**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 à
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 à Male]. To say that P à 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 à Q and Q à P, which is abbreviated
P ó
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 à
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 à Q. Thus, for example, being a father is a
sufficient condition for being male, because Father à 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 à
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 à ‑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 ó 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 pà 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:

__[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.