Psychology Working Outline Assignment
TT24
MorelandCraigChapter12.pdf
2
ARGUMENTATION AND LOGIC
Come now, let us argue it out.
ISAIAH 1:18
1—INTRODUCTION Philosophy, Alvin Plantinga has remarked, is just thinking hard about something. If that is the case, then doing good philosophy will be a matter of learning to think well. That serves to differentiate philosophy from mere emotional expressions of what we feel to be true or hopeful expressions of what we wish to be true. What, then, does it mean to think well? It will involve, among other things, the ability to formulate and assess arguments for various claims to truth. When we speak of arguments for a position, we do not, of course, mean quarreling about it. Rather, an argument in the philosophical sense is a set of statements that serve as premises leading to a conclusion.
Every one of us already employs the rules of argumentation whether we realize it or not. For these rules apply to all reasoning everywhere, no matter what the subject. We use these rules unconsciously every day in normal life. For example: Suppose a friend says to you, “I’ve got to go to the library today to check out a book.” And you reply, “You can’t do that today.” “Why not?” he asks. “Because today is Sunday,” you explain, “and the library isn’t open on Sunday.” In effect, you have just presented an argument to your friend. You have reasoned:
1. If today is Sunday, the library is closed.
2. Today is Sunday.
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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3. Therefore, the library is closed.
Sentences (1) and (2) are the premises of the argument, and sentence (3) is the conclusion. You are saying that if premises (1) and (2) are true, then the conclusion (3) is also true. It is not just your opinion that the library is closed; you have given an argument for that conclusion.
What makes for a good argument? That depends. Arguments may be either deductive or inductive. In a good deductive argument the premises guarantee the truth of their conclusions. In a good inductive argument the premises render the conclusion more probable than its competitors. What makes for a good argument depends on whether that argument is deductive or inductive.
2—DEDUCTIVE ARGUMENTS A good deductive argument will be one that is formally and informally valid, that has true premises, and whose premises, taken together, are more plausible than their contradictories. Let us say a word of explanation about each of these criteria.
First, a good argument must be formally valid. That is to say, the conclusion must follow from the premises in accord with the rules of logic. Logic is the study of the rules of reasoning. Although the word logic is often used colloquially as a synonym for something like “common sense,” logic is, in fact, a highly technical subdiscipline of philosophy akin to mathematics. It is a multifaceted field, consisting of various subfields such as sentential logic, first-order predicate logic, many-valued logic, modal logic, tense logic, and so forth. Fortunately, for our purposes, we need only take a superficial look at the role logic plays in our formulating and assessing simple arguments.
An argument whose conclusion does not follow from the premises in accord with the rules of logic is said to be invalid, even if the conclusion happens to be true. For example,
1. If Sherrie gets an A in epistemology, she’ll be proud of her work.
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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2. Sherrie is proud of her work.
3. Therefore, Sherrie got an A in epistemology.
All three of these statements may in fact be true. But because (3) does not follow logically from (1) and (2), this is an invalid argument. From the knowledge of (1) and (2), you cannot know that (3) is also true. The above is therefore not a good argument.
Second, a good argument will be not only formally valid but also informally valid. As we shall see, there is a multitude of fallacies in reasoning that, while not breaking any rule of logic, disqualify an argument from being a good one—for example, reasoning in a circle. Consider the following argument.
1. If the Bible is God’s Word, then it is God’s Word.
2. The Bible is God’s Word.
3. Therefore, the Bible is God’s Word.
This is a logically valid argument, but few people will be impressed with it. For it assumes what it sets out to prove and therefore proves nothing new. A good argument will not only follow the rules of formal logic but will also avoid informal fallacies.
Third, the premises in a good argument must be true. An argument can be formally and informally valid and yet lead to a false conclusion because one of the premises is false. For example,
1. Anything with webbed feet is a bird.
2. A platypus has webbed feet.
3. Therefore, a platypus is a bird.
This is a valid argument, but unfortunately premise (1) is false. There are animals other than birds that have webbed feet. Therefore, this is not a good argument for the truth of the conclusion. An argument that is both logically valid and has true premises is called a sound argument. An unsound argument is either invalid or else has a false premise.
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Fourth, a good argument has premises that are collectively more plausible than their contradictories (or denials). For an argument to be a good one, it is not required that we have 100 percent certainty of the truth of the premises. If the conjunction of the premises is more plausible than not, then the conclusion of a deductive argument is guaranteed to be more plausible than not.
Some of the premises in a good argument may strike us as only slightly more plausible than their denials; other premises may seem to us highly plausible in contrast to their denials. But so long as the conjunction of the premises is more plausible than not, we should believe in the conclusion of a valid deductive argument.
Thus a good argument for God’s existence need not make it certain that God exists. Certainty is what most people are thinking of when they say, “You can’t prove that God exists!” If we equate “proof” with 100 percent certainty, then we may agree with them and yet insist that there are still good arguments to think that God exists. For example, one version of the axiological argument may be formulated:
1. If God did not exist, objective moral values would not exist.
2. Objective moral values do exist.
3. Therefore, God exists.
Someone may object to premise (1) of our argument by saying, “But it’s possible that moral values exist as abstract objects without God.” We may happily agree. That is epistemically possible, that is to say, the premise is not known to be true with certainty. But possibilities come cheap. The question is not whether the contradictory of a particular premise in an argument is epistemically possible (or even plausible); the question is whether the contradictory is as plausible or more plausible than the premise. If it is not, then one should believe the premise rather than its contradictory.
In summary, then, a good argument will be formally and informally valid and have true premises that are collectively more plausible than not. In order to assist readers in formulating and assessing arguments, we shall now explain each of these features in somewhat more detail.
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2.1 Logically Valid
2.1.1 Sentential Logic 2.1.1.1 Nine Rules of Logic
Sentential or propositional logic is the most basic level of logic, dealing with inferences based on sentential connectives like “if . . . , then,” “or,” and “and.” There are only nine rules of inference readers must learn, along with a few logical equivalences, in order to carry out the reasoning governed by this domain of logic. Equipped with the nine rules, readers will be able to assess the validity of most of the arguments they will ever encounter.
Rule #1: modus ponens
1. P → Q
2. P
3. Q
In symbolic logic one uses letters and symbols to stand for sentences and the words that connect them. In (1) the P and the Q stand for any two different sentences, and the arrow stands for the connecting words, “if . . . , then . . .” To read premise (1) we say, “If P, then Q.” Another way of reading P → Q is to say: “P implies Q.” To read premise (2) we just say, “P.” The reason letters and symbols are used is because sentences that are very different grammatically may still have the same logical form. For example, the sentences “I’ll go if you go” and “If you go, then I’ll go,” though different grammatically, obviously have the same logical form. By using symbols and letters instead of the sentences themselves we can make the logical form of a sentence clear without being distracted by its grammatical form.
The rule modus ponens tells us that from the two premises P → Q and P, we may validly conclude Q. This rule of inference is one that we use unconsciously all the time, as the following examples should make clear.
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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Example 1:
1. If John studies hard, then he will get a good grade in logic.
2. John studies hard.
3. He will get a good grade in logic.
Example 2:
1. If John does not study hard, then he will not get a good grade in logic.
2. John does not study hard.
3. He will not get a good grade in logic.
Notice that our two examples are both valid arguments (they are both in accord with the rule modus ponens), but they reach opposite conclusions. So they cannot both be sound; at least one of them must have a false premise. If we wanted to figure out which one of these examples is a sound argument, we would need to look at the evidence for the premises. Based on John’s past performance, for example, we discover that when he studies hard for a class, he gets a good grade. That gives good grounds for thinking that premise (1) of example 1 is true. Moreover, we observe that John is putting in long hours studying for his logic class. So we have good grounds for thinking that premise (2) of example 1 is true as well. So we have good grounds for thinking example 1 to be a valid argument with true premises. So it is a sound argument for the conclusion that John will in fact get a good grade.
What about example 2? If John were a real genius, it might be the case that he would get a good grade in logic even if he did not study hard. Maybe if he studies hard he will get a good grade, and if he does not study hard he will get a good grade. But we observe, in fact, that John is not that smart. If he does not work hard, he fails to achieve his goals. So we have good reason to believe that premise (1) of example 2 is true. But then we come to premise (2). And this premise is clearly false, for John is no slacker
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but studies hard for his logic class. Therefore, example 2 is not a sound argument because it has a false premise. It is valid but unsound.
Rule #2: modus tollens
1. P → Q
2. ¬Q
3. ¬P
Once again the P and the Q stand for any two sentences, and the arrow stands for “if . . . , then . . .” The sign ¬ stands for “not.” It is the sign of negation. So premise (1) reads, “If P, then Q.” Premise (2) reads, “Not-Q.” The rule modus tollens tells us that from these two premises, we may validly conclude “Not-P.” The following examples should make this rule clear.
Example 1:
1. If Joan has been working out, then she can run the 5K race.
2. She cannot run the 5K race.
3. Joan has not been working out.
Example 2:
1. If it is Saturday morning, then my room mate is sleeping in.
2. My roommate is not sleeping in.
3. It is not Saturday morning.
Modus tollens involves negating a premise. If the premise is already a negation, then we have double negation, which is logically the same as an affirmative sentence. Thus ¬¬Q is equivalent to Q. So from the premises
1. ¬P → Q
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2. ¬Q
we can conclude
3. ¬¬P
which is logically the same as
4. P
In this case the initial conclusion ¬¬P becomes itself a premise from which to draw the further conclusion in (4). Another example of double negation in action would be
1. P → ¬Q
2. Q
In order to use modus tollens we first convert (2) to
3. ¬¬Q
which is the negation of ¬Q. That allows us to use modus tollens to conclude to
4. ¬P
Modus ponens and modus tollens help to bring out an important feature of conditional sentences: The antecedent “if” clause states a sufficient condition of the consequent “then” clause. The consequent “then” clause states a necessary condition of the antecedent “if” clause. For if P is true, then Q is also true. The truth of P is sufficient for the truth of Q. At the same time P is never true without Q: if Q is not true, then P is not true either. So in any sentence of the form P → Q, P is a sufficient condition of Q, and Q is a necessary condition of P.
