Notes: Figure, Mood and the Possible Types of Syllogisms (Part II)

3.      Incomplete & Compound Syllogism

Enthymeme

Enthymeme is called an incomplete syllogism in which one or the other proposition is not stated explicitly.

Such an incomplete syllogism is closer to the way we generally argue in everyday life.

If standard–form is the criterion, then it is not logically valid unless what is implicitly understood is taken into consideration. That is, it must be formally completed.

Examples

·         First Order Enthymeme

You have hurt your neighbour.

Therefore you have sinned against God.

(Major premise implicitly understood: Those who hurt their neighbours sin against God).

·         Second- Order Enthymeme

Those who hurt their neighbours sin against God.

Therefore you have sinned against god.

(Minor premise implicitly understood: You have hurt your neighbour).

·         Third Order Enthymeme

Those who hurt their neighbour sin against God.

And you have hurt your neighbour.

(Conclusion implicitly understood: Therefore you have sinned against God).

First, Second & Third Order

First-order Enthymeme: Major premise is implicitly understood.

Second-order Enthymeme: Minor premise is implicitly understood

Third -order Enthymeme: Conclusion is implicitly understood

If two propositions are adequate to convey the information, where is the need to have full-fledged syllogism? This question can be answered in two ways. When we deal with learned or well-informed persons or with ourselves, enthymeme will surely serve the purpose.

A full – fledged syllogism is needed when we have to educate not so well – informed, if not ill – informed persons.

We should not fail to notice close similarity between enthymeme and svarthaanumana and paraarthaanumana (inference for self and inference for others). The question can be answered in this way also.

Syllogism is formal and enthymeme is informal. Choice is subjective.

Sorites

If an argument consists of three or more than three premises, then such an argument is called sorites.

Also called polysyllogism.

·         Two kinds of sorites:

·         Aristotelian sorites and

·         Goclenian sorites.

The primary rules which govern sorites are the rules of the categorical Syllogism only.

Structure

In Aristotelian sorites the first premise is minor and the last premise is major.

In consecutive premises M is predicate in the first premise and in the next premise subject.

In sorites there are two or more than two conclusions which are implicit. Every such hidden conclusion functions as the premise.

Therefore a sorites consists of at least three syllogistic arguments and hence it consists of a chain of syllogisms which are interrelated. In order to arrive at the final conclusion these hidden conclusions also must be reckoned.

Consider this example: Aristotlean Sorites

Premises 1. All A are B. 2. All B are C. 3. All C are D. 4. All D are E ⸫All A are E.

Hidden conclusions (a and b) a. All A are C. 3 All C are D. b. All A are D. All D are E

Example: Goclenian Sorites

Premises 1. All A are B. 2. All C are A. 3. All D are C 4. All E are D ⸫All E are B

Hidden conclusions (a and b) a. All C are B. 3 All D are C. b. All D are B 4 All E are D

In this kind the predicate of the conclusion is the predicate of the first premise. Therefore the first premise is major. The subject of the conclusion is the subject of the last premise. Therefore the last premise is the minor.

The rules of this kind are as follows.

·         Only the first premise (major) can be negative.

·         Only the last premise (minor) can be particular.

One point should become clear at this stage. One kind of sorites is the reversal of the other. If we disregard the positions of premises, then the difference between these two kinds becomes insignificant.

 

4.      Conjunctive Syllogisms

Hypothetical syllogisms are based on "if/then" sentences.

Disjunctive syllogisms are based on "either/or" sentences.

Conjunctive syllogisms are based on "both/and" sentences.

Conjunction is a logical operation in which an operator (in this case the conjuctive, "and," symbolized by " . "), is used to connect exactly two propositions in such a way that the resulting compound proposition is true if and only if both component propositions are true, and false if either or both of the conjuncts are false.

A truth-functional proposition whose component statements are connected by the truth-functional operator "and" is called a conjunction; the component statements of a conjunction are called "conjuncts."

For example, if a man tells his wife, "Tomorrow morning I will wash the car, and I will fix the fence," when is he telling the truth?  If he does both tasks the next morning, he has told the truth (or at least predicted it!). If he only washes the car but does not fix the fence, or if he fixes the fence and does not wash the car, he has not lived up to his commitment and the statement is falsified.

A compound that is the conjunction of two statements is true if and only if each of the component statements is true.

