Deduction, Validity, Soundness

Lecture II – 01/25/11

Necessary and Sufficient Conditions Revisited!

X is a sufficient condition for Y if X is all that is needed for the occurrence of Y.

Being President is a sufficient condition for being a U.S. citizen.

Y is a necessary condition for X if X cannot occur without the occurrence of Y.

Being an animal is a necessary condition for being a tiger.

Necessary and Sufficient Conditions Revisited!

X is a sufficient condition for Y if X is all that is needed for the occurrence of Y.

If Bill is President, then Bill is a U.S. citizen.If Bill is X, then Bill is Y.You cannot flip X and Y here because If Bill is a U.S.

Citizen then Bill is President is asserting the (crazy) hypothetical that all U.S. citizens are the President. BUT, all Presidents are U.S. citizens, so X is a sufficient condition for Y.

Concerning this conditional, we can also say that being a U.S. citizen is a necessary condition of being president, though, again, not a sufficient condition.

Necessary and Sufficient Conditions Revisited!

Y is a necessary condition for X if X cannot occur without the occurrence of Y.

If Molly is a tiger, then Molly is an animal.If Molly is X, then Molly is Y.This expresses the sufficiency of X for Y. It is a correct

way to write a conditional; however, we want to best express the necessity of being an animal to being a tiger.

If Molly is not an animal, then Molly is not a tiger.If Molly is not Y, then Molly is not X.This formulation coincides with our definition.

Deductive Arguments

So, now that we are clear on what an argument is, we can start looking at different sorts of arguments.

Probably the broadest classification of an argument is whether it is deductive or inductive.

A deductive argument is one that claims that its conclusion is certain and necessary. It is impossible for the conclusion of a deductive argument to be false, if the premises are true. There is no hedging, probability or wavering with deductive arguments.

Deductive Arguments

Questions to ask to determine if an argument is deductive:

(1)Are there any special indicator words that tip us off?

(2)What is the strength of the link between the premises and conclusion?

(3)Is the argument a particular instance of a deductive argument form?

Deductive Arguments

(1) Are there any special indicator words that tip us off?

Necessarily, certainly, absolutely and definitely are words that, if they appear in an argument’s conclusion, usually indicate that it is a deductive argument.

However, such words are less frequent than typical conclusion indicators and can appear in inductive arguments, so learn to rely on other methods of identification.

Deductive ArgumentsAll planets are round.Earth is a planet.It follows necessarily that the Earth is round.

No cats are also birds.Ruffin is a cat.Ruffin is not a bird.

The team with the best defense almost always wins the Super Bowl.

The Steelers have the best defense.The Steelers will certainly win the Super Bowl.

Deductive Arguments

(2) What is the strength of the link between the premises and conclusion?

If the conclusion follows necessarily from the premises, then the argument is deductive. In a deductive argument, the premises essentially force you to admit that, if they are true, the conclusion must also be true.

Deductive Arguments

All men are mortal.Socrates is a man.Socrates is mortal.

No men wear dresses.All men wear pants.Some men wear shorts.

Deductive Arguments

(3) Is the argument a particular instance of a deductive argument form?

Certain argument forms, that is, the way certain arguments are structured, always indicate that the argument is deductive.

categorical syllogisms – each statement begins with “all,” “no” or “some”

hypothetical syllogisms – has at least one conditional premise

disjunctive syllogisms – has at least one disjunctive (either/or) premise

Deductive ArgumentsIf I work in a library, then I am surrounded by books.I work in a library.I am surrounded by books.

All 80s metal band members have long hair.All long-haired persons are cool.All 80s metal band members are cool.

Either Max is a mighty hero or he is a wimpy square.Max is not a wimpy square.Max is a mighty hero.

ValidityWe are now able to classify arguments according to their

inferential strength and/or their form.Next, let us figure out how to sort them in terms of merit. The

property that distinguishes a good deductive argument from a bad one is validity.

A valid argument meets one of the following two conditions:(1)If the premises of the argument are true, then the

conclusion has to be true on pain of contradiction. [modal definition]

(2)The argument is an instance of a valid logical form. [formal definition]

If it is possible that the premises are true and the conclusion is false, then the argument is invalid.

ValidityAre the following valid?

All fleas are flies.All flies are fries.All fleas are fries.

If it’s raining outside, then the pavement is wet.The pavement is wet.It’s raining outside.

If I can lift 5,000 lbs., then I am strong.I can lift 5,000 lbs.I am strong.

All Presidents are U.S. citizensI am a U.S. citizenI am president.

Nothing is better than eternal happiness.A grilled cheese sandwich is better than nothing.A grilled cheese sandwich is better than eternal happiness.

The Border

ValidityA property of form

TruthA property of fact

Zero Toleranc

e

Zero Toleranc

e

Consider the following…

If I am a logic professor, then I am a ham sandwich.

I am a logic professor.----------------------I am a ham sandwich.

All gleebs are gloobs.

All gloobs are gwergs.

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All gleebs are gwergs.

Titus is a boxer mix.

Titus is brown.

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Titus loves people.

All women have blond hair.

Superman is a woman.

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Superman has blond hair.

Validity and UniversalityAgain, validity is a matter of form or structure rather than truth value. Knowing this, we can remove the particular content of any valid argument and be left with a universally

valid standard argument form.

If I can lift 5,000 lbs., then I am strong.I can lift 5,000 lbs.I am strong.

If X, then Y.XY

All gleebs are gloobs.

All gloobs are gwergs.All gleebs are gwergs.

All A are BAll B are CAll A are C

No matter what you allow the variables to stand for, these arguments will be valid because the reasoning process is valid. The structure of the argument is such that it is, always, everywhere and at any time, valid in any and all of its particular instances.

SoundnessAn argument is sound if it is (a) valid and (b) has all true premises, unless the false premise is superfluous to the sound

reasoning process. Are the following sound?

All birds have wings.All airplanes have wings.Some birds are airplanes.

All men are mortal Socrates is a man.Socrates is mortal.

If I am a monkey, then I wear glasses.I am indeed a monkey.I wear glasses.

Either today is Tuesday or it is another day of the week.Today is not another day of the week.Pigs fly.Today is Tuesday.

Deductive Arguments

Valid ArgumentsSound Arguments