There are other ways of expressing sufficient and necessary conditions besides the expression “if . . . , then . . .” For example, we frequently express a necessary condition by saying “only if . . .” Your professor says, “Extra credit will be permitted only if you have completed all the required work.” He is saying that completing the required work is a necessary condition of doing extra credit work. Therefore, if we let P = “You may do
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extra credit work” and Q = “You have completed the required work,” we can symbolize his sentence as P → Q. This is tricky because when the beginner sees the words “only if,” he might think that we should symbolize the clause that comes after them as P. But that is incorrect. When he sees the words “only if,” he should think immediately “necessary condition” and realize that he should symbolize what comes afterward as Q.
This distinction between necessary and sufficient conditions is vitally important because ignoring it can lead to great misunderstandings. For example, you might conclude from your professor’s above statement that if you complete the required work, then you may do extra credit work. But that is not, in fact, what he said! He stated a necessary condition of your doing extra credit work, not a sufficient condition. He asserted P → Q, but he did not assert Q → P. There may be other conditions that have to be met as well before one may do extra credit work. So if you concluded on the basis of his statement that you could do extra credit work after completing the required work, you would be guilty of an invalid inference, which might prove ruinous to your grade! So in a sentence, the clause that follows a simple “if” is the antecedent clause symbolized P, a sufficient condition. The clause that follows “only if” is the consequent clause symbolized Q, the necessary condition.
We now draw attention to a very common logical fallacy: Affirming the consequent.
Example 1:
1. If George and Barbara are enjoying soft-boiled eggs, toast, and coffee, then they are having breakfast.
2. George and Barbara are having breakfast.
3. They are enjoying soft-boiled eggs, toast, and coffee.
Example 2:
1. If God is timeless, then he is intrinsically changeless.
2. God is intrinsically changeless.
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3. He is timeless.
What is wrong with this reasoning is that in both examples (1) states only a sufficient, not a necessary, condition of (2). If George and Barbara are eating those things, then they are having breakfast. But it does not follow that if they are having breakfast, then they are eating those things! If God is timeless, then he is intrinsically unchanging. But that does not imply that if he is intrinsically unchanging, he is therefore timeless.
If P → Q, modus ponens tells us that if we affirm that the antecedent P is true, then the consequent is also true. Modus tollens tells us that if we deny that the consequent Q is true, then the antecedent P must also be denied. Thus, if P → Q, it is valid reasoning to either affirm the antecedent or deny the consequent and draw the appropriate conclusion. But we must not make the mistake of affirming the consequent. If P → Q, and Q is true, we may not validly conclude anything.
Rule #3: Hypothetical Syllogism
1. P → Q
2. Q → R
3. P → R
The third rule, hypothetical syllogism, states that if P implies Q, and Q implies R, then P implies R. Since we do not know in this case if P is true, we cannot conclude that R is true. But at least we can know on the basis of premises (1) and (2) that if P is true, then R is true.
Example 1:
1. If it is Valentine’s Day, Guillaume will invite Jeanette to dine at a fine restaurant.
2. If Guillaume will invite Jeanette to dine at a fine restaurant, then they will dine at L’ Auberge St. Pierre.
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3. If it is Valentine’s Day, then Guillaume and Jeanette will dine at L’ Auberge St. Pierre.
Example 2:
1. If Jeanette orders médallions de veau, then Guillaume will have saumon grillé.
2. If Guillaume has saumon grillé, he will not have room for dessert.
3. If Jeanette orders médallions de veau, then Guillaume will not have room for dessert.
We can use our three logical rules in conjunction with one another to draw more complicated inferences. For example, we can use modus ponens (MP) and hypothetical syllogism (HS) to see that the following argument is valid.
1. P → Q
2. Q → R
3. P
4. P → R (HS, 1, 2)
5. R (MP, 3, 4)
The first three steps are the given premises. Steps (4) and (5) are conclusions we can draw using the logical rules we have learned. To the right we abbreviate the rule that allows us to take each step, along with the numbers of the premises we used to draw that conclusion. Notice that a conclusion validly drawn from the premises becomes itself a premise for a further conclusion.
Here is another example:
1. P → Q
2. Q → R
3. ¬R
4. P → R (HS, 1, 2)
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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5. ¬P (MT, 3, 4)
The more rules we learn, the more complicated the arguments we may handle.
Rule #4: Conjunction
1. P
2. Q
3. P & Q
Here we introduce the symbol &, which is the symbol for conjunction. It is read as “and.” This rule is perspicuous: If P is true, and Q is true, then the conjunction “P and Q” is also true.
Example 1:
1. Charity is playing the piano.
2. Jimmy is trying to play the piano.
3. Charity is playing the piano, and Jimmy is trying to play the piano.
Example 2:
1. If Louise studies hard, she will master logic.
2. If Jan studies hard, she will master logic.
3. If Louise studies hard, she will master logic, and if Jan studies hard, she will master logic.
As example 2 illustrates, any sentences can be joined by &. When the premises in our arguments get complicated, it helps to introduce parentheses to keep things straight. For example, you would symbolize the conclusion (P → Q) & (R → S).
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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The symbol & symbolizes many more words than just and. It symbolizes any conjunction. Thus the logical form of sentences having the connective words but, while, although, whereas, and many other words is the same. We symbolize them all using &. For example, the sentence “They ate their spinach, even though they didn’t like it” would be symbolized P & Q. P symbolizes “They ate their spinach,” Q symbolizes “they didn’t like it,” and & symbolizes the conjunction “even though.”
Rule #5: Simplification
1. P & Q
2. P
1. P & Q
2. Q
Again, one does not need to be a rocket scientist to understand this rule! In order for a conjunction like P & Q to be true, both P and Q must be true. So simplification allows you to conclude from P & Q that P is true and that Q is true.
Example 1:
1. Bill is bagging groceries, and James is stocking the shelves.
2. James is stocking the shelves.
Example 2:
1. If Susan is typing, she will not answer the phone; and if Gary is reading, he will not answer the phone.
2. If Gary is reading, he will not answer the phone.
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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The main usefulness of this rule is that if you have the premise P & Q and you need either P by itself or Q by itself to draw a conclusion, simplification can give it to you.
For example:
1. P & Q
2. P → R
3. P (Simp, 1)
4. R (MP, 2, 3)
Rule #6: Absorption
1. P → Q
2. P → (P & Q)
This is a rule that one hardly ever uses but that nonetheless states a valid way of reasoning. The basic idea is that since P implies itself, it implies itself along with anything else it implies.
Example 1:
1. If Allison goes shopping, she will buy a new top.
2. If Allison goes shopping, then she will go shopping and buy a new top.
Example 2:
1. If you do the assignment, then you will get an A.
2. If you do the assignment, then you do the assignment and you will get an A.
The main use for absorption will be in cases where you need to have P & Q in order to take a further step in the argument. For example:
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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1. P → Q
2. (P & Q) → R
3. P → (P & Q) (Abs, 1)
4. P → R (HS, 2, 3)
Rule #7: Addition
1. P
2. P v Q
For this rule we introduce a new symbol: v, which is read “or.” We can use it to symbolize sentences connected by the word or. A sentence that is composed of two sentences connected by or is called a disjunction.
Addition seems at first to be a strange rule of inference: It states that if P is true, then “P or Q” is also true. What needs to be kept in mind is this: in order for a disjunction to be true only one part of the disjunction has to be true. So if one knows that P is already true, it follows that “P or Q” is also true, no matter what Q is!
Example 1:
1. Mallory will carefully work on decorating their new apartment.
2. Either Mallory will carefully work on decorating their new apartment, or she will allow it to degenerate into a pigsty.
Example 2:
1. Jim will make the honor roll.
2. Either Jim will make the honor roll or his dad will fly to the moon.
Addition is another one of those “housekeeping” rules that are useful for tidying up an argument by helping us to get some needed part of a premise. For example:
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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1. P
2. (P v Q) → R
3. P v Q (Add, 1)
4. R (MP, 2, 3)
Rule #8: Disjunctive Syllogism
1. P v Q
2. ¬P
3. Q
1. P v Q
2. ¬Q
3. P
This rule tells us that if a disjunction of two sentences is true, and one of the sentences is false, then the other sentence is true.
Example 1:
1. Either Mary will grade the exams herself or she will enlist Jason’s aid.
2. She will not grade the exams herself.
Example 2:
1. Either Amy worked in the garden or Mack spent his Saturday morning doing paperwork.
2. Mack did not spend his Saturday morning doing paperwork.
3. Amy worked in the garden.
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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The important thing to remember about logical disjunctions is that both of the sentences connected by or could be true. In other words, the alternatives do not have to be mutually exclusive. In example 2 both sentences in premise (1) could be true. Therefore, one cannot conclude that because one of the disjuncts is true, the other is false. Both could be true. So disjunctive syllogism allows you to conclude only that if one part of a true disjunction is false, then the other disjunct is true.
