Why is this argument logically valid?

Question

Premises:

If Ron went to the store, Ron would be home by now.

Ron isn't home yet.

Conclusion:

Ron didn't go to the store.

I am studying the first logic course in the university. I don't understand why the above is a logically valid argument. The definition of a logically valid argument is that "it is not possible for all the premises to be true and the conclusion false."

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 argument?

Answer 1

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 argument?

When you say this, it seems to me that you are conflating two different levels of logic: sentences being true and arguments being valid.

So first of all, let's make our definitions of arguments being valid, sentences being true, and sentences having truth-values:

1. "An argument is valid" means:

   • If its premises are true, then its conclusion must be true also.
   • To be precise with the scope of modality, the condition above is equivalent to this: it is necessary that if the premises are true then the conclusion is also true.
   • Your definition of validity is just another way to put the same definition above, namely: it is not possible that the premises are true and at the same time the conclusion is false.

2. "A sentence is true" means:

   • We assign a truth-value of "True" to the sentence.
   • What we precisely mean by "assignment" and "truth-value" depends on which interpretation or semantics of logic we adopt. There are many kinds of semantics, i.e., different ways to interpret and use logical symbols like '->', 'and', 'or', 'p', 'q', etc. (In fact this description also conflates some details, but let me pass over at this point.)

3. "A sentence has a truth-value" means:

   • Under some interpretation of logical symbols (based on some semantics), the sentence has some particular value assigned.
   • We usually assign "True" or "False" to a sentence. This is an intuitive and natural semantics, but it is not the only one available. See many valued logic for example.

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I think you are right to say that the first premise (conditional) does not by itself have a truth-value; it really does not have a truth-value until we assign some particular value to it.

Yet notice: to check whether the argument is valid, we start by assuming that the premises are true -- assigned a truth-value of 'True' -- and then see whether the conclusion necessarily follows. If the conclusion necessarily follows, then the argument is valid: the form of the argument, if the premises are really true, guarantees that the desired conclusion follows.

You might also be interested in the notion of arguments being sound. An argument is sound if and only if the argument is valid and the argument's premises are actually true.

Hope it helps.

(In fact, I should add: we need not adopt any 'interpretation' or semantics to show that a given argument is valid; we may manipulate logical symbols formally (that is, without considering their "meanings" we assign to them) and show that the desired conclusion does or does not follow from the premises.)

Answer 2

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 argument?

The full argument is the following:

Premises:

If Ron went to the store, Ron would be home by now.

Ron isn't home yet.

Conclusion:

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 it.

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. :

1. Ron went to the store. (The antecedent of the conditional)
2. 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.

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Reference

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/