"P and Q" is true if and only if "P" is true, and "Q" is true. Conjunctive syllogisms are the only kind that yield two conclusions (or more) from only one premise.

By simply joining together two propositions or terms in the first premise, we can separate them and affirm each as a separate conclusion. If both terms together are true, then each one separately is true also.

NOTE: THIS DOES NOT INDICATE ANY RELATION BETWEEN THE TERMS. THEY MIGHT HAVE NOTHING AT ALL TO DO WITH EACH OTHER, BUT THE FACT THAT BOTH ARE TRUE MAKES THE SYLLOGISM VALID.

Both P and Q are true. Therefore, P. Therefore, Q.

P . Q \ P \ Q

Combination of Conjunctive Statements

Four possible combinations of conjuctive statements based on their truth value:

1.      Chemistry is a science (T), and termites cannot read (T).

2.      Cats can talk (F) and humans can talk (T).

3.      Washington was the first U.S. president (T), and Lincoln is still alive (F).

4.      Trout are mammals (F), and beagles are reptiles (F).

Examples of conjunctive statements

Roses are red and violets are blue.

Roses are red.

Violets are blue.

Negative Conjunctive Syllogism

It is possible to negate the entire conjunct: "not [p and q]."

It is necessary to negate the entire conjunct since in a conjunction, it is the whole proposition that is asserted to be true.

In negating a conjunction, it does not necessarily mean, however, that both p and q are false. If both p and q are false, then the whole is false.

However, if only one of the conjuncts is false, then the whole is false as well. For example, if we say: "2 + 2 = 4 and 2 + 3 = 7," then the whole conjunction is false (note the inclusive quotation marks), because the truth of the conjunction depends on the truth of both of its parts. In the above example, the second part is false.

Thus, if a conjunction is negated, then one or both of the conjuncts must be false.

If not "P and Q" Then, either "not P" or "not Q" or "not P and not Q"

Conjunctions as a whole must be denied. If one part of the statement is false, then the conjunct is false as a whole (even if one part happened to be true). Both conjuncts must be true for the conjunction as a whole to be true.

Thus, if you are arguing against a conjunction, you need only show that at least one of the terms is false in order to show that the conjunct as a whole is false.

Here is a good example:

1. Deism = God exists (P) and miracles are not possible (Q).

2. But, miracles are possible (not Q; i.e., miracles are possible, since God exists)

3. Therefore, the entire conjunct that God exists and miracles are not possible (i.e., Deism = P and Q) is false.

The world view of deism requires for its articlation two premises: that God exists and miracles are not possible. Without either, you do not have deism. But to assert that God exists and to deny miracles is contradictory since you have already accepted the biggest miracle of all, namely, creation.

Therefore, it is possible to deny the second conjunct ("no" to no miracles) on the basis of the

first conjunct (God). But to deny one of the two conjuncts necessary to deism is to deny deism as a whole. How 'bout that! Try other examples from physics and psychology.

Example 1

·         The molecule of water requires two atoms of hydrogen and one atom of oxygen (H2O).

·         This molecule has two atoms of hydrogen and two atoms of oxygen (H2O2).

·         Therefore, this molecule is not water, but is hydrogen peroxide(?).

Example 2

·         Human beings consist of a body and soul.

·         This being has no soul.

·         This being is not human.

5.      Dilemma

What is?

The dilemma form of an argument attempts to force one to affirm at least one of two positions, neither of which he/she wants to affirm.

A dilemma certainly makes one think about the implications of what one believes.

A dilemma sets forth two hypothetical statements in a conjunctive manner in the first or major premise.

Then in the second or minor premise, a disjunctive "either/or" statement either affirms that one or the other of the antecedents is true (constructive), or denies that one or the other of the consequents is true (destructive).

The conclusion forces one to choose between (1) the consequents on the basis of affirmed antecedents or (2) the denied antecedents on the basis of denied consequents.

The dilemma consists of three propositions of which two constitute premises and third one is the conclusion.

Constructive dilemmas

In a constructive dilemma, the first premise consists of two conjoined conditional or hypothetical statements, and the second premise asserts the truth of one of the antecedents.

The conclusion, which follows logically by means of two modus ponens steps (affirming the antecedents in the hypotheticals), asserts the truth of at least one of the consequents.

