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The Jonathan Dolhenty Archive |
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DEDUCTION: Part 4 by Jonathan Dolhenty, Ph.D.
Any sort of categorical proposition may be used in the construction of an argument. There will be as many varieties of the categorical syllogism as there are varieties of categorical propositions. Since most arguments seem to consist of a vast array of different propositions, sometimes constructed in a complex arrangement, and our minds do not seem to usually function in a nice, neat, simple manner, it's important to study a variety of syllogistic constructions. It's so easy to fall into fallacious reasoning when our arguments are constructed and arranged in a complex manner. Each kind of syllogism, moreover, presents its own special problems and needs to be analyzed carefully and the various parts of the argument studied separately. This is what we'll do in this chapter. There is a general rule we can use to help us in our attempt to deal with complex arguments. This rule has various sections and they are presented here:
Complex and Modal Syllogisms Complex Propositions in Syllogisms The single complex proposition is not difficult to work with. The special requirement necessary is that any complex terms used in the propositions be held strictly together and used in the same combination throughout the argument. Study the following argument and see if you can spot a problem:
The problem in this argument, of course, is we actually have four terms in the argument instead of the required three terms, as demanded by General Rule Number 1. The four terms are: (1) boys with red hair, (2) hot-tempered, (3) my students, and (4) boys. "Boys with red hair" is a complex term, whereas "boys" is not. The term "boys" does not have the same meaning as "boys with red hair." There are two different "objects" being referred to. The middle term "boys with red hair" is not retained in its completeness. The argument, to be correctly formulated, should read:
Now we have only three terms, satisfying General Rule Number 1. Remember, however, whether the first premise is true or not is not at issue here. We're merely supposing its truth for the sake of the argument. Most of us have probably known red-haired boys who were not hot-tempered. Sometimes we hear or read an argument which appears to be complex but really is not. It may merely contain superfluous phrases or incidental clauses which are essentially unnecessary to the argument. We can take out these unnecessary words and clean up the argument to make it more understandable and capable of being analyzed. Study the following argument:
This argument contains some words which are not essential to the argument. If we take these words out, we can simplify the argument, arrange it properly, and then easily analyze it. We can eliminate the following as unnecessary:
Here is the argument reduced to a syllogism:
We can see that the syllogism contains only three terms:
The middle term (M) is "food containing vitamin C." The minor term (S) is "fresh fruits and vegetables," and the major term (P) is "helps to prevent scurvy" or "good as a preventative against scurvy," (these mean the same thing even though stated slightly differently). One situation where special caution is necessary is an argument which includes a qualifying clause or complex term containing a negation. If we aren't careful, we might conclude that an argument is invalid when it really is valid but seems invalid because of loose language. We may have to alter the wording to make it easier to test the validity of such an argument. Consider this argument:
The minor premise, "Some tennis players do not train," appears to be negative, but it actually is not. The minor premise really means, "Some tennis players are athletes who do not train." The basic part of the proposition is, "Some tennis players are not athletes," and the qualifying clause, "who do not train," belongs to the term "athletes." The minor premise," Some tennis players are athletes who do not train," is affirmative, not negative. Modal Propositions in Syllogisms A modal proposition is a composite single sentence in which the copula is so modified as to express the manner (mode) in which the predicate belongs to the subject. For example, the proposition, "It is necessary at times that a country tighten its economic belt," would be better rendered, "All countries at times must tighten their economic belt." As premises, modal propositions are best resolved by changing the sentence in such a way that the mode is made a part of a term, instead of being a statement. Here again, the meaning of a term becomes important. Consider this argument:
This argument would be better stated:
Sometimes arguments are formulated wherein each premise contains a different kind of mode. It's best to avoid this situation. For example, an argument might contain a premise which includes what is called a mode of necessity and another premise which includes what is called a mode of possibility. This may lead to a problem with the conclusion of the argument. Consider this argument:
What is to be the conclusion? "It is possible" or "It is necessary"? We have two different modes and this presents us with a difficulty as to which is to appear in the conclusion. We can get out of this difficulty by a change in the wording. Let's put the mode of necessity into the verb of the major premise: "All firemen must be brave." Then, let's make the mode of possibility an adjective in the minor premise: "Some firemen are possible heroes." Now, let's look at the argument:
We can see now that there's no problem with drawing the proper conclusion in the argument. Overtly Multiple Syllogisms There are some syllogisms which may contain overtly multiple propositions. Overtly multiple categorical propositions are those which are plainly composed of two or more sentences. When dealing with such propositions, the following procedures will apply:
There are five types of overtly multiple propositions: copulative, adversative, relative, causal, and comparative. The Copulative Type This type of multiple syllogism contains one or two copulative premises. In most cases, the sense of the propositions is so clear that no further reduction is necessary. Consider the following argument:
There are actually three arguments here and each could be resolved out and arranged as an independent syllogism. Normally, since the argument is not that complicated, we wouldn't bother to do this. For the purposes of illustration, however, we'll go ahead and construct the three syllogisms possible:
The Adversative Type This adversative proposition is similar to the copulative. Again, the component parts can be resolved into separate syllogisms, although this is seldom necessary unless the argument is very complicated. In the example below, the adversative portion of the sentence, "though difficult," is an integral part of the major term and should remain with it throughout the argument.
