Deduction and Hypothetical Syllogisms

July 10, 2000

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Deduction

The other method of reasoning, the deductive method, begins with an accepted generalization–an already formulated or established general truth and applies it to discover a new logical relationship. That is, through deduction we can come to understand or establish the nature of something strange or uncertain by associating or grouping it with something known or understood.

Deductive arguments are formed in two ways:

1. General to particular. This is the kind most people think of when they think of deduction. For example, the classic syllogism:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

2. General to General. Another kind of deduction arrives at new generalizations through the syllogism. For example:

All trees have root systems.
All root systems need nitrogen.
Therefore, All trees need nitrogen.

In both of these examples we can discern three parts: a beginning statement of the generalization, an ultimate conclusion, and an intermediate step which associates the first statement with the conclusion. Notice also that three different identities or concepts (called “terms”) are present: in the second example they are “trees,” “root systems,” and “[things that] need nitrogen.” Deduction, then, associates or relates two terms by means of a common third (or middle or intermediate) term, so that we can understand how the first two terms are related to each other. The final statement of relationship is called a conclusion, and it expresses either convergence (identity) or divergence (non-identity)–either “X is Y” or “X is not Y,” or some similar statement. Another way to express this process is to say that deduction begins with two judgments sharing a common concept (term), and that by relating the two judgments to each other through the common term, a third judgment necessarily follows. Thus we have a three-part argument: judgment, judgment, conclusionary judgment. When this three-part deductive argument is arranged into a proper form (or structure), the argument is called a syllogism.

But before we get into syllogistic analysis, a little more needs to be said about deduction as a whole. We said earlier that deduction begins with an accepted generalization. Such a statement raises two questions: (1) Where do these generalizations come from and (2) Why are they accepted or assumed to be true?

The generalizations used in deductive thinking come from several sources:

Inductive thinking
Other deductive arguments (of the general to general type)
Revelation
Assumption (a priori givens that cannot be proved but that are assumed. All knowledge must begin with belief.)

In all four of these cases, the immediate source may be authority rather than personal experience. That is, the inductive conclusion, the deductive argument, the revelation, or the assumption may have been achieved by a third party who presents the generalization to us for acceptance on the basis of authority, in which case we take it on faith. You may not be able to do a large scale inductive experiment to find out whether a certain generalization is true, so you look in a book and accept the generalization of the authority.

To clarify the second question about why the generalizations are accepted, it should be said that a given generalization in an argument is assumed to be stabilized or true or agreeable to all parties unless challenged. Any given generalization may be false or unacceptable for the purpose of argument, so that any conclusion deduced from it, while perhaps formally valid (structurally all right), will be untrue. The generalizations used in deduction are often the products of induction, and are thus subject to every danger and error of the inductive process.

The dangers of deduction, then, are two:

1. The premises (generalizations) are not true, or are not adequate representations of reality, either because they have been derived through erroneous inductions or because they are false on the face. (ARGUMENT NOT TRUE)2. A formal error of procedure has been committed. (ARGUMENT NOT VALID)

The advantage deduction has over induction is mostly one of form: if the premises (or generalizations) are granted, and if the procedure is correct, then the conclusion necessarily follows. (Note that the strength of a deductive conclusion cannot be changed, though the conclusion can be overthrown.) Again, however, the quality of the conclusion is directly dependent on the quality of the generalizations on which it is based; and since inductively obtained generalizations are almost always somewhat tentative, we can seldom make absolutely inarguable deductions. Sherlock Holmes frequently makes valid deductions based on rather wild premises (which, of course, nearly always turn out to be correct); we cannot hope to be so lucky, so we have to be more careful.

All of us in our ordinary thinking combine induction and deduction to help us understand our world. We continually add facts together and then subtract the totals from each other to reach some final conclusion. The study of logic assures that our conclusions will be as accurate and sound as possible. As consumers, voters, researchers, jurors, writers, and so forth we especially want our conclusions to be worthwhile and trustworthy and reasonable (with all that implies), and for this some care and exactness and a good scrutinizing ability are necessary.

