SORITES, ISSN 1135-1349
Issue #10. May 1999. Pp. 15-18.
Broadening and Deepening Yoes: The Theory of Conditional Elements
Copyright © by SORITES and Joseph S. Fulda
Broadening and Deepening Yoes: The Theory of Conditional Elements
Joseph S. Fulda
In a well-written, interesting paper,Foot note 2_1 Yoes takes the classical view of indicative conditionals -- that they are truth-functional. He then deals with certain problematic cases of if statements by citing Russell to the effect that «grammar can hide logical form» and arguing that some if statements are simply not conditionals. To the question of which are and which are not, Yoes answers, somewhat satisfyingly, «Standard formal logic answers: when it satisfies Modus Ponens, Modus Tollens, etc. Some `if's do not satisfy these formal properties, and therefore, according to this standard, are not conditionals at all. Thus do the formal properties define the conditional...»
The problem with this view, once mine, is that it is a self-affirming view of the truth-functionality of the conditional -- a view that a non-classicist would say simply begs the question: Any putative counterexample to the classical view of the conditional can be written off with enough ingenuity in pragmatics as simply not a conditional. As succinctly stated by Yoes himself: «trained logical intuition sees a conditional behind every `if'.» «Trained intuition» is oxymoronic, yet quite apt here, for simple intuition would surely lead to Yoes' conclusion or some such or, on the other hand, to the rejection of the truth-functionality of the conditional. With sufficient thought, however, one finds that every if statement, while not necessarily a conditional (þ) has at least a conditional element (a notion to be elaborated on shortly) and we intend to show this by re-analyzing the examples from the literature Yoes cites with this in mind. In some cases, our refined view of if statements is even compatible with Yoes' particular sentential analyses -- this article is put forward as a refinement of, not a reply to, Yoes.
Simply put a conditional element is some part of, some central part of, the standard definition of the conditional, enough to justify the use of if. In an earlier paperFoot note 2_2 which sought to unravel the paradoxes of material implication -- both the disturbing fact that all conditionals with false antecedents are true and the disturbing fact that conditionals with unrelated antecedents and consequents could be true -- I explored alternative truth tables for the conditional that at least preserve the core meaning of implication -- that TþT þ T and that TþF þ F. This resulted in four species of þ: þ1, þ2, þ3, and þ4, with truth tables as below:≥
P Q P þ1 QP þ2 QP þ3 QP þ4 Q
F T T T F F
F F T F T F
Truth Table 1 Truth Table 2 Truth Table 3 Truth Table 4
Each of these captures some of the meaning of if, although þ1 alone is þ, þ2 being just the consequent, Q, þ3 being the biconditional P þ Q, and þ4 being conjunction, P þ Q. The thesis of that paper was that many of the problems attending material implication were the result of misreading biconditionals as simple conditionals, but as Yoes' cited examples show, misreading conjunctions or null-antecedent ifs as simple conditionals can also cause confusion.
It is also interesting to note that Yoes' condition for being a genuine conditional is met by each of these four variants of þ: All satisfy, for example, both Modus Ponens and Modus Tollens. Thus, P þ Q, P þ Q and P þ Q, ¬Q þ ¬P; likewise, Q, P þ Q and Q, ¬Q þ ¬P; and, finally, P þ Q, P þ Q and P þ Q, ¬Q þ ¬P.
Yet another way a conditional element can be present was discussed in two prior papers in this journalFoot note 2_3 -- universally general propositions are taken in predicate logic to be quantified conditionals. Quantified conditionals are not simple conditionals; they are normally best regarded as conjunctions of simple conditionals, but again the conditional element justifying the use of if is there: «If it looks like a duck, waddles like a duck, and quacks like a duck, then it is a duck» is an if statement properly transcribed as follows: (þx)((Lx þ Wx þ Qx) þ Dx).
Before re-analyzing the problematic if statements Yoes cites, I wish to consider a clearer example of the same sort: «If you want to talk law, then your client doesn't even have standing to bring this suit at law.» Now Yoes would presumably read this as «You may want to talk law and your client doesn't have standing to bring this suit at law,» viz. as a conjunction with a modal. This is not mistranslated, but neither is it the best translation, although it is one that has a conditional element, þ4. One reason it is not the best translation is because although the speaker clearly believes that the listener may want to talk law, there is no modal, express or implied, in what he states. Nothing in the language, that is, so much as suggests that an assertion is being made in the protasis. A better reading, therefore, is that the statement being made is simply the consequent: This, too, has a conditional element, þ2. On this reading, the sentence affirms only the proposition «Your client doesn't even have standing to bring this suit at law» with the «antecedent» being simply a rhetorical flourish. I have no doubt that Yoes considered and rejected this alternative, because dismissing a clause as a mere rhetorical flourish is something no logician -- at least no logician who believes in truth-functional semantics -- can possibly be comfortable with. Better to posit a modal not textually supported (but consistent with the text) than to let the clause go entirely. Since, however, the rhetorical flourish is textually supported, I cannot agree.
However, this translation is also not the best; it is incomplete and exactly for the reason that Yoes might object to it altogether: It doesn't make full sense out of the antecedent. Even rhetorical flourishes have their syntax and there is no «if» without a corresponding «then.» On our (limited) account, the «then» does not correspond to the «if.» An extended account is therefore necessary. The sentence is elliptical and the best reading is «If you want to talk law, then let's talk law: Your client doesn't even have standing to bring this suit at law.» Suddenly, all the problems evaporate. Judged at the level of surface grammar, we have a level one conditional (which is probably optative and therefore not propositional -- truth-functional), followed by a level zero atomic proposition. Judged as English, we have a rhetorical flourish which is incomplete as almost all of them are, followed by an assertion which is stated in full as is also the norm. Judged as a simple conditional, we resort -- Yoes' way or my limited account -- to the theory of conditional elements. Judged as a complex (layered) conditional, we have (so far) no need to do so here. (But the theory is needed anyway as exemplified by the material cited in n.2 and n.3.)
Without further ado, we take up Yoes' cases cited from the literature:
(1) If it rained, it did not rain hard.
Limited account: It did not rain hard.
Extended account: If it rained, it rained, but it did not rain hard.
(2) There are biscuits on the sideboard if you want one.
Limited account: There are biscuits on the sideboard.
Extended account: If you want a biscuit, I'll tell you where they are: There are biscuits on the sideboard.
Note that, especially for (1), the translation does not appear so readily to be multi-layered: «but» is normally «and» with a non-truth-functional twist, of course, and so we are back to Yoes. The statement as a whole may well be regarded as a conjunction, but not as a conjunction of a modal and the consequent but rather as a conjunction of a «conditional» rhetorical flourish and the «consequent.»
The author would like to acknowledge the clarifying comments of Neil Nelson and to dedicate this paper to his sister, Aviva.
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