Further Immediate Inferences: Categorical Equivalences Chapter 5.3 Further Immediate Inferences: Categorical Equivalences

It’s easier than it looks. General considerations: All terms have term complements, which consist of everything that is not in the class (i.e., everything that, when considered with the class itself, “completes” or “fills up” the universe of all things). Term complements are indicated by a bar over the term. The ‘xy’ in Axy, Exy, Ixy, Oxy are just variables (place holders) to refer to general cases of the subjects and predicates, respectively, each proposition type. As in normal, informal speech, there are different, equivalent ways of saying things to clarify what one means; similarly, there are different, equivalent ways of expressing each of the categorical proposition types to clarify what one means. The different, equivalent ways of expressing those propositions are the results of three processes: Conversion Obversion, and Contraposition Contraposition, conversion or obversion?

Conversion Only E and I propositions can be converted successfully. Converting an A or O proposition is a logical error (except for conversion by limitation) Accomplished in one step Reverse the subject/predicate order. xy  yx Hence, Exy  Eyx, etc. No dogs are reptiles.  No reptiles are dogs.

Contraposition Only A and O propositions can be contraposed. Contraposing an E or I proposition is a logical error (except for contraposition by limitation). Accomplished in two steps Reverse the subject/predicate order. xy  yx Change each class to its complement. yx  (not-y)(not-x) Hence, Axy  A(not-y)(not-x). All dogs are mammals.  All non-mammals are non-dogs.

Handy demonic devices? convErsIon contrApOsition Irrelevant appeal? “Hey! Look at what I can do!” “Yeah, but what has that got to do with anything?” “He’s got a point, you know.”

The Obversion Two-Step Fr. ob “against” + vertere “to turn”; hence, a proposition seemingly “turned against” itself. Propositions of any type (A, E, I, O) can be obverted successfully, i.e., they are equivalent in meaning to the original proposition. Accomplished in two steps Change quality (from affirmative to negative, or vice versa) A  E, E  A; I  O, O I Change the predicate to its complement xy  x(not-y); x(not-y)  x(not-not-y) (a.k.a. ‘y’) Hence, Axy  Ex(not-y), etc. All dogs are mammals.  No dogs are non-mammals. Obversion Two-Step = Star Trek meets Monty Python The Obversion Two-Step

FYI You can run any proposition through all its possible permutations in four moves (usually beginning with obversion) and obtaining the original proposition on the final move--a handy way of checking whether you’ve done everything correctly. E.g., A obverts to E, convert the E, obvert to A, contrapose to original A Axy  Ex(-y)  E(-y)x  A(-y)(-x)  Axy Four moves = Trinity/Millsap football laterals All dogs are mammals.  No dogs are non-mammals.  No non-mammals are dogs.  All non-mammals are non-dogs.  All dogs are mammals. (original prop.)

Negatives Negativity can take different forms. Analytic: not, no, none, nothing, etc. Synthetic: un-, in-, im-, dis-, etc. Negativity can indicate different things. How classes relate (I.e., E or I) The “presence” of complements A syntactic convolution: “none but . . .,” etc. A contradiction: “It is false that . . . ,” etc.

Dealing with negatives If there are any simple operations that can be performed as a result of double negation, do so. Consider whether and how the negative affects the “quality” (positivity v. negativity) of the proposition. If there appears to be a contradiction, transform the proposition as demonstrated in the Square of Opposition. If there still appears to be a reference to a complementary class, indicate it via bars (recalling that, in this case as well, “two ‘wrongs’ do make a ‘right’”). Nixon’s Law: If two wrongs don’t make a right, try three. 5. Apply as appropriate the operations of categorical equivalence: obversion AND conversion or contraposition.