If P then Q and If R then S Either P or R Therefore Q or S

(P ... Q) and (R ... S) P v R \ Q v S

Examples of Constructive Dilemmas

Example 1:

·         If this punch contains lemon, then Vince will like it, and if this punch contains lime, then Steve will like it.

·         This punch contains either lemon or lime.

·         Therefore, either Vince or Steve will like it.

Example 2:

·         If God exists, I have everything to gain by believing in Him.

·         And if God does not exist, I have nothing to lose by believing in Him.

·         Either God exists or God does not exist.

·         Therefore, I have everything to gain or nothing to lose by believing in Him. (Pascal's Wager!)

Destructive dilemma

The destructive dilemma is similar in form to the constructive dilemma, but the second premise, instead of affirming the truth of the one of the antecedents, denies one of the consequents.

The conclusion, which follows validly from two modus tollens steps, results in the denial of at least one of the antecedents.

If P then Q and If R then S Either not Q or not S Therefore not P or not R

(P ... Q) and (R ... S) ~ Q v ~ S \ ~P v ~R

Examples of Destructive Dilemma

·         If this punch contains lemon, then Vince will like it, and if this punch contains lime, then Steve will like it.

·         Either Vince will not like it or Steve will not like it.

·         Therefore, either this punch does not contain lemon or this punch does not contain lime.

Simple and Complex Constructive Dilemma

Simple Constructive Dilemma

In the simple constructive dilemma, the conditional premise infers the same consequent from all the antecedents presented in the disjunctive propositions.

Hence, if any antecedent is true, the consequent must be true.

Example: This form is illustrated by the reflections of a man trapped in an upper story of a burning building. If I jump, I shall die immediately from the fall And if I stay I shall die immediately from the fire.

I must either jump or stay – there is no other alternative. Therefore I shall die immediately.

Complex Constructive Dilemma

In the complex constructive dilemma, the conditional premise does not infer to the same consequent from all the antecedents presented in the disjunctive propositions.

Example:- If I win a million dollars, I will donate it to an Hamza Foundation. If my friend wins a million dollars, he will donate it to a wildlife fund. I win a million dollars or my friend wins a million dollars. Therefore, either an Hamza Foundation will get a million dollars, or a wildlife fund will get a million dollars.

The dilemma derives its name because of the transfer of disjunctive operator.

6.      Avoiding Dilemmas

Use of dilemma is restricted to some situations. When neither unconditional affirmation of antecedent nor unconditional denial of consequent is possible, logician may use this route.

It indicates either ignorance or shrewdness. When we face dilemma, we only try to avoid, but not to refute.

There are three different ways in which we can try to avoid dilemma:

·         Escaping between the horns of dilemma

·         Taking the dilemma by horns

·         Rebuttal of dilemma

All these ways only reflect escapist tendency. Only an escapist tries to avoid a problematic situation. Therefore, in logic they do not carry much weight.

Escaping between the horns of dilemma

·         Two consequents mentioned may be incomplete. If it is possible to show that they are incomplete then we can avoid facing dilemma. This is what is known as ‘escaping between the horns of dilemma’.

·         It can be shown that one "go between the horns" by showing that there is a valid third alternative to the two options specified in the minor or second premise. If a third alternative can be found, then neither of these disjuncts need be true.

·         It should be noted that even when third consequent is suggested it does not mean that this new consequents is actually true. In other words, the new consequent also is questionable.

·         Example:

1. If God willed the moral law arbitrarily, then he is not essentially good.

2. If God willed the moral law according to an ultimate standard beyond himself, then He is not God (because there is something beyond Him).

3. But God willed the moral law either arbitrarily or according to a standard beyond him.

4. Therefore, either He is not good, or He is not God.

Taking the dilemma by horns

·         In this method of avoiding dilemma, attempts are made to contradict the hypothetical propositions, which are conjoined.

·         In other words, you can dispute the implications of either of the hypotheticals in the first or major premise.

·         A hypothetical proposition is contradicted when antecedent and negation of consequent are accepted.

·         However, in this particular case it is not attempted at all.

·         Moreover, since the major premise is a conjunction of two hypothetical propositions, the method of refutation is more complex.

