The square of opposition is often used to visualize
the relations between the four basic categorical
propositions in Aristotelian logic. It shows, for
example, that the propositions "All S are P" and
"Some S are not P" are contradictory, meaning that
one of them has to be true while the other is false.
Aristotelian logic encompasses a great variety of topics. They include metaphysical theses about
ontological categories and problems of scientific explanation. But in a more narrow sense, it is identical
to term logic or syllogistics. A syllogism is a form of argument involving three propositions: two
premises and a conclusion. Each proposition has three essential parts: a subject, a predicate, and a copula
connecting the subject to the predicate.[107] For example, the proposition "Socrates is wise" is made up of
the subject "Socrates", the predicate "wise", and the copula "is".[108] The subject and the predicate are the
terms of the proposition. Aristotelian logic does not contain complex propositions made up of simple
propositions. It differs in this aspect from propositional logic, in which any two propositions can be
linked using a logical connective like "and" to form a new complex proposition.[109]
In Aristotelian logic, the subject can be universal,
particular, indefinite, or singular. For example, the
term "all humans" is a universal subject in the
proposition "all humans are mortal". A similar
proposition could be formed by replacing it with the
particular term "some humans", the indefinite term
"a human", or the singular term "Socrates".[110]
Aristotelian logic only includes predicates for
simple properties of entities. But it lacks predicates
corresponding to relations between entities.[111] The
predicate can be linked to the subject in two ways:
either by affirming it or by denying it.[112] For
example, the proposition "Socrates is not a cat"
involves the denial of the predicate "cat" to the
subject "Socrates". Using combinations of subjects
and predicates, a great variety of propositions and
syllogisms can be formed. Syllogisms are
characterized by the fact that the premises are linked
to each other and to the conclusion by sharing one
predicate in each case.[113] Thus, these three
propositions contain three predicates, referred to as
major term, minor term, and middle term.[114] The
central aspect of Aristotelian logic involves
classifying all possible syllogisms into valid and
invalid arguments according to how the propositions
are formed.[112][115] For example, the syllogism "all
men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. The syllogism "all cats are
mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid.[116]
Classical logic is distinct from traditional or Aristotelian logic. It encompasses propositional logic and
first-order logic. It is "classical" in the sense that it is based on basic logical intuitions shared by most
logicians.[117] These intuitions include the law of excluded middle, the double negation elimination, the
principle of explosion, and the bivalence of truth.[118] It was originally developed to analyze
mathematical arguments and was only later applied to other fields as well. Because of this focus on
Classical