As you can see, the conclusions of both inductive thinking and deductive thinking can be wrong. In the case of induction, the leap can be too far or incorrect: probability always includes the negative possibility. In the case of deduction, the premises used in the argument might not be true after all. Truly, “To know is to risk being wrong,” for “Now we know only in part. . . .”

Hypothetical Syllogisms

Hypothetical syllogisms are different from standard syllogisms and thus have their own rules. In a hypothetical syllogism the first premise (or major proposition) presents an uncertain condition (“if A, then B”) or a problem (“either A or B”; “S and T cannot both be true”) which must then be properly resolved by the second premise so that a valid conclusion can follow. The resolution of the problem is always in the form of affirmation or denial. In this article, the three types of hypothetical syllogism we will cover are the conditional syllogism, the disjunctive syllogism, and the conjunctive syllogism.

The Conditional Syllogism

The major proposition of this syllogism presents a conditional argument to the effect that if one thing is true, then another is also true. For example, “If this is liquid nitrogen, then its temperature is more than 400 degrees below zero.” The truth of the antecedent (the first statement) implies or establishes the truth of the consequent (the second statement). It follows that (1) if the first thing is not true–that is, is denied (“This is not liquid nitrogen”)–then no conclusion can be drawn about the second thing, and (2) if the second thing is not true or is denied (“This is not more than 400 degrees below zero”), then neither is the first true, since the second thing would be true if the first had been true. Further, if the second thing is true, that does not of itself prove the first to be true–the antecedent proves the consequent true, but the consequent cannot prove the antecedent true (as in this case, other substances have temperatures below minus 400, like liquid hydrogen). Thus we have two valid and two invalid forms of the conditional syllogism. The valid forms are:

If A, then B
And A; therefore B
(Affirm antecedent; then affirm consequent)

If A, then B
But not B; therefore not A
(Deny consequent; then deny antecedent)

The invalid forms and their respective fallacies are:

If A, then B
And B; therefore A
Fallacy of Affirming the Consequent (ACq)

If A, then B
Not A; therefore not B
Fallacy of Denying the Antecedent (DA)

As you can see, it is the order of affirmation or denial that determines the validity of the procedure. Affirming the consequent first or denying the antecedent first is the invalid method.

The conditional syllogism can be a simple three-term argument such as “If that man is the mailman, the mail has arrived; and that man is the mailman, so the mail has arrived.” Or it can be a more complex argument with more than three terms. The form of the syllogism can be extremely loose, with a “then” statement in front of or behind an “if” statement; and the conclusion statement may use either “therefore” or “then,” “thus,” “so,” or “hence.” Note also that a negative statement can be an affirmation, as if you were to say, “I affirm that the towel is not in my locker.” In such a case, to keep the negative would be to affirm the statement and to drop the negative would be to deny the affirmation of it. This yields twelve more forms for the conditional syllogism. For instance:

If not A, then not B
But B; therefore A (valid)

If not A, then not B
But A; therefore B (DA)

If not A, then B
But not B; therefore A (valid)

Here are some examples of the various forms:

If life is a struggle, then I am fully involved in life, since I’m struggling to understand logic. And life is a struggle, so I am fully involved in life. (valid)
If Cindy went to the beach, she got sun- burned. But she didn’t get sunburned, so she must not have gone to the beach. (valid)
If Sarah comes in late, we will have to start over. I see we are starting over, so Sarah came in late. (ACq)
D. If this is Wilson’s book, it contains an essay by Swift. But this is not Wilson’s book, so it doesn’t contain a Swift essay. (DA)
If this movie is not about horses, then I will watch it. But I will not watch it, so it is about horses. (valid)
F. If we do not use premium gasoline, the engine will ping under acceleration. And the engine is pinging under acceleration, so we are not using premium gasoline. (ACq)

Exercise 1

Determine the validity of the following:

1. If you like molasses, grandma will buy you a bottle. But you don’t like molasses, so grandma will not buy you a bottle.

