PHIL 240A:  HANDOUT #3

IMPORTANT LOGICAL TERMS AND ABBREVIATIONS WITH SOME IMPORTANT EXAMPLES:

You will need to be familiar with the following terms and abbreviations:

1.  Properties [Terms] and Propositions [Sentences].  Capital letters, "A", "B", "P", "Q", etc. can stand for properties [expressed by terms, such as "wrong"] and propositions [expressed by sentences, such as "Lying is wrong"].

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

(a).  Negation.  The negation of wrong (W) is not wrong (-W).

(b)  Conjunction.  The conjunction of harmful (H) with wrong (W) is harmful and wrong (H&W).  An act is H&W just in case it is both harmful and wrong.

(c)  Alternation.  The alternation of harmful (H) with wrong (W) is harmful or wrong (HvW).  An act is HvW just in case it is harmful or wrong (or both).

2.  Implication.  I say that one property [or proposition] P implies another property [or proposition] Q

[PQ], just in case:  (a) [for properties] it is not possible for something to have property P without having property Q (i.e., it is not possible for something to be [P & -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] be true.  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).

(1) CHOOSE YOUR OWN EXAMPLE OF TWO PROPERTIES, A AND B, SUCH THAT A IMPLIES B BUT B DOES NOT IMPLY A:  USE THE ARROW [""] TO STATE THE RELATION BETWEEN THE TWO PROPERTIES.

3.  Necessary/Sufficient Conditions.

(a) Sufficient Condition.  A property [or proposition]  P is a 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.

(2) CHOOSE YOUR OWN EXAMPLE OF TWO PROPERTIES, A AND B, SUCH THAT A IS A SUFFICIENT CONDITION FOR B, BUT B IS NOT A SUFFICIENT CONDITION FOR A.  USE THE ARROW TO STATE THE RELATION BETWEEN THE TWO PROPERTIES  [HINT:  CAN YOU USE YOUR ANSWER TO THE PRECEDING QUESTION?]

(b) Necessary Condition.  A property [or proposition] Q is a 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.

(3) CHOOSE YOUR OWN EXAMPLE OF TWO PROPERTIES, A AND B, SUCH THAT A IS A NECESSARY CONDITION FOR B, BUT B IS NOT A NECESSARY CONDITION FOR A.  USE THE ARROW TO STATE THE RELATION BETWEEN THE TWO PROPERTIES.  [HINT:  CAN YOU USE THE TWO PROPERTIES YOU USED IN YOUR ANSWER TO THE PRECEDING QUESTION?.]

(c) Necessary and Sufficient Condition.  A property [or proposition] P is a necessary and sufficient condition for a property [or proposition] Q, just in case P implies Q and Q implies P‑‑which I abbreviate with a double arrow, P ó Q.  Whenever P ó Q, we will say that there is mutual implication, or that P and Q are necessarily equivalent or that P holds (or is true) just in case Q holds (or is true).

Here are some important necessary equivalences among moral terms:

(1) Act A (e.g., telling the truth) is Morally Right (R) ó

Act A is one's Duty (D) ó

One (morally) Ought to do act A ó

Act -A (e.g., not telling the truth) is Morally Forbidden (F) ó

Act -A is Not Morally Permissible (-P) ó

One (morally) Ought not to do act -A ó

Act -A is Morally Wrong (W)

(2) Act A is Morally Permissible (P) ó

It is not the case that it is Morally Wrong to do A ó

Act A is not Morally Forbidden (-F) ó

It is not the case that one (morally) Ought to do -A ó

It is not the case that one has a Duty to do -A

(4) CHOOSE YOUR OWN EXAMPLE OF TWO PROPERTIES, A AND B, SUCH THAT A IS A NECESSARY AND SUFFICIENT CONDITION FOR B.  USE THE DOUBLE-ARROW ["ó"] TO STATE THE RELATION BETWEEN THE TWO PROPERTIES.

Contrapositives.  There is one necessary equivalence that we will make use of more than once in this course, so you should be familiar with it.  As mentioned above, if being a father implies being a male (F à M), then not being male implies not being a father (-M à -F).  [-M à -F] is the contrapositive of [F à M].  Any implication is necessarily equivalent to its contrapositive--that is:  [A à B] ó [-B à -A].