·         Example:

1. If one helps the sick, then he is fighting against God who sent the plague.

2. If one does not help the sick, then he is being cruel and inhumane.

3. One must either help the sick or not help them.

4. Therefore, one must either fight against God, or be cruel and inhumane.

Rebuttal of dilemma

·         You can "counter the dilemma" by answering a dilemma with another dilemma. So you can either deny the conjunction, or the disjunction, or the conclusion overall!

·         This is typically done by changing either the antecedents or the consequents of the conjunctive premise while leaving the disjunctive premise as it is, so as to obtain a different conclusion.

·         Example:

If taxes increase, the economy will suffer.

If taxes decrease, needed governmental services will be curtailed.

Taxes must either be increased or decreased.

Therefore, if follows that the economy will either suffer or that needed governmental services will be curtailed.

Further Examples to Illustrate Methods to Avoid Dilemmas

Complex Constructive Dilemma (CCD)

p1: If any government wages war to acquire wealth), then it becomes a rogue government) and if it wages war to expand its territory, then (it becomes colonial).

p2: (Any government wages war either to acquire wealth) or (to expand its territory)

q: It (becomes a rogue government) or (colonial).

Simple Constructive Dilemma (SCD):

p1: If (taxes are reduced to garner votes), then (the government loses revenue) and if (taxes are reduced in order to simplify taxation), then (the government loses revenue).

p2: (Taxes are reduced either to garner votes) or (to simplify taxation)

q : (The government loses its revenue).

Complex Destructive Dilemma (CDD):

p1: If (the nation wages war), then (there will be no problem of unemployment) and if (the nation does not revise her industrial policy), then (it will lead to revolution).

p2: The (problem of unemployment remains unsolved) or (there will not be any revolution).

q : (The nation does not wage war) or (the nation will revise her industrial policy).

Simple Destructive Dilemma (SDD):

p1: If (you are in the habit of getting up early), then (you are a theist) and if (you are in the habit of getting up early), then (you are a labourer).

p2: (you are not a theist) or (you are not a labourer).

q : (you are not in the habit of getting up early).

The first way of avoiding the dilemma, i.e., escaping between the horns of dilemma can be illustrated using 1 (CCD).

It is possible to argue that, when the government wages war, the motive is neither to acquire wealth nor to expand its territory in which case, the government is neither rouge nor colonial.

The motive may be to spread its official religion or personal vendetta or it may be to protect its interests. If the last one is the motive, then, it becomes difficult to find fault with such government.

Any one of the proposed alternatives or all alternatives to disjuncts may be false. There is no way of deciding what the situation is.

Likewise, consider fourth argument to illustrate the second method. I may concede that a person gets up early only because he wants to maintain health. So the purpose is not to worship God. Nor is he a labourer. Again, this is also an assumption. Rebutting of dilemma requires a different type of example.

Consider this one:

p1: If (teacher is a disciplinarian), then (he is unpopular among students) and if (he is not a disciplinarian), then (his bosses do not like him).

p2: (Teacher is a disciplinarian) or (he is not a disciplinarian).

q : (Teacher is unpopular among students) or (his bosses do not like him).

A witty teacher may respond in this way.

p1 : If (teacher is not a disciplinarian), then (he is popular among students) and if (he is a disciplinarian) then (his bosses will like him.)

p2 : (Teacher is not a disciplinarian) or (he is a disciplinarian)

q : (Teacher is popular among students) or (his bosses will like him)

Only a student of logic discovers that these conclusions of i and ii are not contradictories (you will learn about it in the forthcoming units) in the strict sense of the term. Hence, there is really no rebuttal.

Further, the dilemma, which an individual faces in day-to-day life, is very different. For example, moral dilemma has nothing to do with the kinds of dilemma which we have discussed so far.

Since the dilemma is a medley of both types of conditional propositions, i. e., hypothetical and disjunctive, it should follow the basic rules of hypothetical and disjunctive syllogisms.

It should affirm disjunctively the antecedents in the minor or deny disjunctively the consequents in the minor.

The dilemma is powerful if in the major there is a strong cause-effect relationship between the antecedent and the consequent and in the minor the alternatives are exhaustive and mutually exclusive. Again, the former is debatable.

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[This notes is prepared primarily on the basis of the IGNOU Study material on Philosophy- Logic and certain other materials. These notes are provided here for academic reference for students. No copyright to the above is being claimed.]