Logically Valid Arguments

When developing an argument, the goal is for it to sound, part of which means being valid. So lets start with validity first.

As mentioned in a previous blog post, an argument consists of a set of premises from which a conclusion is supposed to follow. A valid argument is just an argument from which the conclusion actually follows from the premises.

What does this mean? We talked a little about his previously but let’s go into more detail.

Structure

Some arguments are valid because of their structure. What kind of structure are we talking about? Logical structure.

When we talk about structure, we are talking about, roughly, how the pieces of language are fit together rather than what those pieces of language mean. For example, all these claims mean very different things:

Kelsey is a writer.

Daassa is a lovebird.

Lewis is a Sagittarius.

But when it comes to structure, all three of these claims have the same structure: they are all simple subject-predicate sentences. We can reveal this structure by writing:

A is a B.

Now let’s take a look at the following argument to see what logical structure is:

P1) If my cat is a ginger cream, then my cat is a ginger.

P2) My cat is a ginger cream.

C) Therefore, my cat is a ginger.

When looking at structure, we want to find repeating elements and replace them with capital letters so as not to distract ourselves from the structure. Specifically, we want to find the biggest repeating elements we can find.

This last point is important because if we focus just on the smallest repeating elements, we might be capturing structure at too fine grain at level. For example, we may notice that “my”, “cat”, “ginger”, and “cream” are repeating elements of the previous argument. Taking the advice of the previous paragraph, we would end up with this:

If A B is a C D, then A B is a C.

A B is a C D.

Therefore, A B is a C.

This is a good start! Now we aren’t distracted by the meanings of the words involved. But there is such a thing as too much structure. When is structure too much? When even the structure itself includes repeating elements. Looking at the argument, we can see that there are still two repeating elements: “A B is a C D” and “A B is a C”. Thus we can simplify the structure and make the argument clearer by replacing these elements themselves with capital letters:

If E, then F.

E.

Therefore, F.

Now that’s simple!

The rule of thumb for this is: replace repeating elements with capital letters until there are no more repeating elements that are not themselves capital letters.

Back to Validity

Now that we have isolated the structure, we can see if the argument is valid. We do this by means of truth tables.

Remember: a logically valid argument is one in which there is no situation in which the premises are true and the conclusion is false. This is precisely what a truth table can tell us.

We want to start by listing all the atomic claims in our argument. Atomic claims are represented by the capital letters. In our example, this would be E and F. We assume that a claim is either true or false, so there are four truth possibilities for our claims: either both are true, both are false, E is true while F is false, or F is true while E is false:

E | F

——

T | T

T | F

F | T

F | F

Once we have listed our atomic propositions, we want to add the premises of our argument. The second premise is easy to list since it is just E. But our first premise requires a bit more explaining. How do we know what truth values to assign to “If E then F”?

We know because in the logical system we are using, called classical logic, claims which are constructed out of atomic claims are true or false in virtue of solely the truth value of the atomic claims which make them up.

For conditionals like our first premise, they are true just in case the antecedent (the claim right after ‘if’) is true and the consequent (the claim right after ‘then’) is false.

This means that our first premise will only be false given the truth values of the atomic claims listed in the second row of our truth table. Our truth table thus far looks like this:

E F | If E then F ——— E

——|——————————

T T |———T—————T

T F |———F—————T

F T |———T—————F

F F |———F—————F

The thing to notice right off the bat is that only on the first row of our truth table are both premises true. That means that all we need to figure out is if the conclusion is also true on that first row. Remember: when testing validity we only care about the scenarios in which all the premises are true.

Adding the conclusion to the end completes the truth table:

E F | If E then F ——— E—| F

——|—————————— |—

T T |———T—————T—| T

T F |———F—————T—| F

F T |———T—————F—| T

F F |———F—————F—| F

We can see that on the first row, the conclusion is true. Thus there are no situations in which the premises are true and the conclusion is false. Thus, this argument is valid.