug. However, the program does not execute quickly. Therefore, it has a bug."

2. Use propositional logic to prove that the following argument is valid. Use statement letters C, W, R, and S.

"The crop is good, but there is not enough water. If there is a lot of rain or not a lot of sun, then there is enough water. Therefore, the crop is good and there is a lot of sun."

Reference

saylor academy