The Logic of Aristotle, and beyond.

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Logic is the study of the principles and methods of argumentation. An argument in logic is a set of statements. Some of the statements serve as premises, (or statements of evidence,) and others serve as the conclusions that can be drawn from the premises.

A syllogism is the most common type of argument form in deductive logic.
Aristotle, the famous Greek philosopher, first developed the syllogism. On the whole, his formulation of it has remained unchanged. Often called "The Father of Logic," he taught that the syllogism was the main instrument for reaching scientific conclusions. Logic studies the forms of arguments and the rules which distinguish those which are valid, or correct, from those which are invalid, or incorrect. For example, we may validly argue as follows:

   All men are mortal.   (First Statement) 
   Socrates is a man.    (Next Statement)
   Socrates is mortal.   (Conclusion Statement)

But the following argument is invalid:
   All weeds are plants.       (These are both)
   The flower is a plant.   (examples of)
   All weeds are flowers.      (syllogisms)

   (As in arithmetic addition, 
   a line is drawn with the result or conclusion below.)
The correctness of an argument of the type illustrated depends on its form, not on the actual truth or falseness of the premises.

Kinds of Logic

Logic tells us what would be true if the premises were true. In deductive logic, the conclusion is a necessary consequence of the premises. In inductive logic, the conclusion is only more or less probable on the basis of the premises.

Deductive Logic.

Some rules that describe valid deductive arguments are:

(1) If statement A implies statement B, and if B implies C, then A will imply C. For example, if the statement, "It is raining," implies "The ground is wet," and if "The ground is wet" implies "The rivers will be swollen," then "It is raining" implies "The rivers will be swollen."

(2) If statement A implies statement B, and if we can show that A is true, then B will also be true. For example, the statement that 6 is an even number implies that 6 is divisible by 2. If we know that 6 is an even number, we can then assert that 6 is divisible by 2.

(3) If statement A implies statement B, and if we can show that B is false, then we know A is false also. For example, the statement is made that "Today is Wednesday." This implies that tomorrow is Thursday. If we know that tomorrow is not Thursday, then we can infer that today is not Wednesday.

The most common type of argument form in deductive logic is the syllogism. A syllogism consists of two premises and a conclusion made up of statements of the following forms: All A is B, some A is B, some A is not B, and no A is B. The rules for a valid syllogism include:

(1)A syllogism must have exactly three terms. For example, a person gives the syllogism: "All laws are made by Congress. V=at is a law of falling bodies. Therefore, V=at was made by Congress." This syllogism is invalid because the word law is ambiguous, or obscure. Law can mean a physical law, such as the law of falling bodies. Or it can mean a legislative law. Therefore, this syllogism contains four terms instead of three.

(2) Two negative premises yield no conclusion.

(3) Two positive premises yield a positive conclusion.

(4) From a positive and a negative premise, only a negative conclusion follows. For example, "All fish do not live on land. Some turtles live on land. Therefore, some turtles are not fish."

(5) The term that occurs in both premises must be modified by the words all or none at least once. For example, "All books printed in 1660 are valuable. These books were printed in 1660. Therefore, these books must be valuable." This is a valid syllogism because the form is correct, although the first premise is not necessarily true.

(6) A term that occurs in the conclusion modified by all or none must also be modified by all or none in the premises.

Inductive Logic. Conclusions are based upon premises, so logic also studies the grounds for belief This part of logic is called inductive logic. For example, a person might infer from the statement, "All men are courageous, and courageous men are gentle," the conclusion that "All men are gentle." One could then logically ask, "What are the grounds for your belief that all men are courageous?"

Grounds for belief may be based on generalizations, analogies, or causal connections. Scientific experimentation is a way of controlling observations in order to secure reliable grounds for belief. Experiments may be set up to test each of the grounds for belief.

A person may observe the planet Mars in a number of positions in the sky and infer that it is moving in an elliptical path. This is a generalization, or a general principle which makes an assertion about all members of a class of objects.

A person may argue that student A is good at mathematics because he is like student B in temperament. He argues using an analogy, or a comparison of two or more things which agree in some respects.

A person may observe that he gets restless as the temperature rises. He may then draw the conclusion that the heat makes him restless. He is arguing from a causal connection, or from cause and effect.

Symbolic Logic

In recent years, students of logic have begun to use symbols instead of words to stand for logical units. The result is called symbolic logic. Logicians, or scholars who study logic, have used symbolic logic in order to make deductive logic a purely mechanical procedure like mathematics. Symbolic logic has extended the knowledge of logic a great deal.
Symbolic logic works in this manner:
Instead of writing;
"If A is true and A implies B, then B is true,"
a logician writes;
The → means implies, or;
if ________________then________________
Instead of writing...
"All men are courageous,"
the logician writes...
The symbol (x) means for all x,
the symbol Mx means x is a man,
and Cs means C is courageous.
This brings out the meaning of "All men are courageous."
But the proof that John, who is a man, is courageous, takes this form:
(x)(Mx→Cx) and Mj(John is a man)→Cj.
The rule states that we can replace the variable x by a constant in the form (x)(Mx→Cx)
and Mj→Cj.
Next, Truth Tables

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