As mentioned, when the premises in one’s arguments are complicated, it helps to introduce parentheses to keep things straight. For example, one would symbolize the sentence “If Amy replants the bushes, she will water them or they will die” by P → (Q v R). This is quite different from (P → Q) v R. The latter would symbolize the disjunction “If Amy replants the bushes, she will water them; or they will die.”
In figuring out whether more complex arguments are valid, it is important to remember that one cannot use a logical rule on just part of a step; it must be used on the whole step. So, for example, if one has
1. P → (Q v R)
2. ¬Q
one cannot conclude that
3. R
In order to get to (3) we also need the premise
4. P
Then we can conclude
5. Q v R (MP, 1, 4)
And that allows us to arrive at
6. R (DS, 2, 5)
Finally, keep in mind that the logical form of a sentence may be quite different from its verbal form. Often we do not bother to repeat the subject or the verb of the first sentence in a disjunction; for example, “Either Sherry or Patti will go with you to the airport.” This is logically a disjunction:
Moreland, J. P., & Craig, W. L. (2017). Philosophical foundations for a christian worldview. InterVarsity Press. Created from liberty on 2025-02-04 15:05:44.
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“Either Sherry will go with you to the airport or Patti will go with you to the airport.” But this latter is not the normal way we talk. So sometimes we have to figure out the logical form of a sentence.
We must be careful because not every use of or in a sentence indicates that the sentence is a disjunction. Suppose you come to the plate with the bases loaded and two out, and your coach says, “If you get a single or a walk, we’ll win!” Is he saying, “If you get a single, we’ll win, or if you get a walk, we’ll win!” (P → Q) v (R → Q)? Surely not! For then he could just as well have said, “If you get a single or an out, we’ll win!” That whole disjunction would be true because P → Q is true even if R → Q is false. Rather, we should symbolize the coach’s advice as (P v R) → Q. He’s saying that whichever you get, a single or a walk, is a sufficient condition for us to win the game.
Rule #9: Constructive Dilemma
1. (P → Q) & (R → S)
2. P v R
3. Q v S
According to constructive dilemma, if P implies Q and R implies S, then if P or R is true, it follows that either Q or S is true.
Example 1:
1. If Jennifer buys dwarf fruit trees, she can make peach pies; and if she plants flowers, the yard will look colorful.
2. Either Jennifer buys dwarf fruit trees or she plants flowers.
3. Either Jennifer can make peach pies or the yard will look colorful.
Example 2:
1. If Yvette comes along on the trip, then Jim will be happy; and if Jim goes without Yvette, then he will be lonely.
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2. Either Yvette comes along on the trip or Jim goes without her.
3. Either Jim will be happy or Jim will be lonely.
This rule is useful for deducing the consequences of either-or situations, when we know the implications of each of the alternatives.
With these nine rules one can assess the validity of a vast range of arguments and, of course, formulate valid arguments of one’s own. The following exercises will help readers to apply what they have learned.
2.1.1.2 Exercises over the Nine Rules
Symbolize each argument and draw the conclusion, stating the rule that justifies each step.
A.
1. Either Millie will buy ten shares of Acme, Inc., or she will sell out.
2. She will not sell out.
B.
1. God is timeless only if he is immutable.
2. God is immutable only if he does not know what time it is now.
3. If God is omniscient, then he knows what time it is now.
4. God is omnipotent and omniscient.
C.
1. Only if God is temporal can he become incarnate.
2. If Jesus was God or Krishna was God, then God can become incarnate.
3. Jesus was God.
D.
1. If God is all-good, then he wants to prevent evil.
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2. If God is all-powerful, then he can prevent evil.
3. God is all-good and all-powerful.
4. If God wants to prevent evil and God can prevent evil, then evil does not exist.
E.
1. Keith gets up on time.
2. If Keith gets up on time, he will wake up Ashley.
3. If Keith wakes up Ashley, she will either loaf around or vacuum the house.
4. If she loafs around, Keith will go swimming by himself.
5. Ashley will not vacuum the house.
F.
1. If the butler was the murderer, his fingerprints were on the weapon.
2. Either the maid or the gardener was the murderer if the butler was not.
3. If the gardener was the murderer, there will be blood on the garden fork.
4. If the maid was the murderer, then the master was killed with a kitchen knife.
5. The butler’s fingerprints were not on the weapon.
6. There was no blood on the garden fork.
G.
1. We’ll have a debate if either Parsons or Flew agrees.
2. If we have a debate, it will be videotaped.
3. If the debate will be videotaped or audiotaped, you can get a copy of what went on.
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4. If you can get a copy of what went on, then you don’t need to feel bad about missing the debate.
5. Parsons will agree to debate.
H.
1. If God hears prayer, then he will answer if I pray.
2. God hears prayer.
3. I’ll pray.
2.1.1.3 Some Equivalences
In addition to the nine logical rules we have learned, there are a number of logical equivalences that should be mastered.
P is equivalent to ¬¬P
P v P is equivalent to P
P→Q is equivalent to ¬P v Q
P→Q is equivalent to ¬Q→¬P
Moreover, there is a very handy way of converting a conjunction to a disjunction and vice versa. There are three steps:
Step 1. You put ¬ in front of each letter.
Step 2. You change the & to v (or the v to &).
Step 3. You put the whole thing in parentheses and put ¬ in front.
Example 1: Change P & Q to a disjunction.
Step 1. ¬P & ¬Q
Step 2. ¬P v ¬Q
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Step 3. ¬ (¬P v ¬Q)
Example 2: Change P v Q to a conjunction.
Step 1. ¬P v ¬Q
Step 2. ¬P & ¬Q
Step 3. ¬ (¬P & ¬Q)
Sometimes you have to use double negation:
Example 3: Change ¬P & Q to a disjunction.
Step 1. ¬¬P & ¬Q
Step 2. P v ¬Q
Step 3. ¬ (P v ¬Q)
Using this procedure we can find that
¬P & ¬Q is equivalent to ¬(P v ¬Q)
¬P v ¬Q is equivalent to ¬(P & ¬Q)
Since equivalent statements are logically the same, you can replace a premise with its equivalent. Then you may be able to use the new premise along with other premises to draw further conclusions.
Example 1:
1. If God exists, humanism is not true.
2. If God does not exist, humanism is not true.
3. God exists or he does not exist.
4. Therefore, if God exists, humanism is not true; and if God does not exist, humanism is not true. (Conj, 1, 2)
5. Therefore, either humanism is not true or humanism is not true. (CD, 3, 4)
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6. Therefore, humanism is not true. (Equiv, 5)
Example 2:
1. If God does not foreknow the future, then either he determines everything or he gambles.
2. If God determines everything, then he is the author of sin.
3. If God gambles, then he is not sovereign.
4. God is sovereign, but he is not the author of sin.
5. Therefore, God is sovereign. (Simp, 4)
6. Therefore, God is not the author of sin. (Simp, 4)
7. Therefore, God does not determine every thing. (MT, 2, 6)
8. Therefore, God does not gamble. (MT, 3, 5)
9. Therefore, God does not determine every thing, and God does not gamble. (Conj, 7, 8)
10. Therefore, it is not true that either God determines everything or God gambles. (Equiv, 9)
11. Therefore, God does not not foreknow the future. (MT, 1, 10)
12. Therefore, God does foreknow the future. (Equiv, 11)
2.1.1.4 Conditional Proof
In formulating arguments of one’s own, one of the most powerful logical techniques one can use is called conditional proof. Many times we find ourselves in situations where we want to argue that if something is true, then certain conclusions follow. What we need is a way of introducing a new premise into our argument. We can do this by constructing a conditional proof.
Here is how it works. Suppose we are given the following premises:
1. P → Q
2. Q → R & S
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Suppose we want to argue that if P is true, then S is also true. This cannot be done using just premises (1) and (2). So what we do is introduce P as a conditional premise. It is as though we were to say, “Suppose P is true. Then what?” In order to make it clear that P is just a conditional premise, we can indent it.
1. P → Q
2. Q → R & S
3. P
Then we apply our rules of logic to draw the conclusion. Remember to keep subsequent steps indented to remind us that each inference is based on the condition that P is true.
1. P → Q
2. Q → R & S
3. P
4. Q (MP, 1, 3)
5. R & S (MP, 2, 4)
6. S (Simp, 5)
Finally, the last step is to combine our conditional premise with the conclusion we can draw if we suppose that the conditional premise is true. In other words, we know that if premise (3) is true, then our conclusion (6) is true. So we link the conditional premise (3) with the conclusion (6) by →. This final conclusion is not indented because we know that it is true by conditional proof (CP).
1. P → Q
2. Q → R & S
3. P
4. Q (MP, 1, 3)
5. R & S (MP, 2, 4)
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6. S (Simp, 5)
7. P → S (CP, 3-6)
Conditional proof is very useful in proving conditional statements.
Example:
1. If God exists and the present moment is real, then God is in time.
2. If God is in time, then he knows what is happening absolutely now.
3. If God knows what is happening absolutely now, then there is a moment that is absolutely now.
4. Either there is no moment that is absolutely now or Einstein’s special theory of relativity is wrong.
5. The present moment is real.
6. God exists. (Conditional Premise)
7. Therefore, God exists and the present moment is real. (Conj, 5, 6)
8. Therefore, God is in time. (MP, 1, 7)
9. Therefore, he knows what is happening absolutely now. (MP, 2, 8)
10. Therefore, there is a moment that is absolutely now. (MP, 3, 9)
11. Therefore, there is not no moment that is absolutely now. (Equiv, 10)
12. Therefore, Einstein’s special theory of relativity is wrong. (DS, 4, 11)
13. Therefore, if God exists, then Einstein’s special theory of relativity is wrong. (CP, 6-12)
2.1.1.5 Reductio ad Absurdum
A special kind of conditional proof is called reductio ad absurdum (reduction to absurdity). Here we show that if some premise is supposed to be true, then it implies a contradiction, which is absurd. Therefore we can
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conclude that the premise is not true after all. This is an especially powerful way of arguing against a view, for if we can show that a view implies a contradiction, then it cannot be true.
