Post on 29-Dec-2015
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.