How can syllogisms with contradictory premises be valid?

Question

A syllogism is valid if it is impossible for the premises to be true and at the same time the conclusion to be false.

Consider the following syllogism: P1: This apple is red. P2: This apple is not red. C: Therefore, 1+1=2.

Is this syllogism valid? I've heard that it should be but I am confused. Certainly, the premises are such that they are impossible to be true. But if the premises are contradictory, then it is also impossible for the conclusion to follow from the premises. And if the premises do not guarantee the truth of the conclusion, then shouldn't it be invalid?

Answer 1

The principle your question refers to is called the principle of explosion, or sometimes the latin expression is used, ex contradictione quodlibet, meaning from a contradiction anything follows. It is a feature of classical logic, and also of many other logics, though not all logics. Logics that do not have the principle of explosion are called paraconsistent.

There are two ways to see why the principle of explosion should hold. One is that it can be proved by simple rules. Suppose we start with a contradiction "A and not A". Then we can reach any arbitrary conclusion B as follows:

1. A and not A        (assumption)
2. A                  (follows from 1)
3. A or B             (follows from 2 by addition)
4. not A              (follows from 1) 
5. B                  (follows from 3 and 4 by disjunctive syllogism)

A second way to demonstrate the principle of explosion is to use the account of validity that you quoted in your first sentence. An argument is valid if it is impossible for the premises to be true and the conclusion false. Actually, this is only a rough first pass at explaining validity and there are better accounts, but it will do for our purposes. If the premises of an argument are contradictory then it is impossible for them to all be true, hence a fortiori it is impossible for them to be true and the conclusion false.

The principle of explosion often seems strange to newcomers to logic, but once you get the hang of it, it is not really a problem. In classical logic no contradictions are true, so an argument with contradictory premises can never be sound, i.e. it can never both be valid and have true premises. You can even think of the principle of explosion as a kind of straitjacket that enforces the rule that no contradictions are true. If a contradiction were true, the consequences would be catastrophic because anything would follow. So we must never allow true contradictions.

The principle of explosion is also highly useful in mathematics. Suppose we wish to prove that a theory is consistent. An inconsistent theory entails a contradiction and by explosion proves anything whatsoever. So by contraposition, if there is even one formula that can be shown not to be provable by a theory, then the theory is consistent. This was used by a clever logician called Gentzen to prove the consistency of arithmetic.

Answer 2

• Indirect way : when you want to show that a reasoning is not valid, what do you do? you show it is possible (1) all the premises to be true and (2) at the same time the conclusion to be false. Could you do that here? In order to have (1) and (2) , you need to have (1) ? Can you show it's possible all the premises to be true? No, since one of them is contradictory, so, sure, it is not possible all of them to be true since one of them is contradictory. Conclusion : there is no possible case in which the reasoning is not valid.

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Direct way :

• A reasoning is valid iff the following conditional is true :

for all possible case/ situation/ interpretation , if all premisses are true, then the conclusion is true.

• " all premisses are true" is the antecedent of the proposition expressing the test for validity.

• But : an if- then statement is automatically true when its antecedent is false ( see the truth table of the " if ... then" operator"

• Since the antecedent " all the premisses are true" is necessarily false ( due to the fact that one of the premisses is contradictory) , the whole " if ... then" statement is true in all possible case.

• This is an a fortiori argument : since there is no possible case in which one of the premisses is true , there is, a fortiori, there is no possible case in which all the premisses are true and the conclusion false.

• Note : this shows that validity is not a sufficient condition for a reasoning to be a good one ( though it is a necessary one in case the reasoning is deductive) , or, even more, that validity is not a sufficient condition for a reasoning to be a proof of anything; obviously, a reasoning involving a contradictory premise cannot prove anything , even when its conclusion is true ( since a proof must be based on true statements)

Answer 3

The system you are working in has a few names, one of which is term logic. Term logic is not the same as classical logic and you have to make a few choices when you study it.

The inference you are describing is valid in classical logic. Syllogisms are frequently used as a teaching tool to introduce first-order logic since they are syntactically similar to natural language. However, there is more than one way to interpret term logic and more than one way to analyze it.

This apple refers to exactly one specific apple and so can't be handled directly by the syllogistic framework. However, we can paraphrase it using a universal. The conclusion 1 + 1 = 2 also doesn't have a representation in term logic, but you're using it as an example of irrelevant conclusion so I'll replace it with all numbers are even.

P1: Every instance of this apple is red.
P2: No instance of this apple is red.
C:  All numbers are even.

Whether this syllogism is valid or not depends on your point of view.

From the perspective of modern classical logic (classical first-order logic), this inference is valid because of ex falso quodlibet.

However, because the language of term logic is so limited, historical philosophers were able to write down the valid inferences and there are some gaps that are telling.

For example, the following syllogism is valid according to the semantics of modern classical logic, but isn't listed as a valid syllogism.

All A are B.
All B are C.
---------------
Some A are C.

The following syllogism, which is similar in spirit to your question, is also not attested (note that the last conclusion is arbitrary).

All A are B.
No A are B.
-------------
Some C are D.

Answer 4

Consider the following syllogism: P1: This apple is red. P2: This apple is not red. C: Therefore, 1+1=2. Is this syllogism valid?

No, of course not.

It is obvious to every logical person that the conclusion does not follow from the premises.

Yes, mathematical logic does say that such arguments are valid but this only shows that while mathematical logic is mathematics, it is not logic.

Mathematical logic relies on a redacted definition of validity, worded so as to be compatible with the material implication and the so called "principle of explosion", which is itself definitely not a logical principle, and yet is intrinsic to mathematical logic.

The only true classical logic, namely Aristotle's syllogistic, is grounded in the notion that there is one logic. This has been the fundamental position logicians have held ever since. Even George Boole thought that way. In his first book published in 1847, he talked of his calculus as a model of "deductive reasoning", each element of it corresponding to an element in "the human intellect".

His model was wrong, however, and inevitably different mathematicians soon came to develop their own alternative models, none of which correctly models human deductive logic. This is why mathematicians now claim that logic is arbitrary and that there is no reason that there should be only one logic, even though the grammar of the word "logic" itself doesn't leave room for interpretation. We say "logic", not "a logic" or "logics", which is why mathematicians have to talk of "systems of logic" to talk about the various mathematical theories presented as logic (1st order logic etc.). Logic is the one logic of human deductive reasoning.

This is also why questions similar to this one crop up again and again, and why so many students have a hard time understanding the material implication and that teachers have to use a fallacious argument to convince students that the material implication is nonetheless the proper model of the conditional.

Answer 5

The validity of the syllogism follows from the definition of validity you correctly stated: "A syllogism is valid if it is impossible for the premises to be true and at the same time the conclusion to be false." In "P1: This apple is red. P2: This apple is not red. C: Therefore, 1+1=2." it is impossible for both premises to be true. Therefore, it is impossible for the premises to be true and the conclusion to be false (because it's impossible for the premises to be true). [(P^~P)=>Q] is a tautology. Because a conditional is only true if the antecedent is true and the consequent is false, and the antecedent of that conditional (P^~P) can never be true.