Usually, you will begin with premises for your argument on which you and your opponent agree. Then you add to the list of premises the conditional premise that your opponent thinks is also true, but that you think is false. Then you show how the assumption of that premise leads to a contradiction. Since you have reduced his view to absurdity by showing that it implies a contradiction, you negate the conditional premise and write RAA out to the side.
Example:
1. We have a moral duty to love our fellow humans as ourselves.
2. If God does not exist, then our fellow men are just animals.
3. If our fellow men are just animals, we have no moral duty to love them as ourselves.
4. God does not exist. (Conditional Premise)
5. Therefore, our fellow men are just animals. (MP, 2, 4)
6. Therefore, we have no moral duty to love our fellow men as ourselves. (MP, 3, 5)
7. Therefore, we have a moral duty to love our fellow men as ourselves, and we have no moral duty to love our fellow men as ourselves. (Conj, 1, 6)
8. Therefore, if God does not exist, we have a moral duty to love our fellow men as ourselves, and we have no moral duty to love our fellow men as ourselves. (CP, 4-7)
9. Therefore, God does not not exist. (RAA, 8)
10. Therefore, God exists. (Equiv, 9)
Confronted with this argument, your atheist friend may choose to give up one of his original premises rather than give up his belief in (4). But that should not bother you. Your argument has served to show what it will cost
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him to hold onto his atheism. He will have to give up belief in (1), (2), or (3). But each of these statements seems to be pretty clearly true—at least more clearly true than (4)! When we present an argument using reductio ad absurdum, we try to make the cost of giving up one of the other premises as high as possible, in the hope that our opponent will give up his belief in the conditional premise instead.
2.1.2 First-Order Predicate Logic
In first-order predicate logic we learn how to deal with sentences that predicate some property of a subject. This is important because it will enable us to deal with quantified sentences, that is to say, sentences about groups of things. Quantification deals with statements about all or none or some of a group. We often draw conclusions about such matters in everyday life. But what we have learned so far in this chapter does not enable us to do so validly. For example, suppose we are given the premises
1. All men are mortal.
2. Socrates is a man.
From (1) and (2) it obviously follows that
3. Socrates is mortal.
But we cannot draw such a conclusion using only the nine rules learned so far. For this argument would be symbolized as
1. P
2. Q
3. R
which is clearly invalid. Fortunately, we do not need any new rules of inference to solve this
problem. We just need to learn something about the logical form of quantified statements. We present here just a snippet of quantified logic,
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enough to enable you to deal with most of the arguments you will come across.
2.1.2.1 Universal Quantification
Statements about all or none of a group are called universally quantified statements, since the statement covers every member in a group. When we analyze the logical form of such statements, we discover that they turn out to be disguised “if . . . , then . . .” statements. For example, when we say, “All bears are mammals,” logically we are saying, “If anything is a bear, then it is a mammal.” Or if we say, “No goose is hairy,” logically we are saying, “If anything is a goose, then it is not hairy.”
So we can symbolize universally quantified statements as “if . . . , then . . .” statements. In order to do so, we introduce the letter x as a variable that can be replaced by any individual thing. We symbolize the antecedent clause using some capital letter (usually the first letter of the main word in the antecedent to make it easy to remember). For example, we can symbolize “Anything is a bear” by Bx. We do the same thing with the consequent. For example, “it is a mammal” can be symbolized Mx. The whole sentence is then symbolized as follows:
(x) (Bx → Mx)
You can read this as “For any x, if x is a bear, then x is a mammal.” There are many different ways in English of making such affirmative,
universally quantified statements. All, every, each, any are just a few of the words we use to speak about all the things in a class. Sometimes we just make a generalization; for example, “Bears are four-footed” or “Bears have claws.” This can be tricky because some generalizations are not really universal but are meant to be true of only some members of a class; for example, “Bears live at the North Pole.” We have to try to understand what the person meant when he made the statement in order to discern whether a universal statement was being made or not.
Now we are ready to symbolize an argument involving universal quantification and derive the conclusion.
1. Every vegetable planted by Xiu Li sprouted.
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2. One vegetable she planted was corn.
We symbolize (1) by letting V = “is a vegetable planted by Xiu Li” and S = “sprouted.”
1. (x) (Vx → Sx)
We symbolize (2) by letting c = “corn.”
2. Vc
Now we replace the variable x in (1) with c.
3. Vc → Sc
This has the effect of transforming (1) into a statement about one member of the class, namely, corn. It symbolizes “If corn is a vegetable planted by Xiu Li, then it sprouted.” Now we simply apply our nine rules, and we get:
4. Sc (MP, 2, 3)
Thus we are able to conclude validly that the corn sprouted. Some universal statements are negative. They assert that if anything is a
member of a certain group, then it does not have the property in question. We symbolize such a statement by negating the consequent. So, for example, we can symbolize “No goose is hairy” as
(x) (Gx → ¬Hx)
This is read as “For any x, if x is a goose, then x is not hairy.” Again, there are many ways to express a universal, negative statement in English. No, none, nothing, no one or just negative generalizations can be used to express such statements.
Let us symbolize an argument using a universally quantified, negative premise.
1. No goose is hairy.
2. Red Goose is a goose.
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We symbolize (1) and (2) as
1. (x) (Gx → ¬Hx)
2. Gr
Then we plug in r for the variable x to get
3. Gr → ¬Hr
That allows us to infer
4. ¬Hr (MP, 2, 3)
Often we encounter arguments with more than one universally quantified premise. For example,
1. All bears have claws.
2. Anything with claws can scratch.
3. Brown Bear is a bear.
These are symbolized
1. (x) (Bx → Cx)
2. (x) (Cx → Sx)
3. Bb
We go ahead and plug in b for the variable and then apply our rules of inference:
4. Bb → Cb
5. Cb → Sb
6. Bb → Sb (HS, 4, 5)
7. Sb (MP, 3, 6)
Suppose we did not have premise (3). Then we can take a shortcut and just conclude by hypothetical syllogism that (x) (Bx → Sx).
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2.1.2.2 Existential Quantification
Statements that are about only some members of a group are called existentially quantified statements. They tell us that there really exists at least one thing that has the property in question. For example, the statement “Some bears are white” tells us that there is at least one thing in the world that is both a bear and white. The statement “Some bears are not white” says that there is at least one thing that is a bear and is not white.
We symbolize existentially quantified statements by using the symbol ∃. It may be read as “There is at least one ___ such that . . .” We fill in the blank with the variable x, which can be replaced by any individual thing. So if we let Bx = “x is a bear” and Wx = “x is white,” we can symbolize “Some bears are white” as
(∃x) (Bx & Wx)
This is read as “There is at least one x such that x is a bear and x is white.” Notice that existentially quantified statements are symbolized using &, not → as universally quantified statements are. We must not confuse the two by symbolizing “Some bears are white” as
(∃x) (Bx → Wx).
We can symbolize “Some bears are not white” as
(∃x) (Bx & ¬Wx).
This is read as “There is at least one x such that x is a bear and x is not white.”
Now immediately we see that both the affirmative and negative statements can be true. “Some bears are white and some bears are not white” is not a contradiction. So affirmative and negative existentially quantified statements are not contradictory. So what is the opposite of an affirmative existentially quantified statement? It would be symbolized
¬ (∃x) (Bx & Wx).
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This states that there is nothing that is both a bear and white, or, in other words, that there are no white bears. So it turns out that the opposite of an affirmative existentially quantified statement is a negative universally quantified statement. So
(x) (Bx → ¬Wx) is contradictory to (∃x) (Bx & Wx).
Similarly, the contradictory of a negative, existentially quantified statement would be symbolized
¬ (∃x) (Bx & ¬Wx).
This tells us that there is nothing that is a nonwhite bear. In other words, all bears are white. So the opposite of a negative, existentially quantified statement is an affirmative, universally quantified statement. So
(x) (Bx → Wx) is contradictory to (∃x) (Bx & ¬Wx).
We can construct a diagram for displaying contradictories for universally and existentially quantified statements (figure 2.1).
Figure 2.1. Contradictories for universally and existentially quantified statements
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When we symbolize an argument involving existentially quantified premises, we also plug in a letter symbolizing some individual for the variable x. But in this case we can only use a letter as a substitute for x if we have not already used that letter before to symbolize a previous premise. So if we have an argument involving both universally quantified and existentially quantified premises, we must be sure to symbolize the existentially quantified premise first, regardless of the order of the premises. (Otherwise things can get all messed up!) So, for example, suppose we have the premises
1. All bears are mammals.
2. Some bears are white.
These are symbolized as
1. (x) (Bx → Mx)
2. (∃x) (Bx & Wx).
Now to apply our rules of reasoning we plug in letters symbolizing particular individuals. First, we do the existentially quantified premise to get
3. Ba & Wa.
Then we do the universally quantified premise to get
4. Ba → Ma.
Now we apply our rules:
5. Ba (Simp, 3)
6. Ma (MP, 4, 5)
7. Wa (Simp, 3)
8. Ma & Wa (Conj, 6, 7)
Since at least one mammal, namely, the one represented by a, is white, we can conclude that some mammals are white, or (∃x) (Mx & Wx).
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Arguments having several quantified premises can get very complicated. But realistically, with an understanding of the above rudiments of quantified logic, readers will be able to handle most of the arguments they will confront without a great deal of difficulty.