2. The tires must be replaced if the wear indicators are showing. The tires must be replaced. Therefore, the wear indicators are showing.

3. If you were self-motivated, you would be a good student. But you aren’t self-motivated, so you aren’t a good student.

4. If George wanted a richer, fuller life, he would read good literature. Ah, I see he is reading some good literature now. He must want a richer, fuller life.

5. If lettuce is on sale today, Sally will make grinder sandwiches. Sally is not making grinder sandwiches; therefore, lettuce is not on sale today.

6. If Birnam Wood moves to Dunsinane Hill, then Macbeth is in trouble. Macbeth is indeed in trouble, so Birnam Wood has moved to Dunsinane Hill.

7. If Breenthorpe didn’t get drunk all the time, the people would vote for him. But he does get drunk all the time, so the people won’t vote for him.

8. You will scold the carpenter if he has made you a bad table. Therefore, this carpenter has made you a bad table because I see you are scolding him.

9. If it’s after 10:00, the program has started. Ah, it is 11:30 now, so the program has started.

10. If this ice cream has peanut butter in it, I am allergic to it. Our tests confirm that it contains no peanut butter, so I am therefore not allergic to this ice cream.

The Disjunctive Syllogism

This syllogism presents two alternatives in an “either . . . or” form; one of the alternatives is for formal reasons assumed to be necessarily true, so that to deny one leaves the other as the only possibility. The two possibilities, called disjuncts, are stated in the major premise; one is and must be denied in the minor premise; and the other is affirmed in the conclusion. This is the valid form, which can be shown as follows:

Either A or B
Not A; therefore B
(Deny first disjunct; affirm the second)

Either A or B
Not B; therefore A
(Deny second disjunct; affirm the first)

The opposite procedure of first affirming and then denying is, however, incorrect. Except where the members are explicitly contradictory so that both could not possibly be true, the affirmation of one disjunct (in the minor premise) does not deny the other. For example, to say, “Either the power is off or the bulb is burned out; the power is off so the bulb is not burned out,” would be a fallacy, because, while we assume that one of the disjuncts is definitely true, both might be true–we did not check the bulb and so cannot be sure of its condition. Since the second disjunct has not been investigated, it cannot be denied by default. (Where the members of the disjunct are contradictory, as in “The plant is either alive or dead,” the argument should, to avoid confusion, be changed into the conjunctive form of syllogism and worked from there–see below, section #3.)

The fallacy, then, of first affirming one disjunct and then denying the other looks like this:

Either A or B
And A; therefore not B

Either A or B
And B; therefore not A

Fallacy of Affirming a Disjunct (AD)

Some examples of valid and invalid forms:

This is either a dictionary or a chemistry book. It is not a dictionary, so it is a chemistry book. (valid)
Either the battery is dead or something is wrong with the starter. Yes, the battery is dead, so there cannot be anything wrong with the starter. (AD)
Either the oven does not work or I left out the baking soda. But the oven does work, so I left out the baking soda. (valid)
Either I studied disjunctive syllogisms or I am going to blow this one. I did study disjunctive syllogisms, so I won’t blow this one. (AD)

Remember that, as in the third example, to drop a negative is to deny the affirmation of it.

A final note: You have probably already discovered that while for reasons of form we assume one of the two disjuncts to be true, it is entirely possible in many situations that they are both false and that the problem at hand cannot be reduced to a simple two-part opposition. There may be a third (or fourth, or fifth) alternative, or there may be an error in the expression of degree in one or both of the disjuncts (for example, “The book is either very entertaining and instructive or completely worthless”). To reduce a problem mistakenly to an either/or opposition is a material fallacy which is treated later in this handbook. But this all notwithstanding, in proper cases the disjunctive syllogism is still very useful because it helps us to decide between alternatives about whose truth, nature, and existence we have already agreed.

