As we discussed in the previous tutorial, Aristotle cemented the foundation which would
establish logic as its own proper discipline for millennia to come. Many elements can be identified
in other regions around his time, but this was the first time an explicit, comprehensive, and
dedicated theory was established. Logic itself, though, is not a science, but rather a tool to
be used in science and other areas of inquiry. It represents the strongest form of argumentation,
since a valid conclusion needs to be, beyond any doubt, a necessary consequence of its premises.
And to further avoid any traps language may offer, Aristotle proposed that placeholder symbols such
as a single letter should substitute proper words in the analysis – what we know today as variables,
a concept only introduced to mathematics almost 2000 years later in the 16th century.
Let’s return to our original syllogism. As we saw, it contains two kinds of sentences: premises,
which were the first two lines, and a conclusion, which is inferred from the premises. Each of these
sentences also has three different elements akin to grammatical elements: the subject, which is the
main term being defined; the predicate, which says something about the subject; and the operators,
which quantify and qualify the others. The repeated term which connects both premises,
allowing for the conclusion, is called a middle term. Aristotle names ten kinds of predicates
in his Categories, which in Greek also literally meant “predicates”. Those are substance, quantity,
quality, relation, location, time, position, possession, doing, and undergoing.
Regarding their relationships with the subject, or “predicables” he named four: of definition, genus,
unique property, or accidental property. And as we can see with the Socrates example, subjects
and predicates can be a combination of several different things, so we can substitute some of
these words for the aforementioned variables as follows: “Every ‘A’ is ‘B’/’C’ is ‘A’/Therefore,
C is B.” This is a valid argument, and a valid inference, since it’s simply formulaic in that
if both premises are correct, the conclusion will also necessarily be correct. This doesn’t
mean, of course, that it will be true. This is an important distinction. Validity
is a property of the format of the argument, while truth is a property of every individual sentence.
For example, consider this argument. “Every dog is a bird/Dave is a dog/Therefore Dave is a bird”.
Everything here is false, but it’s still a valid argument, since it follows a proper structure,
and the conclusion does follow the premises, even though they are false premises.
Or consider this. “Every cat is a mammal/My dog is a mammal/Therefore, all dogs are mammals”.
Each of these statements are true individually, however, the conclusion does not follow from its
premises, and thus it’s an invalid argument. Therefore, an argument is true if and only if
every sentence composing it is true and they’re valid, in that the argument: 1) has at least two
premises and a conclusion, 2) the conclusion is a necessary inference from the premises, and 3)
the conclusion is different from the premises. We can see in all of these examples two kinds of
sentences, depending on their operators. There are universal ones such as “Every YouTuber is mortal”,
which refer to every member of a given category being part of some other category, and particular
ones such as “Dave is a YouTuber”, which refers to a single or limited number of members belonging
to a given category. And both these kinds of sentences can be either positive, in that they’re
affirming something, or negative, meaning they are denying it. “No YouTuber is immortal”, for
example, would be a universal negative. This leads us to the following board, where we have all four
kinds of sentences in Aristotelian logic. We can categorize each of these sentences by a letter:
“A” for universal affirmatives, “E” for universal negatives, “I” for particular affirmatives and
“O” for particular negatives. These sentences have clear relationships between themselves,
For example A and O would be clear contradictories, as in “Every YouTuber
is mortal/Professor Dave is immortal”, much like E and I. Additionally, I and O are subalterns of
A and E, respectively, since they only represent a part of their generalized versions. Finally,
A and E are contraries, since they exclude each other, as all members of a class either are or
are not something. For example “YouTubers” cannot be mortal and immortal at the same time. This goes
back to another concept in his Metaphysics, that a thing cannot be and also not be at the same time:
the principle of non-contradiction. I and O, on the other hand, are subcontraries, as they can
coexist. Aristotle also used name mnemonics to refer to certain kinds of syllogisms. If one is
composed of three universally positive sentences, for example “All YouTubers are mortal/All humans
are mortal/Thus, all YouTubers are human”, we could call them an “AAA” sentence, and
its mnemonic would be Barbara, with each of its vowels representing one sentence of the syllogism.
Other examples would be “Darii” for AII, which describes our original example, and “Baroco”
for AOO, as in “All YouTubers are mortal/This rock is not mortal/This rock is not a YouTuber”.
Another interesting pattern arises as we analyze Aristotle’s understanding of the
proper ordering of sentences: the premises must have a certain direction in order for
a conclusion to be their result, a direction from the more “general”
premise to a “less” or at least equally general conclusion. This is what we call a deduction.
A deduction is the opposite of an induction, where a more general conclusion is reached
by more particular premises. Aristotle only briefly referred to induction when speaking
of its importance in cognitive processes regarding scientific knowledge. However,
as he was focused on classifying argumentation itself in every field of knowledge when writing
on logic, it is understandable that he focused on the more objective and certain deduction.
Inductive reasoning depended on more practical and methodological observation and categorization,
and thus wasn’t susceptible to a “pure form analysis” like deductive reasoning
is. Nevertheless, we will see later in the series that inductive reasoning still plays a big role
when we think about the scientific method. Finally, Aristotle was also the first to
veer into what would be known today as “modal logic”. “Modal” because it analyzed the degree
of possibility of a sentence, instead of it being simply “assertoric”, or affirming nothing but its
truth or falsity, as with our previous examples. For example, “Dave is a YouTuber” is simply
asserting some information. However, if someone doesn’t know me or what I do for a living, they
could say “Dave is probably a YouTuber” or “Maybe Dave is a YouTuber”. These would be “contingent”,
or “accidental”, “possible” and not “necessary”, and would have different relationships with their
assertoric counterparts. But Aristotle’s formulations were problematic and largely
abandoned soon afterwards, so we’ll not go farther into his perspective. Rather, we will come back
later to examine more interesting developments. Hopefully we now have a greater understanding
of how logic first operated at its birth. Once again, this formulation went largely
unchallenged until Frege’s overhaul only about a hundred years ago, which we will examine later in
the series. Of course, many details and minor concepts were worked out throughout history,
and we will periodically come back to this topic as we study the development of philosophy itself.
For now, let’s return to our timeline and continue examining the history and evolution of philosophy.