2.1.3 Modal Logic
One of the subdisciplines of advanced logic is modal logic, which deals with notions of necessary and possible truth—the modes of truth, as it were. It is evident that there are such modes of truth, since some statements just happen to be true but obviously could have been false—for example, “Garrett DeWeese teaches at Talbot School of Theology.” But other statements do not just happen to be true; they must be true and could not have been false—for example, “If P implies Q, and P is true, then Q is true.” Still other statements are false and could not have been true—for example, “God both exists and does not exist.” Statements that could not have had a different truth value than the one they have are said to be either necessarily true or necessarily false. We can use the symbol □ to stand for the mode of necessity:
□P is to be read as “Necessarily, P” and indicates that P is necessarily true.
□ ¬P is to be read as “Necessarily, not-P” and indicates that P is necessarily false.
Now if P is necessarily false, then it could not possibly be true. Letting ◊ stand for the mode of possibility, we can see that
□¬P is logically equivalent to ¬◊P, which may be read as “Not- possibly, P.”
This is to say that it is impossible for P to be true. The contradictory of ¬◊P is ◊P, or “Possibly, P.” Now if P is necessarily true, it is obviously also possibly true; otherwise its truth would be impossible. So □P implies ◊P; but it precludes the truth of ◊¬P. Indeed, □P is equivalent to ¬◊¬P. That is to say, if P is necessarily true, then it is impossible that P be false. If, on the other hand, it is possible for P to be true and possible for P to be false, then P is a contingent statement, being either contingently true or contingently
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false. Thus we may construct a handy square of opposition (figure 2.2) exhibiting contradictories, contraries, and subimplications.
Figure 2.2. Square of opposition for modal statements
The square shows us that “Necessarily, P” (symbolized either as □P or as ¬◊¬P) is contradictory to “Possibly, not-P” (symbolized as either ◊¬P or as ¬□P), so that if one of these statements is true, the other is false. And “Necessarily, not-P” (□¬P or ¬◊P) is the contradictory of “Possibly, P” (◊P or ¬□¬P), so that if one of these statements is true, the other is false. We also see that “Necessarily, P” is the contrary of “Necessarily, not-P,” so that both these statements cannot be true, though (unlike contradictories) they could both be false (namely, if P is contingent and so is neither necessarily true nor necessarily false). We also see that “Possibly, P” and “Possibly, not-P” are contraries, in that they cannot both be false (for if ◊¬P, for example, were false, then ¬◊¬P would be true, which is equivalent to □P, which implies that ◊P is true, the contrary of ◊¬P), though they could both be true (namely, if P is a contingent statement). Finally, we see that if □P is true, then ◊P is also true, and if □¬P is true, then ◊¬P is true as well.
In recent years an interpretation called possible worlds semantics has been given to modal syntax, which vividly illustrates the key modal notions. A possible world is a way the world might be. One can think of a possible
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world as a maximal description of reality; nothing is left out. It may be thought of as a maximal state of affairs, which includes every other state of affairs or its complement, or as an enormous conjunction composed of every statement or its contradictory. These states of affairs or statements must be compossible, that is, able to obtain together or to be true together, otherwise they would not constitute a possible world. Moreover, such a maximal state of affairs must be actualizable or capable of being actual. Just what that means is unclear. Some philosophers take actualizability to mean strict logical possibility, mere freedom from contradiction. Others demur, regarding such an understanding of actualizability as far too generous. To borrow Plantinga’s example, the statement “The Prime Minister is a prime number” is strictly logically consistent, but such a state of affairs is not actualizable. Plantinga prefers to construe actualizability in terms of broad logical possibility, a notion that he leaves undefined but merely illustrates. The situation is further complicated by the hypothesis of theism, for if God’s existence is necessary then some worlds that seem intuitively to be broadly logically possible may not be actualizable after all because God, necessarily, would not actualize them. For example, a world in which human beings all freely reject God’s plan of salvation and fail to reach heaven seems to be broadly logically possible, but it may not be actualizable because God is essentially too good to actualize such a world. Such problems have led some thinkers to differentiate between broadly logical possibility and metaphysical possibility, or actualizability. In any case, these debates make clear that possible worlds semantics do not explain or ground our modal notions, but at best illustrate them.
In possible worlds semantics, necessary truth is interpreted in terms of truth in every possible world. To say that a statement P is true in a possible world W is to say that if W were actual, then P would be true. So a necessary truth is one that is true regardless of which possible world is actual. Possible truth is construed as truth in at least one possible world. Necessary falsehood is understood as truth in no possible world or, in other words, a statement’s being false in every possible world. Possible falsehood is a statement’s being false in at least one possible world. A statement that is true in some worlds and false in others is contingently true or false.
Care must be taken in dealing with modal statements because it is sometimes ambiguous whether the necessity at issue is de dicto or de re.
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Necessity de dicto is the necessity attributed to a statement (a dictum) that is true in all possible worlds. Necessity de re is the necessity of a thing’s (a res) possessing a certain property, or in other words, a thing’s having a property essentially. If something has a property essentially, then it has it in every possible world in which it is true that this thing exists, even if this thing does not exist in every possible world. So, for example, when it is said, “Necessarily, Socrates is a human being,” it is not meant that the statement “Socrates is a human being” is true in every possible world, for Socrates does not exist in every possible world. Rather, what is meant is that Socrates is essentially human. Sometimes the ambiguity is compounded. For example, “Necessarily, God is good” could be taken to assert either that the statement “God is good” is true in every possible world or that God is essentially good (even if there are possible worlds in which he does not exist) or both.
All of the rules of inference that we learned in our section on sentential logic have their modal counterparts. For example, Modal modus ponens is a valid inference form:
1. □(P → Q)
2. □P
3. □Q
Thus one need not learn a whole new set of rules. The rub comes, however, in arguments having a mixture of modal and
nonmodal premises. Here mistakes are easy, and we wish to alert the reader to a couple of the most frequent modal fallacies to beware of. As we shall later see (chap. 28), extremely important metaphysical and theological conclusions have been drawn on the basis of these seductive fallacies. One common fallacy is the following inference:
1. □(P v ¬P)
2. □P v □¬P
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This inference pattern underlies many arguments for fatalism. For example, it is thought, “Necessarily, either I shall be killed in the bombing or I shall not be killed in the bombing. But then why take precautions, since nothing I do can make a difference?” The fatalist fallaciously assumes that his being necessarily killed or his necessarily not being killed follows from the composite necessity of his being killed or not. Medieval philosophers were aware of this fallacy and labeled it a confusion of necessity in sensu composito (in the composite sense) and necessity in sensu diviso (in the divided sense).
A similar confusion of composite (or undistributed) and divided (or distributed) necessity is involved in the fallacious inference:
1. □(P v Q)
2. ¬Q
3. □P
Someone might fallaciously reason as follows: “Necessarily, either God has willed that x will happen, or else x will not happen. But x did happen. Therefore, necessarily God has willed that x happen.” It does not follow, however, that necessarily God has willed that x happen, but merely that God has willed that x happen. For from (1) and (2) it follows only that P is true, not that it is necessarily true.
Finally, a very common modal fallacy involves modus ponens:
1. □(P → Q)
2. P
3. □Q
This fallacy is involved in reasoning such as the following: “Necessarily, if Christ predicted Judas’s betrayal, then Judas would betray Jesus. Christ did, in fact, predict Judas’s betrayal. Therefore, it was necessary that Judas betray Jesus—which obliterates Judas’s freedom.” But again, from (1) and (2) it only follows that Judas would betray Jesus, not that he would do so
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necessarily. Thus the necessity of Christ’s predictions being accurate and his prediction of Judas’s betrayal do not necessitate Judas’s betrayal. Medieval philosophers also spotted this fallacy and labeled it confusing the necessitas consequentiae (necessity of the consequences or the inference) with the necessitas consequentis (necessity of the consequent). That is to say, the inference of Q from the premises □(P → Q) and P is necessary in accordance with modus ponens; but Q itself, the consequent of the conditional □(P → Q), is not itself necessary.
A wary eye for these modal fallacies will greatly assist the reader in thinking accurately about various philosophical problems.
2.1.4 Counterfactual Logic
Counterfactuals are conditional statements in the subjunctive mood, and they have a logic of their own. Such conditionals are interestingly different from their indicative counterparts. Compare, for example,
1. If Oswald didn’t shoot Kennedy, then somebody else did.
2. If Oswald hadn’t shot Kennedy, then somebody else would have.
The indicative conditional (1) is evidently true in light of Kennedy’s death. But the counterfactual conditional (2) is by no means true; on the contrary it seems very likely that if Oswald had not shot the president, then Kennedy’s motorcade would have proceeded uneventfully. Counterfactuals are so called because the antecedent and consequent of the conditional are contrary to fact. But not all subjunctive conditionals are strictly counterfactual. In deliberative conditionals, for example, we entertain some antecedent with a view toward discerning its consequences, as a result of which we may take the course of action described in the antecedent, so that the consequent becomes true. For instance, as a result of thinking “If I were to quit smoking, then my breath would smell better,” one decides to quit smoking and his breath improves. Nonetheless, the term counter factuals is widely used to cover all subjunctive conditionals.
