Is the following a valid argument: B v C, C v D, ∴A→C ? Why or why not? (see description for full question)

Question

Imagine that you have a set of sentences in QL named A, B, C, and D, and assume that A and B are tautologies, C is contingent, and D is a contradiction. Is the following a valid argument: B v C, C v D, ∴A→C ? Why or why not?

Answer 1

The validity of this argument (under common propositional logic rules) can be shown a variety of ways. Let us simply reason through it.

1. First, recognize that a tautology is always true. Therefore, A and B are true.
2. A contingent claim might be true or false (C).
3. A contradiction is always false (D).

A valid argument is one in which, if the premises are true, the conclusion must be true (i.e., cannot be false). Your argument's premises are:

1) A

2) B

3) B v C

4) C v D

We already know that 1 and 2 are true. If 2 is true, then 3 is also true, no matter what the value for C is, but this leads us to 4. For premise 4 to be true, C must be true. This is because D is a contradiction, and so is always false.

We then evaluate the conclusion: A→C. From our premises, we now that A is true and C is true, so this is: T→T

There is only one circumstance in which the conditional operator evaluates to false under standard methods: if the antecedent is true and the consequent is false. Consequently, given the stated circumstances, if the premises are true, then the conclusion must also be true. The argument is valid.

This could also be demonstrated using a truth table, tree decomposition, natural deduction (an indirect method would help here), etc., but the problem is rather trivial so such tools should not be necessary.