Odi et amo. Quare id faciam, fortasse requiris? Nescio, sed fieri sentio et excrucior.
Definitions are useful tools. They can be used to introduce new terms, precisify extant terms or analyze intuitive concepts. In this paper I discuss a threat to the analysis of pairs of intuitive contraries like "tall"/"short," "hot"/"cold" and "good"/"bad." I use an example from the history of philosophy to show how independently defining each side of such a pair is apt to lead to contradiction, then point out an analogous problem for the response-dependence semantics for "good"/"bad" advanced by Prinz, as well as for the fitting-attitude semantics advocated by Blackburn, Brandt, Brentano, Ewing, Garcia, Gibbard, McDowell and Wiggins.
Explicit definitions have a characteristic form. To define a predicate "F," one formulates a universally quantified biconditional, where "F(x)" occurs alone on the left-hand side and a more or less complex proposition "[PHI](x)" occurs on the right-hand side:
(1) ([for all]x)(Fx [equivalent to] [PHI](x))
Definitions may be criticized for being uninformative [PHI] is not independently understood), though even uninformative definitions may be helpful when contextualized in a broader theory. Worse, definitions may be circular [PHI] contains "F"). Even worse, though, prima facie contraries like "F" and "unF" may be independently defined in a careless way. If they really are contraries, a contradiction lurks just around the corner.
If the definientia of "F" and "un-F" are not themselves contraries, wherever their extensions overlap, a contradiction will crop up, as the following formalization shows:
(2) ([for all]x)(Fx [equivalent to] [PHI]x)
(3) ([for all]x)(un-Fx [equivalent to] [PSI]x)
(4) [logical not] (there exist]x)(Fx [conjunction] un-Fx)
(5) ([logical not]x)([PHI]x [conjunction] [PHI]x)
(6) (there exist]x)(Fx [conjunction] un-Fx)
When one encounters such a contradiction, one has six options beyond outright dialetheism. (1) First, one can give up the assumption that the presumed contraries really are contraries, eliminating 4. Second, one can give up one of the two definitions; that is, one can reject either 2 or 3. Third, one can replace one of the definitions to make the contraries into pure contradictories, i.e., make un-F equivalent to not-F:
(3') ([for all]x)(un-Fx [equivalent to] [logical not]Fx)
Fourth, one could keep the contraries but transform them into trouser words. (2) That is, one could say either
(3'') ([for all]x)(un-Fx [equivalent to] ([PHI]x [conjunction] [logical not]Fx))
(2') ([for all]x)(Fx [equivalent to] ([PHI]x [conjunction] [logical not]Fx))
Fifth, one can try to rejigger the definitions so as to keep both informative without leading to a contradiction. This strategy, however, leaves the theorist vulnerable to empirical refutation if it is discovered that the new definientia are co-instantiated. Finally, one can attack the evidence for co-instantiation.
1. Piety and impiety in the Euthyphro
In the Euthyphro, Socrates points out that the definition of piety as that which is loved by the gods leads to a contradiction because the gods quarrel (Plato 1941, 6e-8a). We can reconstruct his reductio as follows:
(7) Something is pious just in case there is some god that loves it, i.e., ([for all]x)(Px [equivalent to] ([there exists]y)(Gy [conjunction] yLx))
(8) Something is impious just in case there is some god that does not love it, i.e.,
([for all]x)(Ix [equivalent to] ([there exists]y)(Gy [conjunction] [logical not]yLx))
(9) Nothing is both pious and impious, i.e., [logical not] ([there exists]x)(Px [logical not] Ix)
(10) The gods quarrel, i.e., ([there exists]x)( [there exists]y)( [there exists]z)(x [not equal to] y Gx [conjunction] Gy [conjunction] xLz [conjunction] [logical not] yLz)
(11) Hence, there is something (the thing that the gods quarrel about) that is both pious and impious, i. …