For this purpose, we must consider every Conditional Proposition as a Universal-affirmative categorical Preposition, of which the Terms are entire Propositions, viz. the antecedent answering to the Subject, and the consequent to the Predicate.
For example, the Proposition “if A is B, X is Y” may be considered as amounting to this; “The case [or supposition] of A being B, is a case of X being Y.”
And then, to say (as in the Minor Premise and the Conclusion, of a constructive-conditional syllogism) “A is B; and therefore, X is Y,” is equivalent to saying “the present [or the existing] case is a case of A being B: therefore, this is a case of X being Y.”
Again, to say, “If Louis is a good king, France is likely to prosper,” is equivalent to saying, “The case of Louis being a good king, is a case of France being likely to prosper;” and if it be granted as a minor Premise to the Conditional Syllogism, that “Louis is a good king;” that is equivalent to saying, “the present case is the case of Louis being a good king;” from which you will draw a conclusion in Barbara, (viz. “the present case is a case of France being likely to prosper,”) exactly equivalent to the original Conclusion of the Conditional Syllogism: viz. “France is likely to prosper.”
As the Constructive Conditional may thus be reduced to Barbara, so may the Destructive, in like manner, to Celarent: e.g. “If the Stoics are right, pain is no evil: but pain is an evil; therefore the Stoics are not right;” is equivalent to: “The case of the Stoics being right, is the case of pain being no evil; the present case is not the case of pain being no evil; therefore the present case is not the case of the Stoics being right.” This is Camestres, which, of course, is easily reduced to Celarent.
Or, if you will, all Conditional Syllogisms may be reduced to Barbara, by considering them all as Constructive; which may be done, as mentioned above, by “converting by negation” [contraposition] the major Premise. ((See Book II, Section 3))
The reduction of Hypotheticals may always be effected in the manner above stated; but as it produces a circuitous awkwardness of expression, a more convenient form may in some cases be substituted. For example, in the example above, it may be convenient to take “true” for one of the Terms: “that pain is no evil is not true; that pain is no evil is asserted by the Stoics; therefore something asserted by the Stoics is not true.”
Sometimes again it may be better to unfold the argument into two Syllogisms: e.g. in a former example; first, Louis is a good king; the governor of France is Louis; therefore, the governor of France is a good king.” And then, second, “every country governed by a good king is likely to prosper,” etc.
A Dilemma may of course((See Section 5)) be reduced into two or more categorical Syllogisms.
When the Antecedent and Consequent of a Conditional have each the same Subject, you may sometimes reduce the Conditional by merely substituting a categorical Major-Premise for the conditional one: e.g. instead of “if Cassar was a tyrant, he deserved death; he was a tyrant, therefore he deserved death;” you may put for a major, “All tyrants deserve death;” etc.
But it is of no great consequence, whether Hypotheticals are reduced in the most neat and concise manner or not; since it is not intended that they should be reduced to Categoricals, in ordinary practice, as the readiest way of trying their validity, (their own rules being quite sufficient for that purpose;) but only that we should be able, if required, to subject any argument whatever to the test of Aristotle’s Dictum, in order to show that all reasoning turns upon one simple principle.
Source: Richard Whately, “Elements of Logic”. Longmans, Green, Reader and Dyer (London: 1870).
