Enthymemes, Sorites, Reducing of the number of terms categorical syllogism

Before defining any of the bellow let define syllogism and categorical syllogism. syllogism is a deductive argument consisting of two premises and one conclusion. Categorical syllogism is a syllogism consisting of three categorical propositions and containing a total of three different terms (“major terms, minor terms, middle terms”), each of which appears twice in distinct propositions.

Reducing of the number of terms

As described above a categorical syllogism can only have three terms but some time the categorical syllogism contains more than 3 terms. When this happens, we need to use method like Eliminate Synonyms and Complementary Terms. But When we try to reduce the number of terms by applying an operation to a sentence, we need to choose an operation so that the resulting sentence is logically equivalent to the original sentence. This way, we make sure we do not change the argument and turn it into a different argument.
1..Eliminate Synonyms

This method as it name suggest it is simply eliminate synonyms. Consider the following argument:

No wealthy persons are vagrants.

All attorneys are rich people.

Therefore, No lawyers are tramps.

Here we have a number of synonymous terms which can be modified to make the argument clear and concise. For instance, we could translate it into the following argument:

No rich persons are vagrants.

All lawyers are rich persons.

Therefore, No lawyers are vagrants.

This is a standard-form syllogism and is perfectly valid.

2  2. Complementary Terms

Any syllogistic argument involving four terms can be reduced to standard form if one of its four terms is the complement of one of the other three. Any argument containing five or six terms can be reduced to standard form if two or three of its terms are the complements of two or three of the others. These reductions are brought about through the use of conversion, obversion, and contraposition.

Consider this syllogistic argument which contains six terms:

No nonresidents are citizens.

All noncitizens are nonvoters.

Therefore, All voters are residents.

One method of reducing this argument to a standard-form syllogism requires the use of conversion, obversion, and contraposition. Here are the steps:

        I.            using Conversion and then obversion the first premise. "No residents are citizens" becomes "All citizens are residents."

     II.            Then using contraposition (we apply contraposition to the minor premise) the second premise. "All noncitizens are nonvoters" becomes "All voters are citizens."

We now have the following perfectly valid syllogism:

All citizens are residents.

All voters are citizens.

Therefore, All voters are residents.

We have only three terms in the syllogism and all the other general rules for a syllogism are satisfied.

Enthymemes

An enthymeme is an argument that is expressible as a categorical syllogism but that is missing a premise or a conclusion. Enthymemes occur frequently while speaking and writing English for a number of reasons. Some of the reasons are

Ø it is simply boring to express every statement in an argument.

Ø The listener or reader’s intelligence is called into play when he or she is required

       to supply a missing statement, and his or her interest is thereby sustained.

Ø On other occasions the arguer may want to slip an invalid or unsound argument past an unwary listener or reader, and this aim may be facilitated by leaving a premise or conclusion out of the picture.

Many enthymemes are easy to convert into syllogisms.

1.     First the reader or listener must determine what is missing, whether it is the premise or the conclusion. To do so the reader must give attention to the indicator word.

·       If the enthymeme contains an indicator word, then it is missing a premise. This may be seen as follows.

                                                              i.      If an enthymeme contains a conclusion indicator, then the conclusion follows it, which means that the missing statement is a premise.

                                                            ii.      On the other hand, if the enthymeme contains a premise indicator, then the conclusion precedes it, which means, again, that the missing statement is a premise.

·       If, however, an enthymeme contains no indicator words at all then the missing statement could be either a premise or a conclusion. If the two given statements are joined by a word such as “and”, “but”, “moreover” or some similar conjunction, the missing statement is usually a conclusion. If not, the first statement is usually the conclusion, and the missing statement is a premise. To test this latter alternative, it may help to mentally insert the word “because” between the two statements. If this insertion makes sense, the missing statement is a premise.

2.      the next task is to write the missing statement. To do so, one must first identify its major term, minor term, middle term. This can be done by taking account of the terms that are given. Two of the terms in the given statements will match up with each other. Once this pair of terms is found, attention should be focused on the other two terms. These are the ones that will be used to form the missing statement. In constructing the missing statement, attention to the rules for syllogisms may be helpful (if the resulting syllogism is to be valid). For example, if the missing statement is a conclusion and one of the given premises is negative, the missing conclusion must be negative. Or if the missing statement is a premise and the stated conclusion is universal, the missing premise must be universal.

Example of Enthymeme

A.     The corporate income tax should be abolished; it encourages waste and high prices.

This Enthymeme don’t contain any indicator words and also it doesn’t contain any conjunction so to find what is the missing statement so we put because and that work so it is missing the statement is a premise and it is “Whatever encourages waste and high prices should be abolished”

B.     Animals that are loved by someone should not be sold to a medical laboratory, and lost pets are certainly loved by someone.

This Enthymeme don’t contain any indicator words but it has a conjunction so it is missing the statement is a conclusion “Lost pets should not be sold to a medical laboratory.”

C.     mars complete its orbit in more time than the Earth, because mars is further from the sun.

This Enthymeme have an indicator word and the indicator word is “because mars is further from the sun” and it is a premises indicator. And the missing statement is a premises and it is “Any planet further from the sun completes its orbit in more time than the Earth”.

Translating this argument into categorical form, we have

·       All planets further from the sun are planets that complete their orbit in more time than the Earth.

·       All planets identical to mars are planets further from the sun.

·       All planets identical to mars are planets that complete their orbit in more time than the Earth.

Sorites

A sorite is a chain of syllogisms in which the conclusion of one is a premise in another, in which all the conclusions except the last one are unexpressed, and in which the premises are so arranged that any two successive ones contain a common term. Or in simple terms it is a chain of categorical syllogisms in which the intermediate conclusions have been left out.

The rule in evaluating a sorites is based on the idea that a chain is only as strong as its weakest link. If any of the component syllogisms in a sorites is invalid, the entire sorites is invalid.

A sorites is in standard form when each of the component propositions is in standard

form, when each term occurs twice, when the predicate of the conclusion is in

the first premise, and when each successive premise has a term in common with the

preceding one.

Sorites appear in two distinct types: The Aristotelian and the Goclenian. It is the arrangement of the propositions within the sorite which determines what type it is.

1.  Aristotelian sorites arrangement

In the Aristotelian type of sorites the first premise contains the subject of the conclusion and the common term of two successive propositions appears first as a predicate and next as a subject. Here is an example of the Aristotelian sorite arrangement.

Aristotle is a man.

All men are mammals.

All mammals are living beings.

All living beings are substances.

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Therefore, Aristotle is a substance.

Rules for the Aristotelian Sorites

        I.            No more than one premise may be negative. If a premise is negative, it must be the last.

     II.            No more than one premise may be particular. If a premise is particular, it must be the first.

2.  Goclenian sorites arrangement

In a Goclenian sorite the arrangement is different. The first premise contains the predicate of the conclusion and the common term of two successive propositions appears first as subject and next as predicate. Here is an example of the Goclenian sorite

One who has no peace of mind is miserable.

One who lacks much has no peace of mind.

One who has many desires lacks much.

One who has many vices has many desires.

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Therefore, one who has many vices is miserable.

Rules for the Goclenian Sorites

        I.            No more than one premise may be negative. If a premise is negative, it must be the first.

     II.            No more than one premise may be particular. If a premise is particular, it must be the last. with the third premise, the final conclusion follows validly. The sorites is thus composed of two valid categorical syllogisms and is therefore valid.