Categorical Syllogism and Forms

From categorical propositions to categorical syllogism

The goal for logicians has always been the same—classifying a logical argument as valid or invalid—but emotions and internal biases can get in our way sometimes. For instance, in the case of politicians and their supporters, some arguments are easily accepted as valid when our political opponent is criticized, even if the same criticism applies to our favorite politician. Therefore, logicians focus on the form of the argument because words can set off our deeply seated subconscious triggers. Let’s consider the argument provided below. What if we replace a term or a word with an alien/value-neutral term? Doesn’t it suddenly become easier for us to apply our logical minds?

No one knows what the word “jojoobaba” means, but a logician will only assess if the conclusion logically follows or not given that the premises are true. Aristotle wanted to help us in this regard and tried automating this complex process. He restricted his study of arguments to categorical syllogisms.

Syllogism

A syllogism is an argument in which, strictly speaking, two premises are used to infer one conclusion. In sum, a syllogism is a type of argument that employs three sentences in total. The example mentioned above is a syllogism because it contains two premises and one conclusion, but it isn’t a categorical syllogism. What’s a reason why that would be the case?

Categorical syllogism

A categorical syllogism is a syllogistic argument in which all three propositions (two premises and one conclusion) are, strictly speaking, in the standard form of categorical propositions. Secondly, the whole argument speaks of three categories (terms) only, each of which occurs once in exactly two of the propositions. The abovementioned argument isn’t a categorical syllogism because a proper noun or the name of a specific politician doesn’t usually qualify as a general class or category. In contrast, the term “politician” is a common noun and is an entire class or a set that contains many members, including our favorite and not-so-favorite politicians.

Forms of categorical syllogism

This section will introduce a few definitions, but the objective isn’t to overburden our memorization. The purpose is to identify the maximum number of unique forms that a finite structure of a categorical syllogism can take and explore how many of them are valid forms of argumentation. The maximum forms a categorical syllogism can take depends on the mood and figure of the argument. Let’s figure out the mood first.

Mood

Let’s assume that for a categorical syllogism, both premises and conclusion are already in one of the four standard forms of categorical proposition ($A$, $E$, $I$, ...