In basic logic, if an argument contains a self-contradictory premise, how come the argument is valid?

Question

I have started to attend the logic lectures. The lecture notes indicate that a "self-contradiction" is a statement that is always false.

But why will it lead to a valid statement?

Answer 1

Your question at least on a quick reading seems to be using terms a bit willy nilly. Let's start by defining terms.

Background

Statement - a claim that can be either true or false.

Some examples:

• "it is raining"
• "it is not raining"
• "if it is raining, John uses an Umbrella"
• "Jill is either in Denver or Tokyo"

We also need to be a bit careful about the "can be" in the definition. Here, it means something like "capable of being evaluated to either true or false" -- my point being that it's possible a statement is always true (e.g. A or not A) or always false (e.g. A and not A), but the point is that statements don't include things that don't have a "truth-value."

Self-contradictory - I assume this something that posits both A and not A.

Valid - this means that an argument would have a true conclusion were all the premises to be true.

Another term related to validity is "truth-preserving." This is the idea that an argument's premises truth can be carried onto the conclusion.

To your question

In your question you speak as if a statement can be valid or invalid. This is not true in logic. Statements can be well-formed or not. A well-formed statement is one that can be evaluated unambiguously. Any statement that follows the rules of the syntax is a well-formed formula -- not statement is valid because validity is not about statements.

You also speak of a self-contradictory premise. This might make sense, but more commonly this happens with an argument as a whole rather than a single premise.

E.g. "The moon is blue" and "The moon is not blue" would create a self-contradictory premise if it's a single premise or a self-contradictory set of premises if the two claims are separate premises.

To understand why an argument is valid if it has self-contradictory premises, we just need to think through the definition of validity a bit. I will repeat it in pieces:

An argument is valid if and only if it is the case that when the premises are true, the conclusion must be true.

This is often thought of it in a backwards way in the sort of introductory logic and critical thinking classes that exist. In other words,

an argument is valid just so long as there is no case where the premises could be true and the conclusion false.

(In some types of logic, the definition of validity is different than this, but those types are never taught in introductory logic classes in philosophy)

Given a set of contradictory (or self-contradictory) premises, it is never the case that the premises are all true -- because for one to be true ("the moon is blue"), another must be false ("the moon is not blue").

Thus, any such argument is valid on this definition of validity.

Answer 2

I think you're talking about a material implication, which has the form of "if A then B", where A and B are propositions. This statement is false if and only if both A is true and B is false, or "A and ~B". The negation of this is ~(A and ~B) which is logically equivalent to (~A or B). It then follows that the statement "if A then B" is logically equivalent to (~A or B).

Like we said, and by way of truth tables, you can see that the statement "if A then B" (~A or B) is false when and only when A is true and B is false and thus it follows that the statement can be true even when A is false, or even when A is a contradiction.

The fact you can obtain truth from falsity is counterintuitive, so I propose you look at material implications as promises.

Let A be "You work for me" and B be "I will pay you". The promise then says

"If you work for me then I will pay you".

I can only break my promise when you work for me, and I don't pay you. (A is true and B is false). I wouldn't be breaking my promise if you hadn't worked for me and I didn't pay you (Both A and B are false) or even when you hadn't worked for me and I still payed you (A is false and B is true).

Then our analogy that the promise holds even when you didn't work for me translates into the fact that a material implication can be true even when the premise/precedent/antecedent is false, or even a contradiction. This implies that the statement "If penguins can fly then there is no gravity on planet earth" is true.

I hope my answer wasn't too convoluted.

Answer 3

The authors of forall x give this definition of validity (page 8):

An argument is valid if and only if it is impossible for all of the premises to be true and the conclusion false.

They also refer to this as "validity in virtue of form" (page 21):

The validity of the arguments just considered has nothing very much to do with the meanings of English expressions like ‘Jenny is miserable’, ‘Dipan is an avid reader of Tolstoy’, or ‘Jim acted in lots of plays’. If it has to do with meanings at all, it is with the meanings of phrases like ‘and’, ‘or’, ‘not,’ and ‘if..., then...’.

Such validity can be automatically validated using proof checkers. Assuming the proof checker is not poorly constructed if the argument we enter into the proof checker passes, then it is valid in virtue of form.

Consider the sentence, "It is raining." The negation of this sentence could be written as "It is not the case that it is raining." Let us symbolize "It is raining" with "P". Then its negation would be "¬P". A self-contradictory sentence built from these two would be "P ∧ ¬P", that is, both "P" and "¬P".

Using this as a premise can we construct a valid in virtue of form argument? Yes, we can. Here is an example where I am trying to show that "P ∧ ¬P" implies "P".

The first line is the premise. I get the second line by using the conjunction elimination rule (∧E) which allows me to use either side of the conjunction (∧) in my argument. It turns out what I used, "P", was what I wanted to show and so the proof checker verified that I reached my goal.

It may see odd that given a self-contradictory premise, I was able to construct a valid argument showing that one of the conjuncts in the self-contradictory premise followed. It can get even stranger.

With a little more work, I can show that anything whatsoever follows from a self-contradictory premise. Let "Q" be the symbolization for anything whatsoever. This example shows that I can construct a valid in virtue of form argument for "Q":

The first line contained the self-contradictory premise as in the previous example.

On lines 2 and 3 I used conjunction elimination (∧E) to both sides of the premise (line 1) to place these on separate lines.

Once I have them on separate lines, I can use the rule of contradiction introduction (⊥I) to note on line 4 that I have a contradiction (⊥).

On line 5, because I have a contradiction, I can write anything I want. That is called the rule of explosion (X). Since my goal is "Q" and according to the rules I can write anything I want, I write "Q". The proof checker verifies that I have reached my goal.

For more details on these rules, see Chapter 15 "Basic Rules for TFL" in forall x.

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References

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

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/