Counterfactuals come in two sorts: “would” counterfactuals and “might” counter factuals. The former state what would happen if the antecedent were true, while the latter state what might happen if the
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antecedent were true. The sentential connective symbol often used for “would” counterfactuals is □→. A “would” counterfactual is symbolized
P □→ Q,
where P and Q are indicative sentences, and is read “If it were the case that P, then it would be the case that Q.” Similarly, a “might” counterfactual is symbolized
P ◊→ Q,
and is read, “If it were the case that P, then it might be the case that Q.” “Might” counter factuals should not be confused with subjunctive conditionals involving the word “could.” “Could” is taken to express mere possibility and so is a constituent of a modal statement expressing a possible truth. The distinction is important because the fact that something could happen under certain circumstances does not imply that it might happen under those circumstances. “Might” is more restrictive than “could” and indicates a genuine, live option under the circumstances, not a bare logical possibility. In counterfactual logic P ◊→ Q is simply defined as the contradictory of P □→¬Q, that is to say, as ¬ (P □→¬Q). Thus, although P □→¬Q is logically incompatible with P ◊→ Q, it remains true that if P were the case it still could be the case that Q. We can also construct a square of opposition for counterfactual statements (see figure 2.3).
Figure 2.3. Square of opposition for counterfactual statements
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There is no really satisfactory semantics for counterfactual conditionals. But for want of a better alternative, most philosophers use the Stalnaker- Lewis semantics. Since counter factuals are contingent statements (necessary counterfactuals reduce to indicative conditionals), they are true or false relative to a possible world. For convenience, we shall take the actual world as our departure point. One then ranges other possible worlds into concentric spheres of worlds centered on our world on the basis of a similarity relation to the actual world, the most similar worlds being in the nearest sphere. Now we consider the worlds in the nearest sphere in which the antecedent of our counterfactual is true. If in all of the worlds in which the antecedent is true, the consequent is also true, then a “would” counterfactual P □→ Q is true. If in some of the worlds in which the antecedent is true, the consequent is also true, then a “might” counterfactual P ◊→ Q is true.
Such a semantics is inadequate, among other reasons, because it cannot deal with counterfactuals having impossible antecedents (sometimes called counter possibles). Since impossible statements are not true in any possible world, no sphere of worlds, no matter how distant, will contain worlds in which the antecedent is true. But then such counterfactuals all become trivially true because in all the worlds in the nearest sphere in which the antecedent is true the consequent is also true; that is, there is no sphere of antecedent-permitting worlds in which the consequent fails to be true. But such a result is highly counterintuitive. For consider the two conditionals
1. If God did not exist, the universe would not exist.
2. If God did not exist, the universe would still exist.
If God exists necessarily, then the antecedent of (1) and (2) is impossible. But in that case on the customary semantics both (1) and (2) are trivially true. But surely that is not correct. (1) seems to be the sober truth about the world, and (2) seems patently false. Therefore, the customary semantics is not adequate. For want of a better alternative, we may continue to employ the usual semantics, but one should take with a grain of salt philosophical objections to a metaphysical position that are based on the customary semantics for counterfactual conditionals.
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Counterfactual logic is peculiar in that certain rules of inference do not apply to it that apply to sentential logic. For example, although our beloved modus ponens and modus tollens hold for counterfactual conditionals, hypothetical syllogism does not. It is invalid to argue:
1. P □→ Q
2. Q □→ R
3. P □→ R
Thus it would be fallacious to reason, “If Billy Graham had married another woman, he would be having sex with someone other than Ruth. If Billy Graham were having sex with someone other than Ruth, he would be an adulterer. Therefore, if Billy Graham had married another woman, he would be an adulterer.” Both of the first two statements are true, but the conclusion clearly does not follow from them.
In sentential logic P → Q is equivalent to ¬Q → ¬P. But in counterfactual logic, this equivalence fails. It is invalid to argue as follows:
1. P □→ Q
2. ¬Q □→ ¬P
For example, it would be fallacious to think, “If Bonds had homered, the Giants would still have lost. Therefore, if the Giants had won, then Bonds would still not have homered.”
Finally, there is a fallacy in counterfactual logic called “strengthening the antecedent”:
1. P □→ Q
2. P & R □→ Q
Thus it would be fallacious to argue as follows: “If I were to quit smoking, my breath would smell better. Therefore, if I were to quit smoking and start eating raw garlic, my breath would smell better.”
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On the other hand, there are some argument forms that are valid in counterfactual logic that are particularly useful in cases in which hypothetical syllogism cannot be used. For example, it is valid to argue
1. P □→ Q
2. P & Q □→ R
3. P □→ R
Plantinga has employed this argument form in dealing with a theistic version of a problem in decision theory called Newcomb’s paradox.1 You are presented with two boxes, A and B, and are given the choice of having the contents of either both boxes or of just A alone. Here’s the catch: You know there is $1,000 in box B. If you choose only box A, then God will have foreknown your choice and put $1,000,000 in A. But if you’re greedy and pick both boxes, then God will have foreknown this and so put nothing in box A. The money already is or is not in A. What should you choose? Plantinga argues that you should choose only box A on the basis of the following reasoning:
1. If you were to choose both boxes, God would have believed that you would choose both boxes.
2. If you were to choose both boxes and God believed that you would choose both boxes, then God would not have put any money in A.
3. Therefore, if you were to choose both boxes, God would not have put any money in A.
(A parallel argument shows that if you were to choose A alone, then God would have put the $1,000,000 in A. So the one-box choice is the winning strategy.) This reasoning has important application to the problem of divine foreknowledge and human freedom.
Another valid inference form is
1. P □→ Q
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2. Q □→ P
3. Q □→ R
4. P □→ R
Thomas Flint has employed this inference pattern profitably in his work on divine providence.2 He reasons as follows:
1. If Paul were to mow the lawn next Saturday, then God would have foreknown that Paul would mow the lawn next Saturday.
2. If God were to have foreknown that Paul would mow the lawn next Saturday, then Paul would mow the lawn next Saturday.
3. If God were to have foreknown that Paul would mow the lawn next Saturday, then God would prevent it from raining.
4. If Paul were to mow the lawn next Saturday, then God would prevent it from raining.
Such reasoning plays a vital role in a Molinist account of divine providence (chap. 30).
A final valid inference pattern of note blends counterfactual and modal premises:
1. P □→ Q
2. □(Q → R)
3. P □→ R
Again, Flint employs this argument form gainfully in his discussions of divine providence. He reasons:
1. If Paul were to mow the lawn next Saturday, then God would prevent it from raining.
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2. Necessarily, if God prevents it from raining, then it will not rain next Saturday.
3. Therefore, if Paul were to mow the lawn next Saturday, then it would not rain next Saturday.
It should be noted, however, that philosophers who believe that there are nontrivially true counterpossibles (counterfactuals with impossible antecedents) reject this inference pattern. For if this inference pattern is valid, then one can show that □(P → Q) implies that P □→ Q.3 But this implication does not always hold if there are nontrivially true counterpossibles. The key to understanding here is to realize that if P is an impossible (necessarily false) statement, then P necessarily implies anything and everything. So if P is an impossible statement, then it is true that □(P → Q), no matter what Q represents. So, for example, it is true both that “Necessarily, if God does not exist, the universe does not exist” and “Necessarily, if God does not exist, the universe exists anyway.” But if there are nontrivially true counterpossibles, it does not follow from the truth of “Necessarily, if God does not exist, the universe exists anyway” that “If God were not to exist, then the universe would exist anyway.” Thus, if there are nontrivially true counterpossibles, then it is not the case that □(P → Q) implies that P □→ Q. But if that implication fails, then the transitive inference pattern that entails this implication also fails. Still, with ordinary counterfactuals at least, the inference pattern is unobjectionable. These three argument forms may help one to make a transitive argument without appeal to the invalid hypothetical syllogism.
We have only scratched the surface of the field of logic, but our goal has not been to survey the field, even superficially, but rather to provide readers with a basic grasp of a few rules of inference to assist them in assessing arguments they encounter and in formulating good arguments of their own.
2.1.5 Informal Fallacies
A good deductive argument, it will be recalled, must be not only formally valid but also informally valid. In practice, the primary informal fallacy to be on the alert for is the fallacy called petitio principii (begging the
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question). Sometimes this fallacy is also called circular reasoning. If one reasons in a circle, the conclusion of one’s argument is taken as one of the premises somewhere in the argument. Although this does happen, it is very unlikely that any sophisticated thinker will beg the question in so blatant a fashion as that. Rather, question-begging usually occurs in a more subtle way. We can say that a person begs the question if his only reason for thinking a premise in an argument to be true is his belief that the conclusion is true. Consider the following argument for God’s existence:
1. Either God exists or the moon is made of green cheese.
2. The moon is not made of green cheese.
3. Therefore, God exists.
This is a logically valid argument, having the inference form of disjunctive syllogism (P v Q; ¬Q; therefore, P). Moreover, theists will regard the premises as true (recall that for P v Q to be true, only one disjunct needs to be true). Therefore, the above is a sound argument for God’s existence. But such an argument will hardly rival one of Thomas Aquinas’s five ways of proving God’s existence! The reason for the argument’s failure is that it is question-begging: the only reason one would have for thinking that (1) is true is that one already believes that (3) is true. Thus, far from serving as a proof that God exists, the argument will be regarded as unsound or unconvincing by any person who is not already convinced that God exists. This subtle form of question-begging does go on and needs to be exposed.
There are many other informal fallacies in argumentation; but despite their high profile in texts on critical thinking, realistically one is not apt to encounter these often in serious philosophical work. Still, a couple are worth mentioning:
Genetic Fallacy. This is the fallacy of arguing that a belief is mistaken or false because of the way that belief originated. Some sociobiologists, for example, seem to commit this fallacy when they assert that because moral beliefs are shaped by biological and social influences, therefore those beliefs are not objectively true. Or again, some atheists still try to invalidate theistic belief on the basis that it originated out of fear or ignorance. How or
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why a belief came to be held is simply irrelevant to the truth or falsity of the proposition that is the object of that belief.
