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People reasoning in ordinary language rarely express their arguments in the restricted patterns allowed in categorical logic.
But with just a little revision, it is often possible to show that those arguments are in fact equivalent to one of the standard-form categorical syllogisms whose validity we can so easily determine.
Let's consider a few of the methods by means of which we can "translate" ordinary-language arguments into the forms studied by categorical logic.
In the simplest case, we may need only to re-arrange the propositions of the argument in order to translate it into a standard-form categorical syllogism. Thus, for example, "Some birds are geese, so some birds are not felines, since no geese are felines" is just a categorical syllogism stated in the non-standard order minor premise, conclusion, major premise; all we need to do is put the propositions in the right order, and we have the standard-form syllogism:
No geese are felines. Some birds are geese. Therefore, Some birds are not felines.
In slightly more complicated instances, an ordinary argument may deal with more than three terms, but it may still be possible to restate it as a categorical syllogism. Two kinds of tools will be helpful in making such a transformation:
First, it is always legitimate to replace one expression with another that means the same thing. Of course, we need to be perfectly certain in each case that the expressions are genuinely synonymous. But in many contexts, this is possible: in ordinary language, "husbands" and "married males" almost always mean the same thing.
Second, if two of the terms of the argument are complementary, then appropriate application of the immediate inferences to one of the propositions in which they occur will enable us to reduce the two to a single term. Consider, for example, "No dogs are non-mammals, and some non-canines are not non-pets, so some non-mammals are pets." Replacing the first proposition with its (logically equivalent) obverse, substituting "dogs" for the synonymous "canines" and taking the contrapositive of the second, and applying first conversion and then obversion to the conclusion, we get the equivalent standard-form categorical syllogism:
All dogs are mammals. Some pets are not dogs. Therefore, Some pets are not mammals.The invalidity of this syllogism is more readily apparent than that of the argument from which it was derived.
Of course, the premises and conclusion of an ordinary-language argument may not be categorical propositions at all; even in this case, it may be possible to translate the argument into categorical logic. For each of the propositions of which the argument consists, we must discover some categorical proposition that will make the same assertion.
One especially common but troublesome instance is the occurrence of singular propositions, such as "Spinoza is a philosopher." Here the subject clearly refers to a single individual, so if it is to be used as the subject term of a categorical proposition, we must suppose that it designates a class of things which happens to have exactly one member. But then the categorical proposition that links Spinoza with the class designated by the term "philosopher" could be interpreted as an A proposition (All S are P) or as an I proposition (Some S are P) or as both of these together. In such cases, we should generally interpret the proposition in whichever way is most likely to transform the argument in which it occurs into a valid syllogism, although that may sometimes make it less likely that the proposition is true.
Other cases are easier to handle. If the predicate is adjectival, we simply substantize it as a noun phrase in order to make a categorical proposition: "All computers are electronic" thus becomes "Some computers are electronic things," for example. If the main verb is not copulative, we simply use its participle or incorporate it into our predicate term: "Some snakes bite" becomes "Some snakes are animals that bite." If the elements of the categorical proposition have been scrambled, we restore each to its proper position: "Bankers? Friendly people, all" becomes "All bankers are friendly people." And, in a variety of cases your texbook discusses in detail, the statements of ordinary language often contain significant clues to their most likely translations as categorical propositions.
Remember that in each case, our goal is fairly to represent what is being asserted as a categorical proposition.
To do so, we need only identify the two categorical terms that designate the classes between which it asserts some relation and then figure out which of the four possible relationships (A, E, I, or O) best captures the intended meaning.
It is always a good policy to give the proponent the benefit of any doubt, whenever possible interpreting each proposition both
in a way that recommends it as likely to be true and in a way that tends to make the argument in which it occurs a valid one.
Occasionally these methods are not enough to provide for the translation of ordinary-language arguments into standard-form categorical syllogisms.
Next, we examine a few special instances that require a more significant transformation.
In order to achieve the uniform translation of all three propositions contained in a categorical syllogism, it is sometimes useful to modify each of the terms employed in an ordinary-language argument by stating it in terms of a general domain or parameter. The goal here, as always, is faithfully to represent the intended meaning of each of the offered propositions, while at the same time bringing it into conformity with the others, making it possible to restate the whole as a standard-form syllogism.
The key to the procedure is to think of an approriate parameter by relation to which each of the three categorical terms can be defined. Thus, for example, in the argument, "The attic must be on fire, since it's full of smoke, and where there's smoke, there's fire," the crucial parameter is location or place. If we suppose the terms of this argument to be "places where fire is," "places where smoke is," and "places that are the attic," then by applying our other techniques of restatement and re-arrangement, we can arrive at the syllogism:
All places where smoke is are places where fire is. All places that are the attic are places where smoke is. Therefore, All places that are the attic are places where fire is.This standard-form categorical syllogism of the form AAA-1 is clearly valid.
Another special case occurs when one or more of the propositions in a categorical syllogism is left unstated. Incomplete arguments of this sort, called enthymemes are said to be "first-," "second-," or "third-order," depending upon whether they are missing their major premise, minor premise, or conclusion respectively. In order to show that an enthymeme corresponds to a valid categorical syllogism, we need only supply the missing premise in each case.
Thus, for example, "Since some hawks have sharp beaks, some birds have sharp beaks" is a second-order enthymeme, and once a plausible substitute is provided for its missing minor premise ("All hawks are birds"), it will become the valid IAI-3 syllogism:
Some hawks are sharp-beaked animals. All hawks are birds. Therefore, Some birds are sharp-beaked animals.
Finally, the pattern of ordinary-language argumentation known as sorites involves several categorical syllogisms linked together. The conclusion of one syllogism serves as one of the premises for another syllogism, whose conclusion may serve as one of the premises for another, and so on. In any such case, of course, the whole procedure will comprise a valid inference so long as each of the connected syllogisms is itself valid.
Sorites most commonly occur in enthymematic form, with the doubly-used proposition left entirely unstated.
In order to reconstruct an argument of this form, we need to identify the premises of an initial syllogism,
fill in as its missing conclusion a categorical proposition that legitimately follows from those premises, and then apply it as a premise in another syllogism.
When all of the underlying structure has been revealed, we can test each of the syllogisms involved to determine the validity of the whole.
Understanding how these common patterns of reasoning can be re-interpreted as categorical syllogisms may help you to see why generations of logicians regarded categorical logic as a fairly complete treatment of valid inference. Modern logicians, however, developed a much more powerful symbolic system, capable of representing everything that categorical logic covers and much more in addition.