Term logic

In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics. It was revived after the third century CE by Porphyry's Isagoge.
Term logic revived in medieval times, first in Islamic logic by Alpharabius in the tenth century, and later in Christian Europe in the twelfth century with the advent of new logic, remaining dominant until the advent of predicate logic in the late nineteenth century.
However, even if eclipsed by newer logical systems, term logic still plays a significant role in the study of logic. Rather than radically breaking with term logic, modern logics typically expand it.

Aristotle's system

's logical work is collected in the six texts that are collectively known as the Organon. Two of these texts in particular, namely the Prior Analytics and On Interpretation, contain the heart of Aristotle's treatment of judgements and formal inference, and it is principally this part of Aristotle's works that is about term logic. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm. Lukasiewicz's approach was reinvigorated in the early 1970s by John Corcoran and Timothy Smiley – which informs modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009.
The Prior Analytics represents the first formal study of logic, where logic is understood as the study of arguments. An argument is a series of true or false statements which lead to a true or false conclusion. In the Prior Analytics, Aristotle identifies valid and invalid forms of arguments called syllogisms. A syllogism is an argument that consists of at least three sentences: at least two premises and a conclusion. Although Aristotle does not call them "categorical sentences", tradition does; he deals with them briefly in the Analytics and more extensively in On Interpretation. Each proposition of a syllogism is a categorical sentence which has a subject and a predicate connected by a verb. The usual way of connecting the subject and predicate of a categorical sentence as Aristotle does in On Interpretation is by using a linking verb e.g. P is S. However, in the Prior Analytics Aristotle rejects the usual form in favour of three of his inventions:

P belongs to S
P is predicated of S
P is said of S

Aristotle does not explain why he introduces these innovative expressions but scholars conjecture that the reason may have been that it facilitates the use of letters instead of terms avoiding the ambiguity that results in Greek when letters are used with the linking verb. In his formulation of syllogistic propositions, instead of the copula, Aristotle uses the expression, "... belongs to/does not belong to all/some..." or "... is said/is not said of all/some..." There are four different types of categorical sentences: universal affirmative, universal negative, particular affirmative and particular negative.

A - A belongs to every B
E - A belongs to no B
I - A belongs to some B
O - A does not belong to some B

A method of symbolization that originated and was used in the Middle Ages greatly simplifies the study of the Prior Analytics.
Following this tradition then, let:
Categorical sentences may then be abbreviated as follows:
From the viewpoint of modern logic, only a few types of sentences can be represented in this way.

Basics

The fundamental assumption behind the theory is that the formal model of propositions are composed of two logical symbols called terms – hence the name "two-term theory" or "term logic" – and that the reasoning process is in turn built from propositions:

The term is a part of speech representing something, but which is not true or false in its own right, such as "man" or "mortal". As originally conceived, all terms would be drawn from one of ten categories enumerated by Aristotle in his Organon, classifying all objects and qualities within the domain of logical discourse.
The formal model of proposition consists of two terms, one of which, the "predicate", is "affirmed" or "denied" of the other, the "subject", and which is capable of truth or falsity.
The syllogism is an inference in which one proposition follows of necessity from two other propositions.

A proposition may be universal or particular, and it may be affirmative or negative. Traditionally, the four kinds of propositions are:

A-type: Universal and affirmative
E-type: Universal and negative
I-type: Particular and affirmative
O-type: Particular and negative

This was called the fourfold scheme of propositions. Aristotle's original square of opposition, however, does not lack existential import.
The kinds of propositions used for each of the premises and the conclusion combine to create the "mood" of the syllogism. For instance, if the major premise, minor premise, and conclusion are all universal affirmative statements, then the syllogism has the mood AAA.

Term

A term is the basic component of the proposition. The original meaning of the horos is "extreme" or "boundary". The two terms lie on the outside of the proposition, joined by the act of affirmation or denial.
For early modern logicians like Arnauld, it is a psychological entity like an "idea" or "concept". Mill considers it a word. To assert "all Greeks are men" is not to say that the concept of Greeks is the concept of men, or that word "Greeks" is the word "men". A proposition cannot be built from real things or ideas, but it is not just meaningless words either.

Major and minor terms

The "major term" is the predicate of the syllogism's conclusion; the "minor term" is the subject of the syllogism's conclusion.

Singular terms

For Aristotle, the distinction between singular and universal is a fundamental metaphysical one, and not merely grammatical. A singular term for Aristotle is primary substance, which can only be predicated of itself: "Callias" or "Socrates" are not predicable of any other thing, thus one does not say every Socrates one says every human. It may feature as a grammatical predicate, as in the sentence "the person coming this way is Callias". But it is still a logical subject.
He contrasts universal secondary substance, genera, with primary substance, particular specimens. The formal nature of universals, in so far as they can be generalized "always, or for the most part", is the subject matter of both scientific study and formal logic.

Proposition

In term logic, a "proposition" is simply a form of language: a particular kind of sentence, in which the subject and predicate are combined, so as to assert something true or false. It is not a thought, nor an abstract entity. The word "propositio" is from the Latin, meaning the first premise of a syllogism. Aristotle uses the word premise as a sentence affirming or denying one thing or another, so a premise is also a form of words.
However, as in modern philosophical logic, it means that which is asserted by the sentence. Writers before Frege and Russell, such as Bradley, sometimes spoke of the "judgment" as something distinct from a sentence, but this is not quite the same. As a further confusion the word "sentence" derives from the Latin, meaning an opinion or judgment, and so is equivalent to "proposition".
The logical quality of a proposition is whether it is affirmative or negative. Thus every philosopher is mortal is affirmative, since the mortality of philosophers is affirmed universally, whereas no philosopher is mortal is negative by denying such mortality in particular.
The quantity of a proposition is whether it is universal or particular. In case where existential import is assumed, quantification implies the existence of at least one subject, unless disclaimed.

Syllogisms

A syllogism comprises a conclusion derived from two premises. The essential feature of the syllogism is that, of the three terms in the two premises, one must occur twice: this is called the "middle term", and it does not appear in the conclusion. The premise that contains the middle term and the major term is called the "major premise". The premise that contains the middle term and the minor term is called the "minor premise". For instance:
Depending on the roles of the middle term in each of the premises, Aristotle divides the syllogism into three kinds: syllogism in the first, second, and third figure. If the Middle Term is subject of one premise and predicate of the other, the premises are in the First Figure. If the Middle Term is predicate of both premises, the premises are in the Second Figure. If the Middle Term is subject of both premises, the premises are in the Third Figure. The Fourth Figure, in which the middle term is the predicate in the major premise and the subject in the minor, was added by Aristotle's pupil Theophrastus and does not occur in Aristotle's work, although there is evidence that Aristotle knew of fourth-figure syllogisms.
Symbolically, Aristotle's Three Figures may be represented as follows:

	First figure	Second figure	Third figure
	Predicate — Subject	Predicate — Subject	Predicate — Subject
Major premise	A ------------ B	B ------------ A	A ------------ B
Minor premise	B ------------ C	B ------------ C	C ------------ B
Conclusion	A ********** C	A ********** C	A ********** C