Exercise 2

Determine the formal validity of the following disjunctive syllogisms.

1. Either Freentop is a crook, or he is a very crafty individual. I know he is very crafty, so he is not then a crook.

2. Either Bleps are Snords or Vlots are Snords. Vlots are not Snords. Therefore, Bleps are Snords.

3. Either Q is N or T is Z. I just discovered that T is Z. Well then, we know that Q can’t be N.

4. Either we will have tribulation in this world, or life will be uninterrupted bliss. But life is not uninterrupted bliss, so we will have tribulation in this world.

5. Either that statement is wrong or Jones is in error. But that statement is right, so Jones is in error.

6. Either I am perfectly logical or you are logical. And since I am indeed perfectly logical, you are not logical.

7. Either you are in favor of our campus demonstration or you are a repressive fascist. And since you are against our demonstration, you are a repressive fascist.

8. I always write with either a ball point pen or a pencil. Today I decided to avoid pencils, so my letters this afternoon I have written with a ball point pen.

The Conjunctive Syllogism

In the major premise of this syllogism two propositions, called conjuncts, are presented, both of which cannot be true simultaneously. The minor premise proceeds to affirm the true conjunct and the conclusion then denies the remaining one, which must be false by definition. The valid form is:

A cannot be both B and C
A is B; therefore A is not C
(Affirm the first conjunct; deny the second)

A cannot be both B and C
A is C; therefore A is not B
(Affirm the second conjunct; deny the first)

Now, we know by definition that both conjuncts cannot be true. But further, if we briefly look at an example–“You cannot be both a mother and a father”–we can easily understand that perhaps neither conjunct is true. That is, A cannot be both B and C, but A does not have to be either one. A might be D or E. In our example, rather than a mother or a father, the person–you–might not be a parent at all. Therefore, an attempt to affirm one conjunct (the remaining one) by first denying the other is an invalid procedure:

A cannot be both B and C
A is not B; therefore A is C

A cannot be both B and C
A is not C; therefore A is B

Fallacy of Denying a Conjunct (DCj)

Examples of the forms:

A law cannot be both variable and fair. This law is variable, so it is not fair. (valid)
I cannot both go to a movie and finish my term paper. And I must finish my term paper. Therefore, I cannot go to a movie. (valid)
He knew he couldn’t both go to a movie and finish his paper, so he didn’t go to a movie. Thus he must have finished his paper. (DCj)
Fred cannot be both a genius and a fool. And Fred is certainly no genius. I guess that means he is a fool. (DCj)

Final note: As with other kinds of hypothetical syllogisms, the oppositions set up may have shortcomings of degree, representation, and so forth. To say, “You cannot be both sane and insane,” for example, would perhaps lead to some valid conclusion, but the conclusion would probably not be true since the mental health of a person is more accurately understood as a section along a line rather than as one pole or the other.

Exercise 3

Determine the validity of the following conjunctive syllogisms. How well do the oppositions accord with reality?

1. You cannot have both a sloppy lab technique and a good experiment. And since you do have a good experiment, you do not have a sloppy lab technique.

2. A lawnmower cannot be both durable and inexpensive. But this lawnmower is expensive, so it must be durable.

3. You cannot both eat your cake and have it, too. But since you haven’t eaten your cake, you must still have it.

4. Norman has proved repeatedly that he cannot be both a high-speed driver and a safe, accident-free driver. He has just arrived safely from Albuquerque, so he surely drove at a reasonable speed.

5. No person can be both rich and poor. And that Helen is certainly not poor. So, she must be rich.

Exercise 4

In each case, name the type of syllogism involved (conditional, disjunctive, conjunctive) and then tell whether or not it is valid. If invalid, name the fallacy involved.

