PHIL 450A: HANDOUT ON REASONING

 

I assume that students taking this course are familiar with basic logical terminology. Items 1-6 on this handout provide a summary of the necessary terminology. Items 7-12 use that terminology to make some distinctions that will be important in this course.

 

1. Properties [Terms] and Propositions [Sentences]. Capital letters, "A", "B", "P", "Q", etc. can stand for properties [expressed by terms, such as "is morally justified"] and propositions [expressed by sentences, such as "Governments ought to guarantee a protected sphere of liberty."]. Small letters ("p", "q", etc.) can also be used to stand for propositions.

 

Complex properties and complex propositions can be constructed out of simpler ones by the following logical operations:

(a). Negation. The negation of male (M) is not male (-M).

(b) Conjunction. The conjunction of brother (B) and father (F) is brother and father (B&F).

(c) Alternation. The alternation of brother (B) and father (F) is brother or father (BvF). A person is a BvF if he is a B or if he is a F or if he is a B and a F.

(d) A conditional is a proposition of the form: If P, then Q.

 

2. Metaphysically Necessary And Metaphysically Contingent Truths and Falsehoods. If a proposition is such that it could not be false, then the proposition is a metaphysically necessary truth. If it is the negation of a metaphysically necessary truth, it is a metaphysically necessary falsehood. In this course, whenever we use the term "necessary" without modification, it will be assumed that we are speaking of metaphysical necessity. If a proposition is neither necessarily true nor necessarily false, then it is contingent (either contingently true or contingently false).

 

3. Implication and Necessary Equivalence. I say that one property [or proposition] P implies another property [or proposition] Q [P Q], just in case: (a) [for properties] it is not possible for something to have property P without having property Q; or (b) [for propositions] it is not possible for P to be true unless Q is also true (i.e., it is not possible that P & -Q). For example, the proposition that I am a father implies the proposition that I am male, because the property of being a father implies the property of being male [Father Male]. To say that P Q is to say that the conditional [If P, then Q] is necessarily true.

I will say that one property [or proposition] P is necessarily equivalent to another property [or proposition] Q, just in case P Q and Q P, which is abbreviated P Q. Explain why being a father is not necessarily equivalent to being a male.

 

4. Counterexamples. The usual strategy for attempting to show a conditional to be false it so provide a counterexample to it. To find a counterexample to [If Male, then Father] it suffices to show that there is something that is a male but not a father. You should easily be able to think of someone who is a counterexample to [If Male, then Father]. In general, a counterexample to a conditional [If P, then Q] is something that exists that is P but not Q. Philosophers spend a lot of time attempting to construct counterexamples to claims of implication. To construct a counterexample to the claim of implication P Q, it is not necessary to show that there actually exists something that is P but not Q. It is enough to show that it is possible that there be something that is P but not Q. Thus, for example, even if all crows that ever exist are black, if it is possible for there to be an albino crow, then being a crow would not imply being black.

 

 

 

 

 

5. [Metaphysically] Necessary/Sufficient Conditions.

 

(a) Sufficient Condition. A property [or proposition] P is a [metaphysically] sufficient condition for a property [or proposition] Q, just in case P implies Q--that is, P  Q. Thus, for example, being a father is a sufficient condition for being male, because Father Male.

 

(b) Necessary Condition. A property [or proposition] Q is a [metaphysically] necessary condition for a property [or proposition] P, just in case P implies Q--that is, P Q. For example, being male is necessary for being a father. [An equivalent way of thinking of a necessary condition is the following: If Q is necessary for P, then the negation of Q implies the negation of P--that is, ‑Q ‑P (e.g., Because anything that is not male is not a father, being male is a necessary condition for being a father).] Note that if P is sufficient for Q, then Q is necessary for P.

 

(c) Necessary and Sufficient Condition. A property [or proposition] P is a [metaphysically] necessary and sufficient condition for a property [or proposition] Q, just in case P and Q are necessarily equivalent (i.e., P Q). For example, having 5 things is necessary and sufficient for having 12-7 things.

