Term Logic - The Basics

The Basics

The fundamental assumption behind the theory is that propositions are composed of two 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".
• The proposition consists of two terms, in which one term (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 (the "conclusion") follows of necessity from two others (the "premises").

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 ("Every philosopher is mortal")
• I-type: Particular and affirmative ("Some philosopher is mortal")
• E-type: Universal and negative ("Every philosopher is immortal")
• O-type: Particular and negative ("Some philosopher is immortal")

This was called the fourfold scheme of propositions (see types of syllogism for an explanation of the letters A, I, E, and O in the traditional square). Aristotle's original square of opposition, however, does not lack existential import:

• A-type: Universal and affirmative ("Every philosopher is mortal")
• I-type: Particular and affirmative ("Some philosopher is mortal")
• E-type: Universal and negative ("Not every philosopher is mortal")
• O-type: Particular and negative ("No philosopher is mortal")

In the Stanford Encyclopedia of Philosophy article, "The Traditional Square of Opposition", Terence Parsons explains:

One central concern of the Aristotelian tradition in logic is the theory of the categorical syllogism. This is the theory of two-premised arguments in which the premises and conclusion share three terms among them, with each proposition containing two of them. It is distinctive of this enterprise that everybody agrees on which syllogisms are valid. The theory of the syllogism partly constrains the interpretation of the forms. For example, it determines that the A form has existential import, at least if the I form does. For one of the valid patterns (Darapti) is:

Every C is B

Every C is A

So, some A is B

This is invalid if the A form lacks existential import, and valid if it has existential import. It is held to be valid, and so we know how the A form is to be interpreted. One then naturally asks about the O form; what do the syllogisms tell us about it? The answer is that they tell us nothing. This is because Aristotle did not discuss weakened forms of syllogisms, in which one concludes a particular proposition when one could already conclude the coresponding universal. For example, he does not mention the form:

No C is B

Every A is C

So, some A is not B

If people had thoughtfully taken sides for or against the validity of this form, that would clearly be relevant to the understanding of the O form. But the weakened forms were typically ignored...

One other piece of subject-matter bears on the interpretation of the O form. People were interested in Aristotle's discussion of “infinite” negation, which is the use of negation to form a term from a term instead of a proposition from a proposition. In modern English we use “non” for this; we make “non-horse,” which is true of exactly those things that are not horses. In medieval Latin “non” and “not” are the same word, and so the distinction required special discussion. It became common to use infinite negation, and logicians pondered its logic. Some writers in the twelfth and thirteenth centuries adopted a principle called “conversion by contraposition.” It states that

• ‘Every S is P’ is equivalent to ‘Every non-P is non-S’
• ‘Some S is not P’ is equivalent to ‘Some non-P is not non-S’

Unfortunately, this principle (which is not endorsed by Aristotle) conflicts with the idea that there may be empty or universal terms. For in the universal case it leads directly from the truth:

Every man is a being

to the falsehood:

Every non-being is a non-man

(which is false because the universal affirmative has existential import, and there are no non-beings). And in the particular case it leads from the truth (remember that the O form has no existential import):

A chimera is not a man

to the falsehood:

A non-man is not a non-chimera

These are Buridan's examples, used in the fourteenth century to show the invalidity of contraposition. Unfortunately, by Buridan's time the principle of contraposition had been advocated by a number of authors. The doctrine is already present in several twelfth century tracts, and it is endorsed in the thirteenth century by Peter of Spain, whose work was republished for centuries, by William Sherwood, and by Roger Bacon. By the fourteenth century, problems associated with contraposition seem to be well-known, and authors generally cite the principle and note that it is not valid, but that it becomes valid with an additional assumption of existence of things falling under the subject term. For example, Paul of Venice in his eclectic and widely published Logica Parva from the end of the fourteenth century gives the traditional square with simple conversion but rejects conversion by contraposition, essentially for Buridan's reason. —Terence Parsons, The Stanford Encyclopedia of Philosophy