“Greatest Hits” of Formal Logic

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“Greatest Hits” of Formal Logic. Overview of deductive reasoning. Formal Logic is the science of deductive reasoning. Definition: “reasoning from known premises, or premises presumed to be true, to a certain conclusion.” In contrast, most everyday arguments involve inductive reasoning.

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“Greatest Hits” of Formal Logic

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“Greatest Hits” of Formal Logic

Overview of deductive reasoning

Formal Logic is the science of deductive reasoning
• Definition: “reasoning from known premises, or premises presumed to be true, to a certain conclusion.”
• In contrast, most everyday arguments involve inductive reasoning.
• reasoning from uncertain premises to probabalistic conclusions
• “inference-making”
Formal logic cannot establish the truth of the premises. The truth of the premises must be presumed, or taken as a given.

Some premises may be proven or authenticated by scientific testing, reference to external sources, etc.

Some premises may be granted or stipulated by all the parties to an argument

Some premises may have been established as the conclusion of a previous argument

DNA testing and paternity

If a DNA sample is collected and analyzed properly and,

If the DNA is an exact match with the alleged father,

Then that person is the father.

Structural validity versusmaterial truth
There is no middle ground. A deductive argument can’t be “sort of” valid.

By contrast, everyday arguments enjoy degrees of probability--plausible, possible, reasonable, believable, etc.

In deduction, proofs are always valid or invalid.
• The fundamental property of a valid, deductive argument is that IF the premises are true, THEN the conclusion necessarily follows.
• The conclusion is said to be “entailed” in, or contained in, the premises.
• If all pigs have curly tails
• And Nadine is a pig
• Then Nadine has a curly tail
The terms used in a syllogism must be defined precisely
• If the meanings of key terms are vague or ambiguous, or change during the course of a deductive argument, then no valid conclusion may be reached.
• Major premise: All pitchers hold water
• Minor premise: Tom Glavin is a pitcher
• Conclusion: Therefore, Tom Glavin holds water

(the term “pitcher” has two different meanings in this argument, so no valid conclusion can be reached)

Example of a valid deductive argument

major premise: All cats have 9 lives

minor premise: “Whiskers” is a cat

conclusion: Therefore, Whiskers has 9 lives

(Note: it doesn’t matter whether cats really have 9 lives; the argument is premised on the assumption that they do.)

“Validity” versus “Soundness”
• An argument is valid if its structure conforms to the rules of formal logic.
• An argument is sound if it is valid, and its premises are true.
• Thus validity is a prerequisite for soundness, but an argument needn’t be sound to be valid.
• If sound, then valid too
• If valid, not necessarily sound
Example of a valid, but unsound argument

major premise: All cats are pink

minor premise: Felix is a cat

conclusion: Therefore, Felix is pink

(Cats aren’t pink, which makes the first premise untrue. Validity, however, presumes the truth of the premises.)

Example of a valid and sound argument

major premise: Anthrax is not a communicable disease

minor premise: Communicable diseases pose the greatest threat to public health

conclusion: Therefore, anthrax does not pose the greatest threat to public health

(The premises are true and the conclusion is valid, that is, it necessarily follows from the premises)

Validity versus soundness
Syllogistic reasoning

The syllogism is a common form of deductive reasoning.

There are different types of syllogisms

categorical (universal premises)

hypothetical (if-then premises)

disjunctive (either-or premises)

major premise

minor premise

conclusion

Categorical syllogisms rely on universal premises

Example of a valid categorical syllogism:

major premise: All Christians believe Jesus is the son of God.

minor premise: Biff is a Christian.

conclusion: Biff believes Jesus is the son of God.

(Note: validity isn’t affected by whether the premises are true or not. Obviously, other religions don’t accept Jesus as the son of God.)

Hypothetical syllogisms use “if-then” premises

Example of a valid hypothetical syllogism:

Major premise: If Biff likes Babbs, then he’ll ask her to the prom.

Minor premise: Biff likes Babbs,

Conclusion: Therefore, he’ll ask her to the prom.

Disjunctive syllogisms use “either-or” premises

Example of a valid disjunctive syllogism:

Major premise: Either Babbs will get her navel pierced, or she’ll get a tongue stud.

Minor premise: Babbs didn’t get her navel pierced.

Conclusion: Therefore, Babbs got a tongue stud.

Practice syllogism

Major premise: Any creature with six legs is an insect.

Minor premise: . Dr. Gass has six legs.

Conclusion: Therefore, Dr. Gass is an insect.

• What kind of syllogism is this? (categorical, hypothetical, or disjunctive)
• Are the premises true?
• Is the conclusion valid?
• Is the argument sound (true premises and a valid conclusion)

Well-known forms of deductive invalidity
• Affirming the consequent
• Invalid Example:
• If A, then B
• B
• Therefore, A
• Invalid Example:
• Students who plagiarize are expelled from school
• Rex was expelled from school
• Rex must have plagiarized
More deductive invalidity
• Denying the antecedent
• Invalid example:
• If A, then B
• Not A
• Therefore, not B
• Invalid example:
• If you exceed the speed limit, you’ll get a ticket.
• I’m not exceeding the speed limit.
• Therefore, I won’t get a ticket.
More deductive invalidity
• Undistributed middle term:
• Valid example:
• All A are B
• All B are C
• Therefore, all A are C
• Invalid example
• All A are B
• All C are B
• Therefore, all A are C

The middle term, B, must serve as the subject of one premise, and the predicate of another premise, but cannot occur in the conclusion

Undistributed middle term:
• Invalid example:
• All humans need air to breathe
• All dogs need air to breathe
• Therefore, all humans need dogs
All rock stars want to become movie stars

Morton wants to become a movie star

Therefore, Morton must be a rock star

affirming the consequent

denying the antecedent

undistributed middle term

valid syllogism

What, if anything, is wrong with this syllogism?

Anyone who has lived in California for more than a few years has experienced an earthquake

Nadine has lived in California for more than a few years

affirming the consequent

denying the antecedent

undistributed middle term

valid syllogism

What, if anything, is wrong with this syllogism?

Anyone who has tried heroin has tried marijuana

Naomi hasn’t tried heroin

Therefore, Naomi hasn’t tried marijuana

affirming the consequent

denying the antecedent

undistributed middle term

valid syllogism

What, if anything, is wrong with this syllogism?

If A, then B

Not A

Therefore, not B

All Christian fundamentalists are opposed to abortion

affirming the consequent

denying the antecedent

undistributed middle term

valid syllogism

What, if anything, is wrong with this syllogism?