Deduction, Induction, & Truth Kareem Khalifa Department of Philosophy Middlebury College.

Deduction, Induction, & Truth

Kareem Khalifa

Department of Philosophy

Middlebury College

Overview

• The Central Issue

• Deductive Validity

• Inductive Strength

• Deductive Validity vs. Inductive Strength

• Validity vs.Truth

• Exercises

The Central Issue

• Recall: an argument is a set of propositions such that one member of that set, the conclusion, can be affirmed on the basis of the others, the premises.

• What does it mean for a proposition to be “affirmed on the basis” of other propositions?

Deductive Validity: The Gold Star

• If the premises guarantee the truth of the conclusion, then the conclusion can always be affirmed on the basis of the premises.

• In other words, there is no way that the premises could be true and the conclusion could be false. That’s a guarantee!

Example of a deductively valid argument

• Premise: If Khalifa is a mammal, then Khalifa is warm-blooded.

• Premise: Khalifa is a mammal.

• Conclusion: Khalifa is warm-blooded.

Deductive Validity: The Official Definition

• A deductive argument is valid when, if all of its premises are true, its conclusion must be true.

• This is the SINGLE MOST IMPORTANT CONCEPT IN THE CLASS!!!!!!!!!

• Failure to define validity properly is an automatic 5 point penalty on anything you do! You’ll also be very confused if you don’t get this concept.

Inductive Strength

• So, deductive validity describes one way in which a conclusion can be affirmed on the basis of its premises: the iron-clad guarantee.

• However, we have many good arguments that do not provide such guarantees, for example…

• All observed peaches have pits. So all peaches have pits.

• Previously, when I flip the switch, the light goes on. So the next time I flip the switch, the light will go on.

• My parents have told me my name is Kareem Khalifa. So my name really is Kareem Khalifa.

• There is a strong correlation between smoking and lung cancer. So smoking causes lung cancer.

Deductive validity versus inductive strength

• Recall: A deductive argument is valid when, if all of its premises are true, its conclusion must be true.

• Compare: An inductive argument is strong when, if all of its premises are true, its conclusion is probably true.

Why would we ever settle for inductive arguments?

• Deductive arguments require certainty; but we often have to reason with incomplete information.

• Conclusions of deductive arguments contain no new information over and above their premises; we often have to reason in order to gain further information.

• Much of our reasoning is sensitive to background knowledge. Only inductive reasoning allows us to adapt our reasons to changes in our background knowledge.

• These considerations need to be weighed against the inherent risk involved in inductive inference.

Invalid vs. Inductively strong arguments

• Deductive validity: no way the premises are true and the conclusion is false.

• Deductive invalidity: some way the premises are true and the conclusion is false.

• Inductively strong: unlikely that the premises are true and the conclusion is false.

• So inductively strong arguments are deductively invalid arguments. How do we distinguish them?

Possible replies

• Copi & Cohen: Deductive arguments (valid or invalid) claim to be valid; inductive arguments (strong or weak) claim to be strong. (30)

• My preference: It doesn’t matter. If the argument is invalid, be aware of– How it is possible that conclusion is false when

premises are true; AND– How probable it is that the conclusion is false when

premises are true.• You should be considering this regardless of

what the argument claims to do!

Validity versus truth

• Recall: a deductive argument is valid when, if its premises are true, its conclusion must be true.

• This does not say that valid arguments actually have true premises or true conclusions.

• Validity only concerns the connection between premises and conclusion. But weak things can be connected by something strong.

Implications for logic

• Propositions are true/false; arguments are valid/invalid.– This is an important conceptual point.

• Deductive logic can tell us if a conclusion necessarily follows from a set of premises, but it cannot tell us if the premises and/or conclusions are true/false.– That’s why there are disciplines other than logic!

• There can be valid arguments with false premises and/or false conclusions.– We saw this last class.

Exercise 1

• Valid, 1 true prem, 1 false prem, false concl.– If Khalifa is a lizard, then Khalifa is a reptile.– Khalifa is a lizard.– Khalifa is a reptile.

Exercise 2

• Valid, 1 true prem, 1 false prem, true concl– If Khalifa is a koala, then Khalifa is a

mammal.– Khalifa is a koala.– So Khalifa is a mammal.

Exercise 3

• Invalid, two true prems, false concl– If Khalifa is a human, then Khalifa is a

mammal.– If Khalifa is a mammal, then Khalifa is warm-

blooded.– So, Khalifa is not a human.

Exercise 4

• Invalid, two true prems, true concl– If Khalifa is a human, then Khalifa is a

mammal.– If Khalifa is a mammal, then Khalifa is warm-

blooded.– Khalifa is right-handed.

Exercise 5

• Valid, with 2 false prems, true concl– If Khalifa is an amoeba, then Khalifa is a

vertebrate.– Khalifa is an amoeba.– So Khalifa is a vertebrate.

Exercise 6

• Invalid, two false prems, true concl– There are exactly two students in PHIL0180.– Middlebury tuition costs two dollars.– 2+2=4.

Exercise 7

• Invalid, 1 true prem, 1 false prem, true concl– If Khalifa is a reptile, then Khalifa is a

vertebrate.– If Khalifa is a vertebrate, then Khalifa is warm-

blooded.– So Khalifa is a vertebrate.

Exercise 8

• Valid, true prems, true concl—called a SOUND argument– If Khalifa is a human, then Khalifa is a

mammal.– Khalifa is a human.– So Khalifa is a mammal.