1. Your suddenly reduced gas mileage can be caused by either low tire pressure, too much weight in the trunk, or clogged fuel injectors. Ah, look at the barbells, anvils, and other excessive weight in the trunk. That must be it. Your tire pressure and fuel injectors are not the cause after all.

2. If the new medicine is working, the tumor will have reduced in size. But the tumor has not reduced in size, so the new medicine is not working.

3. Electricity is obviously getting to the power supply if the fan, which runs off the power supply, is running. And the power supply must be getting electricity since the fan is indeed running.

4. The vegetables will be free from harmful pests if they have been fumigated. The inspector confirms that they are indeed free from harmful pests, so we can conclude that these vegetables have been fumigated.

5. The patient cannot be both completely healthy and have a blood pressure of more than 150/100. The patient is not completely healthy. So he must have a blood pressure of more than 150/100.

6. Either the witness is telling the truth or Frimpson is innocent. But the witness is lying so Frimpson is innocent.

7. The soup cannot have both a salt content greater than five pounds per thousand gallons and a specific gravity of more than 988. Look at the densitometer! The specific gravity of the soup is 992, so there must not be more than five pounds of salt per thousand gallons.

8. If this river is being polluted by the Simpson factory, the fish will be dying. The fish, however, are very healthy and alive. Thus, the river is not being polluted by the Simpson factory.

9. The car is rapidly losing oil pressure. Either the oil pump is failing or oil is leaking out. Hmm. The dipstick shows that the oil level is full. Well, then, the oil pump must be failing.

10. This machine cannot be in both air conditioning mode and heating mode at the same time. I asked John to check as he passed by if it was in heating mode and he said, “No.” So it must be in air conditioning mode.

Exercise 5

In each case, name the type of syllogism involved (conditional, disjunctive, conjunctive) and then tell whether or not it is valid. If invalid, name the fallacy involved.

1. We’ll know that the lodge is open if we see the flag flying. But look, the flag isn’t flying, so the lodge must be closed.

2. The patient has either the flu or food poisoning or amebic dysentery. Ah, here’s the lab report on the amebic dysentery test: Positive–he has it. So at least we know he doesn’t have the flu or food poisoning.

3. Jane is talking to Fred, and look how dilated her pupils are. She’s obviously interested in him because if a girl is interested in a guy, her pupils will dilate when she talks to him.

4. If the suspect had been a stranger, Igor would have barked loudly and attacked him when he came into Igor’s room to take the money Igor was guarding. But Igor didn’t make any noise or attack at all. So the suspect must be an acquaintance.

5. We would expect to find that the plane hit the ground in a nose up attitude if wind shear caused the crash. And look at this chart. The plane did indeed hit the ground in a nose up attitude. So wind shear was the cause of the accident.

6. It’s clear that the book we are looking for cannot be on the shelf and checked out at the same time. I just looked and it isn’t on the shelf, so it must be checked out.

7. If the ship can maintain a speed of twenty knots, then it can outrun the storm. The captain assures me that the ship is capable of a sustained 32 knots. So it must be able to outrun the storm.

8. The errors we are experiencing with the computer system must be caused by either a hardware or a software problem. The technician has just finished running tests on the hardware and it is working all right. So the errors must be caused by a software problem.

9. If the victim was murdered with this gun, it would mean that Johnson is almost certainly guilty. But the victim was murdered with a knife, so Johnson is almost certainly not guilty.

10. We’ve established that the same person cannot be both the bank robber and the getaway driver. This man was photographed driving the getaway car during the crime. He therefore cannot be the bank robber.

Exercise 6

In each case, name the type of syllogism involved (conditional, disjunctive, conjunctive) and then tell whether or not it is valid. If invalid, name the fallacy involved.

1. If this is a military-duty rated part, it will last at least 4000 hours. And–here’s the official certificate–it is indeed a military duty rated part, so is should last at least 4000 hours.

