Consider the question:
The sentence "If Ron went to the store, Ron would be home by now."
does not have a truth value. How do we further determine the validity of
The full argument is the following:
If Ron went to the store, Ron would be home by now.
Ron isn't home yet.
Ron didn't go to the store.
The following quotes will come from forall x: Calgary Remix.
First, what is a "sentence"? (page 4)
In logic, we are only interested in sentences that can figure as a
premise or conclusion of an argument. So we will say that a sentence
is something that can be true or false.
Next the logic that we are using is called "truth-functional logic" (TFL). That is, atomic sentences are assigned truth values. Non-atomic sentences are composed of atomic sentences using sentential connectives. They receive their truth values from the atomic sentences given a characteristic truth table for their sentential connectives: (page 54)
Any non-atomic sentence of TFL is composed of atomic sentences with
sentential connectives. The truth value of the compound sentence
depends only on the truth value of the atomic sentences that comprise
In the given argument there are two premises and one conclusion. All three of these are sentences that have a truth value. The second premise and the conclusion are atomic sentences and so they can be assigned a truth value. The first premise, however, is a conditional, a non-atomic sentence composed of two subsentences, an antecedent and a consequent. :
- Ron went to the store. (The antecedent of the conditional)
- Ron would be home by now. (The consequent of the conditional)
Here is the rule for the conditional connective: (page 55-6)
...we are going to stipulate the following: A→B is false if and only
if A is true and B is false. We can summarize this with a
characteristic truth table for the conditional.
A B A→B
T T T
T F F
F T T
F F T
That is rule showing how the first premise, a conditional, obtains its truth value, the third column in the characteristic truth table, based on the antecedent subsentence (column A) and the consequent subsentence (column B).
From here we can verify the validity of the argument.
Check the truth values of all three sentences, premises and conclusion, of the argument noting that we are assuming that the two premises are true. However, the second premise claims the consequent of the first premise is false and the conclusion claims that the antecedent of the first premise is false. Can we make the conclusion true given the assumption that the premises are true?
Is there a line in the characteristic truth table of the conditional where A (the antecedent) is false, F, B (the consequent) is false, F, and A→B (the conditional) is true, T? There is. It is the fourth line. We verified that all of the three sentences of the argument have the value true. So the argument is valid.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/