“We must determine what a premise is, what a term is, and what sort of deduction is complete and what sort incomplete.’”
(Aristotle, Prior Analytics, transl. Robin Smith, 2012)
We already took a first look at syllogisms in our very first lesson. To be more precise, here’s a definition:
Syllogisms are at the very core of Aristotelian logic; we will look at them in much more depth later in the course. Now that we have an understanding of how to evaluate the truth of a proposition by checking for whether the truth of a proposition corresponds to facts in the world, we can look at two important properties of the syllogism.
[Activity] Before moving on, recall a few of the arguments we looked at in the first lesson. What made each of them “good” or “bad” arguments to you? Discuss with a partner!
A syllogism is valid if the truth of the premises guarantees the truth of the conclusion. In other words, valid syllogisms are those where, if the premises are true and the terms are clear and unambiguous, it is not possible for the conclusion to be false. We will talk about vagueness/clarity and ambiguity in the next lesson. Here, we will focus on the truth of premises and validity of syllogisms. If a syllogism is not valid, then it is called invalid. Consider the following example:
Premise 1 (P1): All snakes are trees.
Premise 2 (P2): All trees are plants.
Conclusion (C): All snakes are plants.
Despite expressing a completely absurd fact, this syllogism is valid. This is because if we assume the premises to be true (i.e. that all snakes are trees and all trees are plants), then the conclusion does follow. Whether the premises are actually true or not does not matter for the notion of validity. Compare with the following:
Premise 1 (P1): All snakes are trees.
Premise 2 (P2): All trees are plants.
Conclusion (C): No snake is a plant.
This argument is invalid because if the premises were true, the conclusion would simply not follow logically.
We are interested in studying validity because we want to know what kinds of syllogisms will get us from true premises to a true conclusion, irrespective of their content. If we manage to elaborate a list of valid syllogisms and invalid ones, we will have a first practical standard for evaluating the strength of an argument – if we find it on the wrong side of our list, then we will know that it is a bad argument and we will know why.
An argument is called sound if (1) it is valid, (2) its premises actually are true, and (3) its terms are clear and unambiguous. Otherwise, it is called unsound. For example, the argument
Premise 1 (P1): All people are mortal.
Premise 2 (P2): Aristotle is a person.
Conclusion (C): Aristotle is mortal.
is sound. Firstly, it is valid – if the premises are true, then so is the conclusion. Secondly, its premises actually are true – the respective propositions do correspond to actual states of affairs in the world, since it is the case that all people are mortal and it is the case that Aristotle is a person. By contrast, the argument
Premise 1 (P1): All snakes are trees.
Premise 2 (P2): All trees are plants.
Conclusion (C): All snakes are plants.
is unsound. Although we have seen that this argument is valid, it is not the case that its premises are true. In particular, P1 is false – it does not correspond to a state of affairs in the world because it is not the case that all snakes are trees! Note that validity is a precondition for soundness; in other words, it is not possible for an argument to be invalid, but sound. To be clear, if an argument is sound, then the conclusion must be true.
[Activity]: With a partner, come up with your own examples of a valid and sound argument, a valid and unsound argument, and an invalid and sound argument. If one of these is impossible, discuss why.