Aristotle and Paraconsistent Logic

Does Aristotle use a Paraconsistent Logic?

Photo by Birmingham Museums Trust on Unsplash

Many people know that Aristotle endorsed the principle of noncontradiction. In the Metaphysics, he writes:

It is impossible for the same thing to belong and to not belong at the same time and in the same respect¹

Yet a philosopher can endorse the principle of noncontradiction while nevertheless still utilizing a paraconsistent logic, and indeed, the majority of logics prior to the 20th century were paraconsistent.

Consistent logics are not just systems of logic that endorse the law of noncontradiction, but are forms of logic where a violation of the principle of noncontradiction leads to trivialism, that is, systems of logic where a contradiction implies that every statement is true.

The Principle of Explosion

Consistent logics are rooted in the principle of explosion, which claims that “from a true contradiction, anything follows.” This principle was first developed by the 12th century French philosopher William of Sisson, although it was not widely accepted until the development of modern formal logic in the early 20th century.

The principle of explosion is rather counterintuitive. According to this principle, from the true contradiction “It is raining, and it is not raining,” we can infer that Napoleon Bonaparte is the current president of the United States. Let me explain how the argument works.

Let’s begin with the premise “It is raining, and it is not raining.” If this is proposition is true, then the proposition “it is raining” is also true. From this, we can use a logical principle called “disjunctive introduction,” which says that if a premise is true, then the disjunction of that premise and/or any other premise is true. So then “It is raining or Napoleon Bonaparte is the current president of the U.S.” Since we know that “it is raining” is true, we know that it’s also true that either one of these propositions is true. Yet since it is just as equally true that “it is not raining,” then from the premise “either it is raining or Napoleon Bonaparte is the current president of the U.S.,” we can infer that Napoleon Bonaparte is the current president of the U.S. This follows from the disjunctive syllogism. So the argument, expressed again, goes:

1. It is raining and it is not raining.

2. It is raining

3. It is raining or Napoleon Bonaparte is the current president of the U.S.

4. It is not raining.

5. Since it is either raining or Napoleon Bonaparte is the current president of the U.S. and it is not raining, the current president of the U.S. must be Napoleon Bonaparte.

The argument can be expressed in symbolic notation as:

• P & not-P

• P

• P or Q

• Not-P

• Therefore Q

We can see, accordingly, why one might think that contradictory premises imply everything. This argument for the principle of explosion works as long as we accept the validity of two other principles of logic: disjunctive introduction and the disjunctive syllogism. In order to reject the principle of explosion, we must reject one of these two logical principles.

Aristotle’s Paraconsistent Logic

Aristotle seems to have a paraconsistent logic. In fact, most philosophers prior to the formulization of mathematical logic at the beginning of the 20th century seem to have operated without the principle of explosion. In the Prior Analytics, Aristotle discusses certain conditions where one can make valid deductions from inconsistent premises, and others where one cannot make valid deductions. Aristotle writes:

For let A stand for good and B and C for science. Now, if someone took every science to be good, and also no science to be good, then A belongs to every B and to no C, so that B belongs to no C: no science, therefore, is a science.²

1. Every science is good

2. No science is good

3. No science is a science

Aristotle then gives a second example:

And similarly also if, taking every science to be good, he took medical knowledge not to be good (for A belongs to every B and to no C, so that no particular science will not be a science).³

We can formulate Aristotle’s second example as follows:

1. Every science is good

2. Medical science is not good

3. Medical science is not a science

Yet this means that the following inference, for Aristotle, would seem to be invalid:

1. Every science is good

2. Medical science is not good

3. Every science is a science

Graham Priest expresses this point through a different example:

1. No man is mortal

2. Some mortals are men

3. All men are men⁴

If no man is mortal, then it’s impossible that some mortals are men. These statements are a contradiction. If we are operating within a consistent logic, then it should be the case that “all men are men” follows from these contradictory premises. That is, this should be a valid argument. Yet in Aristotelian syllogistic logic, this inference is not valid, thereby suggesting that Aristotle operates with a paraconsistent concept of logic.⁵ The principle of noncontradiction, for Aristotle, is a rule for thinking, yet Aristotle does not endorse the rule that if we are allowed to say some contradictory things, then this implies that everything is true.

Paraconsistency and Modally Robust Disjunction

In the essay “What is Dialectic?”, Karl Popper attacks the dialectical philosophy of Marx and Hegel by invoking the principle of explosion. Popper shows how the principle of explosion follows from disjunctive introduction and the disjunctive syllogism and argues: “It is a fact that everyone, and dialecticians are no exception, makes (perhaps unconsciously) use of the aforesaid two rules of deduction.”⁶ Yet Aristotle himself does not accept Popper’s rules of deduction, meaning that Popper only shows his own ignorance of the history of logic with this statement.

Syllogistic logic, in fact, has a completely different view of disjunction, one based primarily upon an idea of “exclusive disjunction,” that is, disjunction understood as an “either/or.” Disjunction, moreover, is understood primarily as a modal determination, that is, as a metaphysical claim about the space of possibilities.

Consider, again, the inference “it is raining and not raining, therefore Napoleon Bonaparte is the current president of the U.S.” What bothers us about this inference is that the current U.S. president seems to have no necessary relation to whether it is raining. Whether it’s raining, not raining, or both seems to have no relationship to who is president.

In older logic, the disjunctive syllogism was what Robert Brandom calls “modally robust,” meaning that disjunction attempts to carve out the entirety of the space of possibilities. The disjunctive proposition “the book is either red or blue,” for example, doesn’t just list two possibilities, but actually claims that it is impossible for the book to be any other color other than red or blue.

If we adopt Brandom’s strategy of understanding contradiction and contrariety in terms of modally robust exclusion, then a disjunctive judgment implies that two things stand in a necessary relationship of exclusion. Disjunctive introduction, accordingly, is no longer valid. We cannot conclude from the statement “it is raining” that “it is raining or Napoleon Bonaparte is the current president of the U.S.,” precisely because rain does not (at a metaphysical level) exclude Napoleon Bonaparte from the presidency. These two facts are, in Leibnizian terms, compossible, that is, it is not impossible for both to be true simultaneously.

Conclusion

The syllogistic logic of Aristotle is paraconsistent, despite Aristotle’s endorsement of the law of noncontradiction. The same can be said of Kant’s understanding of syllogistic logic. A paraconsistent logic, we should recall, does not just mean a system of logic that allows for true contradiction, but includes any system of logic where contradiction does not lead to explosion. Aristotle’s syllogistic logic, accordingly, has much in common with the dialectical logic operating within the Hegelian and Marxist tradition.

Hegel and Marx, of course, reject the law of noncontradiction, which means that they clearly are operating within a paraconsistent logic, since it is not the case that Hegel and/or Marx believe that it’s possible to infer any proposition from the fact that something is both here and not here. But I will leave a more detailed discussion of dialectics for a different article.

1. Aristotle, Metaphysics, 1005b.

2. Aristotle, Prior Analytics, 64a1-4.

3. Ibid., 64a4-7.

4. Example taken from Graham Priest, “What’s So Bad About Contradictions?” The Journal of Philosophy, vol. 95, no. 8 (1998): 411.

5. Ibid.

6. Karl Popper, “What is Dialectic?” Mind 49, no. 196 (1940): 410.