The major term really consists of "important" and "difficult. The conclusion may take one or both parts. It would be equally consistent to conclude:
The Relative Type The relative proposition involves the element of time or place in relation to which something occurs. Consider the following argument:
This type of argument should not present any difficulties. The Causal Type A causal proposition used as a premise has a main sentence and an incidental clause. The incidental clause will usually begin with a term like "because" or "for" and gives the reason why the main sentence is considered to be true. The argument may follow either the main sentence or the incidental clause. Consider this argument:
There are some cases where a causal syllogism may appear to involve four terms, violating General Rule Number 1 (only three terms permitted). When considered closely, however, it will be noticed that one of the terms is simply given as an example of the principle expressed. Consider the following argument:
It appears that there are four terms in this argument: "lonely person," "someone with whom to share things," "selfish person," and "miserable." But let's look closely. The middle term (M) is obviously, "one who doesn't have someone with whom to share things." What are the other terms? It seems in this argument that the words "lonely person" in the major premise should be taken simply as an example of "everyone who doesn't have someone with whom to share things." That part can be eliminated without harming the meaning intended. The above argument, therefore, can be restated as follows:
We need to consider the logical intention in an argument, not the grammatical construction. If a change of wording will produce a valid conclusion, there is no problem with the syllogism. The Comparative Type The comparative proposition should present no difficulty. The premise involves a comparison between two objects under consideration. If the comparative term is part of a complex term, we have to see that it is not separated from the complex term because it is an integral part of it. Consider the following argument:
The middle term (M), "are more important than human luxuries," is a complex term containing a number of words which need to be kept together throughout the argument. The ideas being compared are "Food and water" and "human luxuries." Covertly Multiple Syllogisms Covertly multiple propositions in syllogisms may present some serious difficulties. The multiple character of these propositions may be hidden within some word or phrase which looks innocent on the surface but, if not caught, can lead to an inconsistent argument. This is why this type of proposition is called covert. The real meaning may be hidden in it and not immediately apparent. When this type of proposition appears, it almost always needs to be resolved into its exponents and each constructed into an independent syllogism to test the validity. Four types of covertly multiple propositions will be discussed: exclusive propositions, exceptive propositions, reduplicative propositions, and specificative propositions. Exclusive Propositions in Syllogisms These are particularly tricky. The exclusive sentence usually contains some word like "only," "alone," "none but," and "solely." Consider these propositions:
In these propositions, something is predicated of something else in an exclusive fashion. Traditional logic reduces the exclusive proposition to the standard form for categorical propositions. Thus:
The following examples also show this resolution: Only citizens can vote. - is the same as - All those who can vote are citizens. None but the brave deserve the fair. - is the same as - All who deserve the fair are brave. None but seniors are eligible. - is the same as - All those eligible are seniors. The vote is a constitutional right. - is the same as - Only citizens can vote. Exclusive propositions that are included in an argument should be resolved into their component sentences so as to avoid any possible fallacy which may enter into the argument. Consider this argument:
Is this a legitimate conclusion? It appears valid. It would be perfectly correct if the argument was stated this way:
The word "only" in the first of the two syllogisms may raise some doubt about the validity of the conclusion in that syllogism. Does this innocent-looking word make a difference in the conclusion of the first syllogism? It may not seem so but, on the other hand, how can we be sure? Let's check it out. We can resolve the minor premise, "Only the wicked are wealthy," and construct a double syllogism from the resulting two propositions. Here is what the double syllogism looks like:  Let's examine the result. The first syllogism follows the First Figure. Its rule states: The minor premise must be affirmative and the major premise must be universal. This rule is carried out in the first syllogism. Therefore, there is no inconsistency in the first syllogism. Now, let's consider the second syllogism. This syllogism also follows the First Figure. It must also follow the rule: The minor premise must be affirmative. In the second syllogism, however, the minor premise is negative: "Those who are not wicked are not wealthy." This means we have to have a negative conclusion with a universal major term (since it is the predicate of a negative proposition). In the major premise, the major term is particular (it is the predicate of an affirmative proposition). So we find in the conclusion an illicit major and the argument is invalid. The original conclusion, "Only the wicked are happy," cannot be made with consistency from the premises. The little innocent word "only" must be omitted in the conclusion. The original syllogism should argue:
Let's consider a syllogism where both premises contain an exclusive proposition:
Each of the premises contains two hidden statements. The major premise is composed of these two:
The minor premise is composed of these two:
Here is the formulation for the double syllogism:  Let's examine the double syllogism. We can see that both syllogisms follow the First Figure. The rule for this figure states: The minor premise must be affirmative and the major premise must be universal. We can see that in both syllogisms the minor premise is affirmative and the major premise is universal. In the second syllogism we have a negative conclusion. This does not, however, involve an illicit major since the major term is also a universal term in the major premise (it is the predicate of a negative proposition). Both arguments are valid. We can, therefore, combine the conclusion of the first syllogism, "The wicked are happy," with the conclusion of the second syllogism, "Everyone non-wicked is unhappy." From this we form a single exclusive proposition: "Only the wicked are happy." The original syllogism is valid and comprises a legitimate argument. Consider this syllogism:
