An Invitation to Logic - Amateur Logician

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— George Wick

Logic is about good thinking, about good reasoning. As you master this part of the Trivium, you will become a clearer thinker, arguer, and debater.

You’ll be able to judge arguments better. (Does a conclusion follow?)
You’ll be more of an independent thinker. (Deduce conclusions yourself.)
You’ll be able to form arguments better. (Construct sound arguments.)
You’ll be able to spot fallacies. (Challenge talking-heads and politicians!)

An Invitation, Or Introduction, To Traditional Logic:

Logic — at least its most exciting part — is first and foremost about inference. That is, it deals with how we can derive new propositions from given propositions. It is in this sense that logic can extend our knowledge.

A proposition is simply a sentence that declares something to be, or not to be, the case. It is something that can be true or false. Before we can have any propositions, however, we need terms.

Without understanding terms, nobody can meaningfully declare, “All men are mortal.” Thus we need to understand concepts. A concept is an idea of something, e.g., “mankind” or “mortality.” A lot can be said about concepts, and Aristotelian-Scholastic logic in particular has a lot to tell us. Concepts have “intension” and “extension,” for example.

Once we have concepts, we can form propositions. These are judgements. They relate terms through the “to be” verb (i.e., the copula). In two-name categorical logic, there is a subject and predicate. Such propositions have a “quality” and “quantity.” The predicate is an attribute that is either affirmed or denied of the subject. And the subject is either referred to universally or particularly (or as a singular subject referred to as a specific individual). “All men are mortal” tells us that every human being positively has the attribute of mortality.

Next, we can think about syllogisms. These deal with inferences. We can sometimes take two propositions to derive a third. In a categorical syllogism, there are a total of three terms and three judgements. The conclusion, as a judgement, will have a subject (i.e., the minor term) and predicate (i.e., the major term). A third term (i.e., the middle term) will appear, as a subject or predicate, in both of the judgments used to derive the conclusion. In fact, it’s through the middle term that gets us the third proposition. “All men are mortal. Socrates is a man. Thus, Socrates is mortal.”

Notice, there’s a progression here: the categorical syllogism requires propositions and the propositions require terms. Each step corresponds to different “acts of the mind.”

In traditional logic, we have “Three Acts of the Mind”:
(1) Simple Apprehension / Conceptualization: act that concerns forming ideas.
(2) Judgement: act that objectively affirms or denies an attribute of a subject.
(3) Reasoning / Inference: act deriving (ideally) a new truth from older truths.

Now, when thinking about inference in Aristotelian logic, probably the most important basic point is to preserve extension or distribution. So, what’s important is how much of what is being referred to.

Consider again the argument:
“All men are mortal. Socrates is a man. Thus, Socrates is mortal.”

As we declare that “All men are mortal,” it’s clear that we are referring to all men. The term “men” is called “distributed” because of this. However the term “mortal” is not distributed. This is because we are not referring to every mortal being in the universe.

Consider “Socrates is a man.” We’re referring to Socrates in his entirety, that is, that he is a man. Conversely, man is undistributed, that is, we are not referring to the entire extension of the term “man.” In other words, we don’t know (and are not told!), according to that proposition, that everything that is a man is Socrates!

Finally, “Socrates is mortal.” That follows from our first two propositions. We can infer that new proposition (i.e., conclusion) from those previous two propositions (i.e., premises). We’re told that Socrates belongs to the group called “men” and that all “men” are mortal beings. “Men” is in the middle of the inference.

This is, in a nutshell, what traditional logic teaches.

“Modern” or Mathematical Logic

Aristotle’s logic had dominion for hundreds of years. It’s not quite fair to claim that logic didn’t develop over those years; it did, just research the medieval contributions to logic. Note that the usual presentations of traditional logic in most contemporary textbooks tends to reveal, at best, a simplified version of what is really a robust system.

Still, mathematical logic got its push from Gottlob Frege (1848-1925). It was thought that, because language can be so vague, an exact system with precise symbols was required.

Propositional logic was developed, where various “sentential connectives” allow you to relate different propositions. For example, you can combine two propositions through the “and” sentential connective to form a conjunction.

The only problem, though, is that propositional logic can’t handle syllogisms the way Aristotelian-Scholastic logic can. This required the introduction of predicate logic, with a means to quantify variables. Since then, logic has gone through several revolutions with exotic systems: modal logic, deontic logic, etc.