Argument from Ignorance. This is the fallacy of arguing that a claim is false because there is not sufficient evidence that the claim is true. Our ignorance of evidence for a claim’s truth does not imply the falsity of the claim.
Equivocation. This is the fallacy of using a word in such a way as to have two meanings. This fallacy is committed in the following argument: “Socrates is a Greek; Greek is a language; therefore, Socrates is a language.” The danger of equivocation should motivate us to define the terms in our arguments as clearly as possible. By offering careful definitions and using words univocally, we can blunt charges that we have committed this fallacy.
Amphiboly. This is the fallacy of formulating our premises in such a way that their meaning is ambiguous. For example, the statement “If God wills x, then necessarily x will happen” is amphibolous. Do we mean “□(God wills x → x will happen)” or “God wills x → □(x will happen)”? Again, in order to avoid the errors in reasoning that will result from ambiguous formulation of premises, we need to take great care in expressing them. One of the major tasks of philosophical analysis is not only careful definition of terms but also differentiation of the different meanings a premise in an argument might have and then assessing respectively their plausibility.
Composition. This is the fallacy of inferring that a whole has a certain property because all its parts have that property. Of course, sometimes wholes do have the properties of their parts, but it is fallacious to infer that a whole has a property just because its every part does. This fallacy seems to be committed by those who argue that because every part of an infinite past can be “traversed” to reach the present, therefore the whole infinite past can be traversed.
There are scores of such informal fallacies, but the above are some of the more common ones to look out for and avoid.
2.2 True Premises
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Little needs to be said by way of clarification of this criterion of a good deductive argument. Logical validity is a necessary condition of a cogent argument, but not a sufficient condition. In order for an argument to be sound, not only must it be valid but its premises must also be true. The main point to keep in mind here is that one must not confuse the epistemic status of the premises (their knowability) with their alethic status, or truth value. In order to be sound, an argument’s premises must be true, but their truth could be not merely uncertain but utterly unknown to us. Of course, if we are utterly ignorant of the premises’ truth, the argument will be of little use to us, even if it be, unknown to us, sound. But if we are warranted in believing the premises to be true, then the argument warrants us in accepting the conclusion.
2.3 Premises More Plausible Than Their Denials
An argument may be sound and informally valid and yet not a good argument. In order for the argument to be a good one, the premises need to have a particular epistemic status for us. But what sort of status is that? Certainty is an unrealistic and unattainable ideal. Were we to require certainty of the truth of an argument’s premises, the result for us would be skepticism. Plausibility or epistemic probability might be thought to be sufficient, but plausibility seems to be neither a necessary for nor a sufficient condition of a good argument. It is not necessary because in some cases both the premise and its denial (or contradictory) may strike us as implausible. One thinks of premises concerning the nature of the subatomic realm as described by quantum physics, for example. On the other hand, neither is plausibility sufficient because both the premise and its denial may have equal plausibility or the denial may have even more plausibility than a quite plausible premise. This suggests that what we are looking for is a comparative criterion: the premises in a good argument will have greater plausibility than their respective denials. While this typically suffices for a good argument, in order to guarantee that the conclusion is more plausible than its denial, the conjunction of the premises must be more plausible than
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not. If the conjunction of the premises is more plausible than not, then the argument’s conclusion is guaranteed to be more plausible than not, and so we should believe it.
It is important to understand that in a deductive argument, the probability of the conclusion is not equal to the probability of the conjunction of the premises; rather the probability of the conjunction of the premises sets a lower bound to the probability of the conclusion. The probability of the conclusion could actually be much higher, but it cannot be any lower than the probability of the conjoined premises. So if the combined premises of a valid deductive argument have a probability >50 percent, then the conclusion is guaranteed to be at the very least >50 percent and so we should believe it.
Now plausibility is to a great extent a person-dependent notion. Some people may find a premise plausible and others not. Accordingly, some people will agree that a particular argument is a good one, while others will say that it is a bad argument. Given our diverse backgrounds and biases, we should expect such disagreements. Obviously, the most persuasive arguments will be those that are based on premises that enjoy the support of widely accepted evidence or seem intuitively to be true. But in cases of disagreement we simply have to dig deeper and ask what reasons we each have for thinking a premise to be true or false. When we do so, we may discover that it is we who have made the mistake. After all, one can present bad arguments for a true conclusion! But we might find instead that our interlocutor has no good reason for rejecting our premise or that his rejection is based on misinformation, or ignorance of the evidence, or a fallacious objection. In such a case we may persuade him by giving him better information or evidence or by gently correcting his error. Or we may find that the reason he denies our premise is that he does not like the conclusion it is leading to, and so to avoid that conclusion he denies a premise that he really ought to find quite plausible. Ironically, it is thus possible, as Plantinga has observed, to move someone from knowledge to ignorance by presenting him with a valid argument based on premises he knows to be true!
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3—INDUCTIVE REASONING Up to this point we have focused only on deductive reasoning. In a sound deductive argument the conclusion follows necessarily from the premises: if the premises are true and the inference form valid, then it is impossible that the conclusion be false. It is worth observing that an argument’s having a deductive form is irrelevant to the epistemic status of the premises and conclusion. The difference between a deductive and an inductive argument is not to be found in the degree to which they approach demonstrative proof of some conclusion. A good deductive argument may make a conclusion only slightly epistemically probable if its premises are themselves far from certain, whereas an inductive argument could give us overwhelming evidence for and, hence, confidence in its conclusion. This fact is especially evident when we reflect that some of the premises in a deductive argument may themselves be established on the basis of inductive evidence. Thus, contrary to the impression sometimes given, an argument’s being inductive or deductive in form is not an indication of the certainty of the argument’s conclusion.
An inductive argument is one for which it is possible that the premises be true and no invalid inferences be made, and yet the conclusion still be false. A good inductive argument must, like a good deductive argument, have true premises that are more plausible than their contradictories and be informally valid. But because the truth of their premises does not guarantee the truth of their conclusions, one cannot properly speak of their being formally either valid or invalid. In such reasoning the evidence and rules of inference are said to “under determine” the conclusion; that is to say, they render the conclusion plausible or likely, but do not guarantee its truth. Here is an example of a good inductive argument:
1. Groups A, B, and C were composed of similar persons suffering from the same disease.
2. Group A was administered a certain new drug, group B was administered a placebo, and group C was not given any treatment.
3. The rate of death from the disease was subsequently lower in group A by 75 percent in comparison with both groups B and C.
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4. Therefore, the new drug is effective in reducing the death rate from said disease.
The conclusion is quite likely true based on the evidence and rules of inductive reasoning, but it is not inevitably true; maybe the people in group A were just lucky or some unknown variable caused their improvement.
3.1 Bayes’s Theorem
Although inductive reasoning is part and parcel of everyday life, the description of such reasoning is a matter of controversy among philosophers. One way of understanding inductive reasoning is by means of the probability calculus. Probability theorists have formulated various rules for accurately calculating the probability of particular statements or events given the truth or occurrence of certain other statements or events. Such probabilities are called conditional probabilities and are symbolized Pr (A|B). This is to be read as the probability of A on B, or A given B, where A and B stand for particular statements or events. Probabilities range between 0 and 1, with 1 representing the highest and 0 the lowest probability. Thus a value >.5 indicates some positive probability of a statement or event and <.5 some improbability, while .5 would indicate a precise balance between the two.
Many of the typical cases of inductive reasoning involve inferences from sample cases to generalizations—for example, the probability of Jones’s contracting lung cancer given that he is a smoker—and so have greater relevance to scientific than to philosophical concerns. Still a philosophical position can constitute a hypothesis, and that hypothesis can be argued to be more probable than not, or more probable than a particular competing philosophical hypothesis, given various other facts taken as one’s evidence. In such cases, the philosopher may have recourse to Bayes’s theorem, which lays down formulas for calculating the probability of a hypothesis (H) on given evidence (E).
One form of Bayes’s theorem is the following:
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In order to compute the probability of H|E, we plug in numerical values for the various probabilities in the numerator and denominator. In philosophical, as opposed to scientific, discussions this is usually impossible to do with precision, so we must be content with vague approximations like “highly improbable” (which is represented as <<.5) or “highly probable” (which is represented as >>.5) or “approximately even” (which is represented as ≈.5). Such vague approximations may still prove useful in arguing for one’s hypothesis.
In the numerator we multiply the intrinsic probability of H by H’s explanatory power (E|H). The intrinsic probability of H does not mean the probability of H taken in utter isolation, but merely in isolation from the specific evidence E. The intrinsic probability of H is the conditional probability of H relative to our general background knowledge (B), or Pr (H|B). Similarly, B is implicit in H’s explanatory power (E|H & B). The formula takes B tacitly as assumed. The Pr (E|H) registers our rational expectation of E given that H is the case. If E would be surprising on H, then Pr (E|H) <.5, whereas if we are not surprised to find E, given H, then Pr (E|H) is >.5.
In the denominator of the formula, we take the product of H’s intrinsic probability and explanatory power and add to it the product of the intrinsic probability and explanatory power of the denial of H. Notice that the smaller this latter product is, the better it is for one’s hypothesis. For in the limit case that Pr (¬H) × Pr (E|¬H) is zero, then the numerator and denominator have the same number, so that the ratio is equal to 1, which means that one’s hypothesis is certain given the evidence. So one will want to argue that while one’s hypothesis has great intrinsic probability and explanatory power, the denial of the hypothesis has low intrinsic probability and explanatory power.
