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.