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Why do we need logic?
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### Logic Slides

By D. McManaman

Logic – Basic Terms

Logic: the study of how to reason well.

Validity: Valid thinking is thinking in conformity with the rules. If the premises are true and the reasoning is valid, then the conclusion will be necessarily true.

Non-sequitur: (it does not follow). This means that the proposed conclusion cannot be deduced with certitude from the given premises.

For example:If Jews and Palestinians were of the same religion, there wouldn’t be conflict in the Middle East. Therefore, it is religion that is the source of the conflict.

The categorical proposition: A complete sentence, with one subject and one predicate, that is either true or false.

For example

All cows are smelly

The Subject: that about which something is said.

All giraffes are animals. (giraffes = subject)

The Predicate: that which is said about something.

All giraffes are animals. (animals = predicate)

The copula: connects together or separates the S and the P.

All giraffes are animals. (is/is not)

Standard Propositional Codes.

These codes come from the Latin words "Affirmo" and "Nego".

Affirmo: I affirm. Note the A and the I

Nego: I deny. Note the E and the O

A - universal affirmative: All S is P

I - particular affirmative: Some S is P

O - particular negative: Some S is not P.

E - universal negative: No S is P.

The parts of a categorical syllogism:

a.  The two premises.

All A is B (first premise also known as major premise)Some B is C (second premise aka minor premise)Therefore, Some C is A

b.  The Conclusion.

In the above syllogism, Therefore, Some C is A

The major term: this term is always the P (predicate) of the conclusion.  In the example directly above, A is the major term.

The minor term: this term is always the S (subject) of the conclusion.  In the example directly above, C is the minor term.

The middle term: this term is never in the conclusion but appears twice in the premises. (The function of the middle term is to connect together or keep apart the S and P in the conclusion).

Distribution:This is a very important term in logic. A distributed term covers 100% of the things referred to by the term. An undistributed term covers less than 100% of the things referred to by the term (few, many, almost all).

For instance, All men are mortal.

In this statement, "men" is distributed; for it covers 100% of the things referred by the term "men".

In Some men are Italian, "men" is undistributed; for the term covers less than 100% of the things referred to by the term "men".

Universal Affirmative statements (A statements): the subject is distributed, the predicate is undistributed.

Universal Negative statements (E statements): both the subject and the predicate are distributed.

Particular Affirmative statements (I statements): neither subject nor predicate is distributed (both are undistributed).

Particular Negative statements (O statements): the predicate alone is distributed.

A = All S is P

E = No S is P

I = Some S is P

O = Some S is not P

Note the following (bold and underline = distributed):

Rules of Syllogistic (categorical) reasoning.

In a valid categorical syllogism, the middle term must be distributed at least once.

In a valid categorical syllogism, any term which is distributed in the conclusion must also be distributed in the premises.

A syllogism must have three and only three terms.

Rules of Syllogistic (categorical) reasoningcontinued

From two negative premises, no conclusion can be drawn.

If a premise is particular, the conclusion must be particular.

If a premise is negative, the conclusion must be negative.

Examples of violations

• From two negative premises, no conclusion can be drawn.
• No dogs are cows
• No cows are pigs
• Therefore, no dogs are pigs.
• If a premise is particular, the conclusion must be particular.
• Some Italians are from Calabria.
• All Italians love spaghetti
• Therefore, all those from Calabria love spaghetti.

In a valid categorical syllogism, the middle term must be distributed at least once.

• All Germans love beer
• All Irishmen love beer
• Therefore, all Irishmen are Germans.
• In a valid categorical syllogism, any term which is distributed in the conclusion must also be distributed in the premises.
• No secretary is a principal.

A syllogism must have three and only three terms.

• All Italians like soccer.
• Therefore, some Canadians like soccer.

If a premise is particular, the conclusion must be particular.

Some men are AmericanAll Americans love apple pieTherefore, all men love apple pie.

• Some men are American
• All Americans love apple pie
• Therefore, all men love apple pie.
• If a premise is negative, the conclusion must be negative.
• Some Canadians are not hockey players.
• Some hockey players are professionals
• Therefore, some professionals are Canadian.

2. Determine what kind of statement is the first premise (I.e., A statement, E statement, etc.)

3. Determine what kind of statement is the second premise. (I.e., A statement, E statement, etc.)

4. Determine what kind of statement is the conclusion. (I.e., A statement, E statement, etc.)

5. Place a “d” above all your distributed terms

6. Check to see if your middle term is distributed at least once. If it is, move on to #7.

7. Check your major and minor terms in the conclusion. If one of them is distributed, see if that term is distributed in the premises.

8. Check to see if any other rule is violated. If not, you have a valid syllogism.

Hypothetical Syllogisms
• affirming the antecedent
• If A then B
• A
• Therefore B
• affirming the consequent
• If A then B
• B
• Therefore A
• denying the antecedent
• If A then B
• Not A
• Therefore not B
• denying the consequent
• If A then B
• Not B
• Therefore not A

For example, consider whether this conclusion follows from the given premises:

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie eats cake every day.

Therefore, Johnnie is placing himself at risk for diabetes.

If you think it is valid, you are correct

Consider the following

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie does not eat cake every day.

Therefore, Johnnie is not placing himself at risk for diabetes.

Valid or invalid?

Invalid: He might drink pop every day.

Or, the following:

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie is placing himself at risk for diabetes.

Therefore, Johnnie is eating cake every day.

Invalid: He might be drinking pop every day, or eating chocolate bars, etc.

Or,

If Johnnie eats cake every day, then he is placing himself at risk for diabetes.

Johnnie is not placing himself at risk for diabetes.

Therefore, Johnnie is not eating cake every day.

Valid?

It is valid

A note on mathematical logic

The logic we’ve been studying is called intentional logic, or Aristotelian logic. This logic is qualitatively different than symbolic or mathematical logic. The two are discontinuous. Mathematical logic “submits the object of logic to a thorough mathematicizing treatment. So developed, this modern logic becomes a branch of mathematics without relevance to sciences that are not subalternate to mathematics.” (Joseph Owens)

“That symbolic logic, in its techniques, concepts, or specific propositions, can aid in the solution of any philosoophical problem, is seriously doubted.” M. Weitz, “Oxford Philosophy,” Philosophical Review, LXII, (1953) 221.