One of the difficulties in using the above form of Bayes’s theorem in arguing inductively is that the negation of one’s hypothesis comprises such
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a diversity of alternatives that it may be difficult to show that H is more probable than not. For example, if H is the theistic hypothesis that God exists, then ¬H is not simply naturalism, but also pantheism, polytheism, panentheism, idealism, and the host of their variants. A specific body of evidence E may make theism more probable than, say, naturalism, but not more probable than polytheism. It may be not only very difficult to calculate the probability of ¬H, but it may also be rather beside the point. One’s interest may be, not to show that H is more probable than not relative to a specific body of evidence, but that H is more probable than its chief competitor H1.
If that is our interest, then we can employ the odds form of Bayes’s theorem to calculate the comparative probability of two competing hypotheses H1 and H2.
Here one’s goal is to show that H1’s intrinsic probability and explanatory power exceed that of H2, so that H1 is the more probable hypothesis.
The drawback of all such appeals to Bayes’s theorem in understanding inductive reasoning is that the probabilities can seem inscrutable and thus the conditional probability of one’s hypothesis incalculable. Nonetheless, Bayesian approaches to the so-called problem of evil (see chap. 29) have been fashionable in recent years and merit consideration.
3.2 Inference to the Best Explanation
A different approach to inductive reasoning that is apt to be more useful in philosophical discussions is provided by inference to the best explanation. In inference to the best explanation, we are confronted with certain data to
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be explained. We then assemble a pool of live options consisting of various explanations for the data in question. From the pool of live options we then select the explanation that, if true, best explains the data. Just what criteria go toward making an explanation the best is disputed; but among the commonly acknowledged criteria will be properties such as the following:
1. Explanatory scope. The best hypothesis will explain a wider range of data than will rival hypotheses.
2. Explanatory power. The best hypothesis will make the observable data more epistemically probable than rival hypotheses.
3. Plausibility. The best hypothesis will be implied by a greater variety of accepted truths and its negation implied by fewer accepted truths than rival hypotheses.
4. Less ad hoc. The best hypothesis will involve fewer new suppositions not already implied by existing knowledge than rival hypotheses.
5. Accord with accepted beliefs. The best hypothesis, when conjoined with accepted truths, will imply fewer falsehoods than rival hypotheses.
6. Comparative superiority. The best hypothesis will so exceed its rivals in meeting conditions (1) through (5) that there is little chance of a rival hypothesis’s exceeding it in fulfilling those conditions.
The neo-Darwinian theory of biological evolution is a good example of inference to the best explanation. Darwinists recognize that the theory represents a huge extrapolation from the data, which support micro- evolutionary change but do not provide evidence of macro-evolutionary development. They further freely admit that none of the evidence, taken in isolation, whether it be from microbiology, paleogeography, paleontology, and so forth provides proof of the theory. But their point is that the theory is nonetheless the best explanation, in virtue of its explanatory power, scope, and so on.
By contrast, the charge leveled by critics of the neo-Darwinian synthesis such as Phillip Johnson that the theory presupposes naturalism is best
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understood as the claim that the explanatory superiority of the neo- Darwinian theory is a function of the pool of live options’ being restricted by an unjustified methodological constraint, namely, the philosophical presupposition of naturalism. Johnson is quite happy to agree that the neo- Darwinian synthesis is the best naturalistic explanation available (in contrast to Lamarckianism, self-organization theories, and so on). But he insists that the interesting and important question is not whether the neo- Darwinian theory is the best naturalistic explanation, but whether it is the best explanation, that is to say, whether it is correct. Johnson argues that once hypotheses positing intelligent design are allowed into the pool of live options, then the explanatory superiority of the neo-Darwinian theory is no longer apparent. On the contrary, its deficiencies, particularly in the explanatory power of its mechanisms of random mutation and natural selection, stand in stark relief. What is intriguing is that several of Johnson’s detractors have openly admitted that Darwinism’s explanatory superiority depends on limiting the pool of live options to naturalistic hypotheses, but they claim that such a constraint is a necessary condition of doing science—a claim that is not, as such, scientific, but is a philosophical claim about the nature of science (see chap. 17). In any case, this controversy serves as a vivid illustration of inference to the best explanation, and many misdirected criticisms are launched from both sides due to a failure to understand this pattern of inductive reasoning.
CHAPTER SUMMARY A good deductive argument is formally and informally valid, has true premises, and has premises that are more plausible than their denials. Several rules of inference of sentential logic should be kept in mind:
Rule #1: modus ponens
1. P → Q
2. P
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3. Q
Rule #2: modus tollens
1. P → Q
2. ¬Q
3. ¬P
Rule #3: Hypothetical Syllogism
1. P → Q
2. Q → R
3. P → R
Rule #4: Conjunction
1. P
2. Q
3. P & Q
Rule #5: Simplification
1. P & Q
2. P
1. P & Q
2. Q
Rule #6: Absorption
1. P → Q
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2. P → (P & Q)
Rule #7: Addition
1. P
2. P v Q
Rule #8: Disjunctive Syllogism
1. P v Q
2. ¬P
3. Q
1. P v Q
2. ¬Q
3. P
Rule #9: Constructive Dilemma
1. (P → Q) & (R → S)
2. P v R
3. Q v S
In addition to the nine rules of inference, there are a number of logical equivalences that should be mastered.
P is equivalent to ¬¬P
P v P is equivalent to P
P→Q is equivalent to ¬P v Q
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P→Q is equivalent to ¬Q→¬P
We can convert a conjunction to a disjunction and vice versa by the following procedure:
Step 1. Put ¬ in front of each letter.
Step 2. Change the & to v (or the v to &).
Step 3. Put the whole thing in parentheses and put ¬ in front. In predicate logic we deal with classes of things. Universally quantified
statements are understood to have the logical form of conditional statements. Letting F and G stand for arbitrary predicates, we can symbolize an affirmative, universally quantified statement as (x) (Fx → Gx). A negative, universally quantified statement may be symbolized as (x) (Fx → ¬Gx). Existentially quantified statements typically have the form of conjunctions. An affirmative, existentially quantified statement can be symbolized as (∃x) (Bx & Wx). A negative, existentially quantified statement may be symbolized as (∃x) (Bx & ¬Wx). We plug in some individual for the variable x and then apply our nine rules of inference to draw deductions.
Modal logic is a branch of advanced logic dealing with possible and necessary truth. In possible worlds semantics, necessary truth is interpreted as truth in all possible worlds, and possible truth as truth in some possible world. We need to keep clear the distinction between necessity de dicto, which is the necessity attributed to a statement that is true (or false) in all possible worlds, and necessity de re, which is the necessity of a thing’s possessing a certain property, or a thing’s having a property essentially. We must take care to avoid the following fallacies in modal reasoning:
1. □(P v ¬P)
2. □P v □¬P
1. □(P v Q)
2. ¬Q
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3. □P
1. □(P → Q)
2. P
3. □Q
Counterfactual logic deals with inferences involving subjunctive conditionals, either of the “would” variety or of the “might” variety. In counterfactual logic, hypothetical syllogism, the equivalence known as contraposition, and strengthening the antecedent are all invalid. But several other interesting inference forms are valid, namely:
1. P □→ Q
2. P & Q □→ R
3. P □→ R
1. P □→ Q
2. Q □→ P
3. Q □→ R
4. P □→ R
1. P □→ Q
2. □(Q → R)
3. P □→ R
Some of the most common informal fallacies are begging the question (having no reason for accepting a premise other than one’s belief in the argument’s conclusion), genetic fallacy (arguing that a belief is mistaken or
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false because of the way that belief originated), argument from ignorance (arguing that a claim is false because there is not sufficient evidence that the claim is true), equivocation (using a word in such a way as to have two meanings), amphiboly (formulating our premises in such a way that their meaning is ambiguous), and composition (inferring that a whole has a certain property because all its parts have that property).
A good deductive argument must have true premises but need not have premises that are known with certainty to be true. Rather, in a good argument the premises are more plausible than their denials. If the conjunction of the premises is more plausible than its denial, then the conclusion is guaranteed to be more plausible than its denial.
Good inductive arguments must also have true premises that are more plausible than their contradictories and must be informally valid. But because the truth of their premises does not guarantee the truth of their conclusions, one cannot speak of validity with respect to them. Arguments involving probability calculations should be assessed according to Bayes’s theorem, one form of which is the following:
The odds form of the theorem can be used to assess two rival hypotheses:
We may also think of inductive reasoning as inference to the best explanation. In such an inference we choose from a pool of live options the explanation that, if true, would best explain the facts at hand. We assess which explanation is the best in terms of such criteria as explanatory scope,
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explanatory power, plausibility, degree to which it is ad hoc, accord with accepted beliefs, and comparative superiority vis-à-vis its rivals.
CHECKLIST OF BASIC TERMS AND CONCEPTS actualizable
arguments
Bayes’s theorem
broad logical possibility
conclusion
conditional probability
conditional proof
contingent statement
contradictory
counterfactual
counterpossibles
deductive argument
deliberative conditional
epistemically possible
existentially quantified statement
formally valid
inductive argument
inference to the best explanation
informal fallacy
invalid
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logic
metaphysical possibility
“might” counterfactual
modal logic
necessary truth
necessitas consequentiae
necessitas consequentis
necessity de dicto
necessity de re
necessity in sensu composito
necessity in sensu diviso
possible truth
possible world
possible worlds semantics
premise
probability calculus
propositional logic
quantification
reductio ad absurdum
rules of logic
sentential logic
sound argument
strict logical possibility
subjunctive conditional
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symbolic logic
universally quantified statement
unsound argument
“would” counterfactual
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