 

6. Definitions. A definition of a term T states its meaning in more basic terms. Example: sister =df female sibling. An acceptable definition of a term T provides necessary and sufficient conditions for T. But not every necessary and sufficient condition for a term T provides a definition of T. Use the example of 5 = 12-7 to explain why not.

 

7. Logical Analysis. A complete logical analysis of a property P gives necessary and sufficient conditions for P. A partial logical analysis of a property P gives a necessary condition for P, or a sufficient condition for P. Theaetetus proposed to define knowledge as true belief. By the end of the first week, you should be able to explain why this proposal fails as a complete logical analysis of knowledge. Is it at least a partial logical analysis of knowledge?

 

8. Deductive Implication. Suppose you are given a set of premises and a conclusion. Whenever the logical form of the premises and the conclusion is such that it is not possible for the premises all to be true and the conclusion to be false, we will say that the premises deductively imply the conclusion. Whenever a group of premises deductively imply a given conclusion, we will say that the combination of premises and conclusion is a valid deduction. Here is an example of a valid deduction:

 

PREMISES: EP1. Justified, True, Belief that p Knowledge that p

PEJ1. I am justified in believing that I exist and

PEJ 2. I exist is true.

CONCLUSION: PEJ3. I know that I exist.

 

9. Top-Down Reasoning: Proceeds in the direction of deductive implication, from the premises to the acceptance of the conclusion. In many cases, the premises will include an epistemic principle (e.g., EP1 above) and the conclusion will be an epistemic judgment about a particular case, a particular epistemic judgment (e.g., PEJ2 above).

 

10. Bottom-Up Reasoning: Begins with judgments about particular actual and hypothetical cases and uses them to support the principles or generalizations that best explain those particular judgments. Bottom-Up reasoning supports principles or generalizations that explain our particular judgments and undermines moral principles that do not.


Here is an example of Bottom-Up Reasoning:

 

All crows are black

[Other ancillary premises that need not be specified here]

DEDUCTIVELY IMPLY:

The reports of sightings of crows have all been sightings of something black.

 

In this example, the truth of the conclusion of the argument provides some support for the generalization (that all crows are black) that explains it. Though valid deductive valid arguments play a role in Bottom-Up reasoning of this kind, such reasoning is not itself deductively valid. Why not? (Hint: Does the conclusion of the preceding deductive argument (that the reports of sightings of crows have all been sightings of something black) deductively imply that all crows are black?) This sort of reasoning is often referred to as inductive reasoning.

 

Similarly, in epistemology, someone might hold that the claim that justified true belief is sufficient for knowledge is itself a hypothesis supported (until relatively recently) by the fact that no one has ever come across or been able to imagine a case of justified, true belief that was not knowledge. On this sort of account, the epistemic principle that being a justified, true belief is a sufficient condition for knowledge would have been justified (until relatively recently) by Bottom-Up reasoning from judgments about actual and imagined cases: Because of the analogy with inductive reasoning, I refer to this kind of support for a logical analysis (partial or complete) of a term as quasi-inductive.

 

11. The Proof Paradigm. For most of its history, Western philosophy has assumed that reasoning fits the Proof Paradigm. On the Proof Paradigm, we must have an infallible source of knowledge of the premises of our reasoning. Then reasoning about particular cases is Top-Down, from principles to particular judgments.

 

12. Equilibrium Model of Reasoning. Equilibrium Reasoning is both Top-Down and Bottom-Up. In Equilibrium Reasoning, our main reason for accepting an epistemic principle is usually that it seems to provide a good explanation of particular cases. When we accept an epistemic principle on this basis, we can then reason Top-Down from that principle to a particular epistemic judgment, but the epistemic principle is not regarded as infallible. If we discover a particular epistemic judgment that the principle conflicts with, we must either give up the particular moral judgment or give up the principle. The decision about which to give up is based on what makes the most sense.

In this course we don't prove anything. We use equilibrium reasoning to try to find principles that explain our judgments about particular actual and hypothetical cases. When someone proposes such a principle, we consider its deductive implications and try to find counterexamples to it. If we decide that there is a counterexample to a proposed principle, we don't give up trying to find an adequate principle. We use counterexamples as clues to help us formulate better explanatory principles.