This is a First Figure syllogism and its rule is: "The minor premise must be affirmative." At a glance, the syllogism violates this rule since the minor premise, "The men are not wicked," is not affirmative, but negative. We know this is the minor premise because it's clear that the subject of the conclusion, "The men," which is always the minor term, is also the subject of this premise. We have a negative conclusion, with a universal major term, which is a particular term in the major premise. This seems to be an illicit major in the conclusion. But let's consider things again. The major premise is an exclusive proposition involving two component statements: "The wicked are wealthy," and "Whoever is non-wicked is non-wealthy." We can construct a syllogism on the last of these two propositions and we get the following:  As we can see, the syllogism is valid. It follows the First Figure and satisfies the rule of the First Figure: The minor premise must be affirmative and the major premise must be universal. The conclusion is validly drawn from the premises. This syllogism simply followed the hidden proposition that is implied by the exclusive word "only." A valid argument was the result. This points out that exclusive propositions must be examined closely when they are used as premises in an argument. Exceptive Propositions in Syllogisms In an exceptive proposition, the predicate is denied of some part of the denotation of the subject. Here are some examples:
These propositions can also be translated into standard-form categorical propositions: All students except freshmen may smoke. - becomes - Freshmen alone among students may not smoke. - becomes - All students who may not smoke are freshmen. The latter proposition is the standard "A" (universal affirmative) sentence. The danger involved in exceptive propositions used as a premise in a syllogism is similar to the danger in using exclusive propositions. It may be necessary, therefore, to resolve an exceptive proposition into its component parts and formulate a syllogism for each. Any hidden fallacy can then be uncovered. Consider the following argument:
A quick glance at this argument would lead one to believe it sounds plausible. But what does the major premise actually state? It says that "the wounded members" received a decoration. It says nothing about those members of the squad who were not wounded. It does not say that those not wounded did not receive a decoration. The member who was not wounded may have been the bravest of them all and he may have received a decoration for something else, if not for a wound. The proposition, of course, does not say that this unwounded member did receive a decoration for something else. Nothing is affirmed or denied regarding this one member. Let's resolve the minor premise into its component parts and construct separate syllogisms for each. Then we can examine the results and see if agreement is made with our first impression about the argument. The minor premise is resolved into: "All were wounded members of the squad" and "one was not a wounded member of the squad." Here is the double syllogism:  The first syllogism, as we can see, is legitimate. It follows the First Figure and the rule: The minor premise must be affirmative and the major premise must be universal. The second syllogism is also a First Figure. It does, however, violate the first part of the rule: The minor premise must be affirmative. Because of this, we have an illicit major in the conclusion. Consider: the major term is particular (pP) in the major premise since it is the predicate of an affirmative sentence, and it is a universal term in the conclusion (uP) since it is the predicate of a negative sentence. This violates General Rule Number 2: Neither the major nor minor term may be a universal in the conclusion, if it was only a particular term in the premises. It is inconsistent to say in the conclusion that "All, except, one were decorated." The argument does not permit any conclusion about one member. He may or he may not have been decorated for some reason other than for being wounded. The final conclusion is not only inconsistent and false, but even the conclusion of the first syllogism cannot be accepted as stated. The one member of the squad who was not wounded belongs to the squad and he may have been decorated for some other reason than being wounded. But he may not have been decorated as well. We don't know. All we can properly conclude is that members of the squad who were wounded were decorated, but that has already been said in the major premise. We could change the above situation by modifying the major premise into an exclusive proposition: "Only the wounded members of the squad received a decoration." The proposition is still multiple and it can be resolved into these component parts: "All wounded member of the squad received a decoration" and "No unwounded members of the squad received a decoration." An acceptable conclusion could be made from these sentences without any problem. Each syllogism will follow the First Figure and both will conform to the rule: The minor premise must be affirmative and the major premise must be universal. Here is the new double syllogism:  Reduplicative and Specificative Propositions in Syllogisms Just a brief word or two about reduplicative and specificative propositions. Once understood, they should present no difficulty. A reduplicative proposition contains an expression which duplicates the subject or predicate and implies the reason or cause for the connection between subject and predicate. The following examples illustrate this type of proposition:
A stipulative proposition contains an expression which stipulates something about the subject or predicate. The following examples illustrate this type of proposition:
These should present no difficulty if it is remembered to keep the reduplicative or specificative phrase with the term it qualifies. They should not be separated. Consider this argument:
The above syllogism is quite correct because it has kept the stipulation or qualifying phrase "in as much as he is rational" with the term "Man" with which it is associated. Now, however, consider this argument:
See the flaw in the argument? NEXT - Part 5: The Hypothetical Syllogism
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