2. The Soviets can be receiving secret information about our new fighter either by satellite photography, telephone eavesdropping, computer signal interception, or an agent inside the fighter plant. And look, here’s a report from our Moscow operatives: conclusive proof that the Soviets are gathering data about our fighter from satellite photos. Well, I’m glad we can be sure that there’s no eavesdropping or inside agent problem.

3. This West German study says if a man kisses his wife passionately each day as he leaves for work, he will live at least five years longer than men who don’t. Old Harry Freen lived ten years longer than this group of non kissers, so that proves he kissed his wife passionately every day.

4. If Ted is working on his philosophy paper, he will be using his computer. Look, he’s using his computer now. That means he’s working on his philosophy paper.

5. Fred cannot both assemble the radio kit and ride his bike at the same time. He is not assembling the radio now, so he must therefore be riding his bike.

6. We will not get a call from the boss if inventory remains below 12,000 units. Inventory has risen to 17,500 units, so we will get a call from the boss.

7. After extensive investigation, we now know that this leaf browning means that the trees need either iron chelate or ammonium sulfate. Here’s the soil test: there is no iron in the soil at all. The trees therefore must need iron and not ammonium sulfate.

8. Mrs. Blake will yell at Norman again if he got another speeding ticket. Listen! Mrs. Blake is screaming at Norman at the top of her lungs. He clearly got another speeding ticket.

9. This shampoo cannot be both acidic and alkaline. The label says, “This product is not acidic.” Therefore, it must be alkaline.

10. This unlabeled canned meat cannot be both chicken and tuna. The grocer told me that he was certain it was not tuna. Therefore, it must be chicken.

Exercise 7

In each case, name the type of syllogism involved (conditional, disjunctive, conjunctive) and then tell whether or not it is valid. If invalid, name the fallacy involved.

1. We are losing pressure in the space shuttle. Only three things could cause that: we are out of oxygen, the pressurizing controller is not working, or we have an air leak. But the oxygen tanks are over half full and the pressurizing controller checks out okay, so we must have an air leak.

2. We know that the erratic performance of this computer must be caused by a flaw in any of these six modules, because they are in the section that Harvey dropped on the floor. And guess what? The second module I tested was defective. I was lucky to find the problem so early; now I can relax, knowing the others are okay.

3. We know that this urn cannot be both a valuable old antique and a modern creation. The art dealer has just determined that it is not a modern creation. Therefore it must be a valuable old antique.

4. Either that statement is true or the evidence is contradictory. But that statement is false, so the evidence is contradictory.

5. If the taller candidate in an election loses almost all of the time, then Senator Bribola will almost certainly win. But the taller candidate actually wins almost all of the time, so Senator Bribola will almost certainly lose.

6. Hmm. Look how high strawberry prices are. If Krimble had planted strawberries, he would have made a good profit on this field. But he planted tomatoes instead, so he isn’t going to make a good profit on the field.

7. This woman cannot be both the district attorney and the public defender. And she has just identified herself as the public defender. So she must not be the district attorney.

8. Jane will be in the garage if she is working on her bicycle. Ah, I see that she is indeed in the garage now, so she must therefore be working on her bicycle.

9. The satellite transponder cannot be both in self test mode and in transmission mode at the same time. Since we are receiving data now, it is obviously in transmission mode and therefore is not in self test mode.

10. Brian would marry Sally if he wanted to marry for money. But I conclude that he won’t have anything to do with Sally because he has stated that he doesn’t want to marry for money.

Review

Terms and Concepts

hypothetical syllogism
conditional syllogism
fallacy of affirming the consequent
fallacy of denying the antecedent
disjunctive syllogism
fallacy of affirming a disjunct
conjunctive syllogism
fallacy of denying a conjunct

Test Yourself

In each case, determine whether the argument is valid or not and then choose the single best answer.

1. If Jane is planning a trip to the mountains, she will be packing the picnic basket. Look, she’s packing the picnic basket now. That proves she’s planning a trip to the mountains.