Question

A set of premises and a conclusion are given. Use the valid argument forms discussed in this chapter to deduce the conclusion from the premises, giving a reason for each step. Assume all variables are statement variables.

(a) p v q
(b) q → r
(c) p∧s → t
(d) ~r
(e) ~q→u∧s
(f) ∆t
Make selections from the ones below to show the first steps of a proof and the reason for the conclusion.
Write the intermediate steps, including reasons, and submit as a free response. (Submit a file with a maximum size of 1 MB)

Answer 1

The valid argument forms and steps above, we deduce the conclusion Δt from the given premises.

Let's use the given premises to deduce the conclusion ∆t. We'll use valid argument forms to justify each step:

p∨q (Premise)

Reason: Given premise.

q→r (Premise)

Reason: Given premise.

∼r (Premise)

Reason: Given premise.

(∼q→u)∧(∼q→s) (Premise)

Reason: Given premise, and applying the equivalence

P→Q≡∼P∨Q.

∼q→u (Simplification from 4)

Reason: Simplification.

∼q→s (Simplification from 4)

Reason: Simplification.

∼q (Modus Tollens: 2, 3)

Reason: Modus Tollens on premises 2 and 3.

u (Modus Ponens: 5, 7)

Reason: Modus Ponens on premises 5 and 7.

s (Modus Ponens: 6, 7)

Reason: Modus Ponens on premises 6 and 7.

p (Disjunctive Syllogism: 1, 7)

Reason: Disjunctive Syllogism on premises 1 and 7.

p∧s (Conjunction: 10, 9)

Reason: Conjunction on premises 10 and 9.

t (Modus Ponens: 11, 3)

Reason: Modus Ponens on premises 11 and 3.

Δt (Conjunction: 12, 8)

Reason: Conjunction on premises 12 and 8.

Therefore, using the valid argument forms and steps above, we deduce the conclusion Δt from the given premises.