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A Brief Lesson in Categorical Logic

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                 A Brief Lesson in Categorical Logic

Part I.

Aristotelian Logic or Categorical Logic is like mathematics in that its conclusions follow the premises by necessity. It is called deductive reasoning. Scientists reason from evidence and try to see patterns and make generalizations. Their reasoning is not by deduction, but induction. They have strong or weak arguments whose conclusions are probably true, and include a margin of error, a level of confidence, or a degree of certainty. Abductive reasoning leads to the best explanation or provides the best explanatory hypothesis.

Analysis tries to understand the whole by examining the parts. Thinking that comprehends by putting something together, requiring organizing and composition is called synthesis .

Aristotle’s four standard form logical statements are

__A_____ _____E____ ____I___ _____O____.

The first is the universal affirmative.  All S are P.

The second is the universal negative.   No S are P.

The third, is the particular positive.  Some S are P.

And fourth, is the particular negative.    Some S are not P.

The value of a proposition is either true or false. (A proposition is a simple sentence in terms of logic.)  A syllogism is either valid or invalid. Immediate inferences are either equivalent or non-equivalent.  When the premises of an argument are true and the form of the argument is valid, we have a sound argument. Inductive arguments are either weak or strong.

The most simple and aesthetic argument discovered by Aristotle having two premises and a conclusion is called the syllogism. For example:

All flowers are plants with petals.   All   M are   P

All roses are flowers.                           All   S are   M.

Therefore all roses are plants with petals:  All  S are   P.

S stands for the Subject term. P stands for the Predicate term. And M stands for the Middle term.

All the inferences and truth values for the proposition “All snakes are reptiles” presented by

If you have truth values of one of the standards sentences then you can either get all of the other truth values or just the contradictory. True universals and false particulars give you all the values, false universals and true particulars give you only the value of the contradictory.

DISTRIBUTION: a proposition distributes a term if it refers to all sub-classes of a class designated by that term; if it does not do so, the term is undistributed in that proposition.

DISTRIBUTION of the terms of the four standard form propositions:

Those terms marked with an asterisk (*) are distributed:

All *S   are  P

No  *S   are *P

Some S   are  P

Some S are not *P.

Four immediate inferences in deductive reasoning are

a) converse, b) obverse, c) inverse, and  d) contra-positive.

Truth Values for the standard form A:

All cats are animals. The proposition – true

All animals are cats. The converse – false

Some animals are cats. Limited Converse – true

No cats are non-animals. The Obverse – true

No non-animals are cats. Converse – true

All non-animals are non-cats. Contra-positive – true

Part II.

Aristotle’s mediated inference is called a syllogism. It has three terms that come up precisely twice each in its two premises and the conclusion. This argument operates by the exclusion of the middle term. (This is not the axiom, however, which refers to the middle between true and false.)

FIGURES AND MOODS OF SYLLOGISMS

BARBARA.           Major premise     A   All    M are    P   VALID

Figure I is valid  Minor Premise     A      All    S are    M

AAA Fig. I         Conclusion               A     All    S are    P

CELARENT.          Major premise     E     No   M are   P     VALID

Figure I is valid  Minor Premise     A     All  S are    M

EAE Fig. I         Conclusion               E        No   S are    P


Note: Syllogisms with an * are not valid when also considering existential import.

Observations: AAA  has the only universal affirmative conclusion and is a unique syllogism.

There are never two particular premises in a valid syllogism.

By inferring the obverse of the major premise and conclusion of AAA it can be reduced to EAE. Because all 24 syllogisms were derived by Aristotle from these two syllogism, therefore all syllogisms can be derived from AAA.

The three rules for the validity of syllogisms:

1. Not more than one negative premise is possible.

2. The middle term has to be distributed at least once.

3. Whatever term is distributed in the conclusion has to be distributed in one of the premises.

Logic can be defined as “the study of methods and principles used to distinguish correct from incorrect reasoning.”[1] Logic means correct thinking. Therefore a logical fallacy is an oxymoron because it is tantamount to saying “right reasoning – blunder in thought.”

The three Laws of Thought: (Axiomatic for Logic)

1. The “Principle of Identity” asserts that if any statement is true, then it is true.

2. The “Principle of Contradiction” asserts that no statement can be both true and false.

3. The “Principle of Excluded Middle” asserts that any statement is either true or false.[2]

(Dr. Peter D. S. Krey for Critical Thinking, Spring, 2006)


[1]From Irving Copi and Carl Cohen, Introduction to Logic, 10th Edition, (Upper Saddle River, NJ: Prentice Hall, 1998), p. 3.

[2]From Irving Copi, Introduction to Logic, 2nd Ed., (Macmillan Company, 1961), p. 271.

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Written by peterkrey

December 17, 2011 at 9:26 pm

Posted in Logic

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