Part I: Lesson

“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)

A helpful way to think about propositions is as follows: 

• Propositions: a proposition is whatever a declarative sentence expresses, and it can either be true or false.
• What is the Structure of a Proposition?

From the perspective of Aristotelian logic, propositions take a specific form: first a subject, followed by a copula, which introduces the predicate. 

1. Propositions can be True or False

Unlike terms, propositions can either be true or false, and are always one of the two. We call true and falsetruth-values. A proposition must be either true or false. That is, it cannot be both true and false, and it cannot be neither true nor false.

2. How do we figure out if a proposition is true or false?

In Aristotelian logic, we say that a proposition is true if and only if it corresponds to the facts of the matter that it describes. For example, “Aristotle is mortal” is true if and only if it is actually the case that Aristotle is mortal – if we go out in the world, “check” if Aristotle is mortal, and discover that he actually is mortal, then “Aristotle is mortal” is true. When we do this “checking”, we say that we evaluate the proposition. This notion of “evaluation” picks up on the notion of truth-values. True and false are here taken to be “values.”

Sometimes, instead of checking, we’re instead interested to see what we can deduce from a proposition if it is true or if it is false. We suppose that it’s true (or false) and see where that leads us. Put differently, sometimes we don’t want to evaluate a proposition directly, but instead we want to hypothetically give it a truth-value and see what we can learn. In such cases, we say that we assign a truth-value to a proposition. We can think, “What follows if ‘America lost the Revolutionary War’ is true?” or “What follows if ‘all men are mortal’ is false?” 

[Activity] What would it take for “Aristotle is mortal” to be false?

[Activity] Think about the connection between the two truth-values. If we know that “Aristotle is mortal” is true, is there another proposition that we can subsequently deduce is false?

3. Negation

The connection hinted at in the last activity turns out to be really important in logic. It has to do with negation. The negation of a proposition is what we get when we take a proposition and say that whatever it expresses is not the case. For example, the negation of “Aristotle is mortal” is “It’s not the case that Aristotle is mortal.” In this case, we can achieve the same result by denying the predicate of the subject. In other words, the negation of “Aristotle is mortal” is “Aristotle is not mortal.” It is worth noting that this method of simply denying the predicate does not always yield the negation of a proposition. We will deal with this in Section III: Aristotle’s Logic of Propositions. 

Looking at our initial definition, we can see that the negation of a proposition is also a proposition itself: it’s either true, or false. With this in mind, it’s easy to see the connection between a proposition and its negation: when the proposition has one truth-value, its negation has the other one. Specifically, since “Aristotle is mortal” is true, “It’s not the case that Aristotle is mortal” is false. Very importantly, the “double-negation” of a proposition yields the same proposition. For instance, “It’s not the case that it’s not the case that Aristotle is mortal” is really just saying “Aristotle is mortal.” This result is very important in logic.