(5) GIVE AN EXAMPLE OF AN IMPLICATION AND ITS CONTRAPOSITIVE.  EXPLAIN WHAT IT MEANS TO SAY THAT THEY ARE NECESSARILY EQUIVALENT.

4.  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 must provide NECESSARY AND SUFFICIENT conditions for T.

(6) USE SOME OF THE NECESSARY EQUIVALENCES LISTED ABOVE TO EXPLAIN WHY NOT ALL STATEMENTS OF NECESSARY AND SUFFICIENT CONDITIONS PROVIDE ACCEPTABLE DEFINITIONS:

5.  Counterexamples.  To say that P à Q is to say that it is not possible that something is [P & -Q].  To evaluate such a claim, we consider whether it is possible that something is [P & -Q].  If we come up with a possibility in which something is [P & -Q], that provides a counterexample to the claim that P à Q. Because so many philosophical claims involve claimed implications, much philosophical argument involves the consideration of potential counterexamples to claimed implications Fifth/Sixth Commandment in the Christian Bible:  Thou shalt not kill.  One way of interpreting that commandment is as a claim of implication:

(5/6) Killing à Wrong [K à W].

Suppose you want to argue that (5/6) is false.  The most promising strategy is to attempt to construct a counterexample to it—that is, to describe how it could be possible that [K & -W] (i.e., how it might be possible that an act of killing would not be wrong).

DO YOU BELIEVE THAT THERE ARE ANY COUNTEREXAMPLES TO (5/6)?

6.  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.

WHY IS THIS A NATURAL EXTENSION OF THE NOTION OF IMPLICATION INTRODUCED ABOVE?

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:  MP1.  It is always wrong to kill another person. (This is a Moral Principle)

P2.  I am a person.

P3.  If you shoot me, I will die.

CONCLUSION:  PMJ1.  It is wrong for you to shoot me now (This is a Particular Moral Judgment).

7.  Top-Down Reasoning:  Begins with moral principles and other acceptable premises and uses them to support a moral judgment about a particular actual or hypothetical case (a particular moral judgment).

8.  Bottom-Up Reasoning:  Begins with judgments about particular actual and hypothetical cases and uses them to support the moral principles that best explain those particular moral judgments.  Bottom-Up reasoning supports moral principles that explain our particular moral judgments and undermines moral principles that do not.  So moral principles are never regarded as infallible.

EXPLAIN HOW THE PREVIOUS EXAMPLE OF A VALID DEDUCTION COULD BE UNDERSTOOD AS AN EXAMPLE OF TOP-DOWN REASONING.  EXPLAIN HOW IT COULD BE UNDERSTOOD AS AN EXAMPLE OF BOTTOM-UP REASONING.

Here is an example of how Bottom-Up Reasoning can lead one to give up a moral principle:

PREMISES:  MP1.  It is always wrong to kill another person.  (Moral Principle)

P2.  I am a person.

P3.  If you shoot me, I will die.

P4.  I am trying to kill you.

CONCLUSION:  PMJ2.  It is wrong for you to shoot me now (even though I am trying to kill you).

Let PMJ2' be the judgment that it is not wrong for you to shoot me now if I am trying to kill you.  If you accept PMJ2', you must reject one of the premises of the above valid deduction.  The premise that seems to need revision is the moral principle MP1.

9.  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 fundamental moral principles (e.g., God, our Reason, etc.).  Then reasoning about particular cases is Top-Down, from moral principles to particular moral judgments.

ON THE PROOF PARADIGM, IT COULD NEVER BE RATIONAL TO GIVE UP A MORAL PRINCIPLE.  WHY NOT?

10.  Equilibrium Reasoning.  Equilibrium Reasoning is primarily Bottom-Up.  In Equilibrium Reasoning, our main reason for accepting a moral principle is that it seems to provide a good explanation of particular cases.  When we accept a moral principle on this basis, we can then reason Top-Down from that moral principle to a particular moral judgment, but the moral principle is not regarded as infallible.  If we discover a particular moral 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.

Example of Equilibrium Reasoning:  The previous examples suggest a moral principle that would explain both PMJ1 and PMJ2':

MP2:  Unless it is Done in Self-Defense, It is Always Wrong to Kill Another Human Being: [KHB&-SD] à W

CAN YOU THINK OF A COUNTEREXAMPLE TO MP2?