Today’s test will be on the conditional connective. As always, I will just give you a sentence in English and you will have to symbolize it in the language of sentential logic. Remember that symbolization is a procedure whereby you extract the logical form of a sentence. This is not just a translation procedure and there is no straightforward algorithm for it; it sometimes involves a difficult process of interpretation. But it is crucially important for logic, for the logical techniques that we are going to develop will apply to well-formed formulas of the language of sentential logic and only indirectly to the sentences of English. It will apply, that is, to the logical forms of the sentences of English

Varzi, Achille C.

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1Beth Too, But Only IfAchille C. VarziDepartment of Philosophy, Columbia University, New York[Final version published in Analysis  65:3 (2005), 224–229]Instructions. Today’s test will be on the conditional connective. As always, Iwill just give you a sentence in English and you will have to symbolize it in thelanguage of sentential logic. Remember that symbolization is a procedurewhereby you extract the logical form of a sentence. This is not just a translationprocedure and there is no straightforward algorithm for it; it sometimes involvesa difficult process of interpretation. But it is crucially important for logic, forthe logical techniques that we are going to develop will apply to well-formedformulas of the language of sentential logic and only indirectly to the sentencesof English. It will apply, that is, to the logical forms of the sentences of English.The sentence you have to symbolize is this:(*) If Alf went to the movie then Beth went too, but only if she found ataxi cab.Please supply your answer with a brief description of the reasons why youthink your symbolization captures the meaning (i.e., the truth-conditions) of(*). For uniformity, please use the following key:A = Alf went to the movie.B = Beth went to the movie.C = Beth found a taxi cab.Student 1. This looks like a conditional, but it isn’t. The word ‘but’ is, from alogical point of view, equivalent to ‘and’, and there is no question that in thiscase the ‘and’ has a distributive reading. So ‘but’ expresses the conjunctionconnective and (*) can be rewritten as(**) If Alf went to the movie then Beth went to the movie, and Beth wentto the movie only if she found a taxi cab.Accordingly, the symbolization is:(1) (A → B) ∧ (B → C).2Student 2. The word ‘but’ is not doing any work here. We might be tempted tosymbolize (*) as(1) (A → B) ∧ (B → C),but this sentential expression would have the wrong truth-conditions. For ex-ample, if Alf went to the movie and Beth did not succeed in finding a cab (i.e., ifA is true and C false), then (1) would be false regardless of whether or not Bethwent to the movie, whereas the intuitive truth-conditions for (*) would seem todepend on what Beth did (at least in case Alf went to the movie). Thus, I aminclined to think that ‘but only if’ here means just ‘only if’ and expresses a nec-essary condition for the truth of the sentence that comes before, which ismarked by the comma and is itself in conditional form. Thus, a better renderingof (*) is(**) If Alf went to the movie then Beth went to the movie, only if shefound a taxi cab.Accordingly, the symbolization is:(2) (A → B) → C.Student 3. The word ‘but’ is not doing any work here. One might be temptedto symbolize (*) as(1) (A → B) ∧ (B → C),but this sentential expression would have the wrong truth-conditions. It wouldbe F whenever A is T and C is F, contrary to intuition. So I take it that ‘butonly if’ here just means ‘only if’. What is the antecedent of this ‘only if’?Prima facie, the comma suggests that the antecedent is the whole sentence thatcomes before the connective itself. Thus, since that sentence is in conditionalform, one might be tempted to symbolize (*) as(2) (A → B) → C.However, this sentential expression would also have the wrong truth-condi-tions. It would be T whenever C is T, contrary to intuition. Perhaps this is justanother example of the paradoxes of material implication, but I don’t think so. Idon’t think so because—as Michael Dummett observed—we hardly have anyuse, in natural language, of conditionals in which the antecedent is itself a condi-3tional—and I don’t think (*) is an exception. (A conditional might occur as apremise in an argument, but that is another matter.) So I do not think that ‘onlyif’ is the main connective of (*). It is, rather, the connective of what I take to bethe consequent of (*). The whole sentence can be re-written more perspicu-ously as(**) If Alf went to the movie, then Beth went to the movie only if shefound a taxi cab.And this can be symbolized as follows:(3) A → (B → C).Student 4. This is tricky. We might be tempted to symbolize (*) as(1) (A → B) ∧ (B → C)or as(2) (A → B) → Cor as(3) A → (B → C),but all of these sentential expressions have the wrong truth-conditions. In par-ticular, (3) is just as bad as (2), since both would get the value T as long as Cgets the value T, whereas intuitively Beth’s finding a cab is not a sufficient con-dition for the truth of (*). This is not just a sign of the inadequacy of the mate-rial conditional. Rather, I think that the source of the difficulty lies in the‘but only if C’ clause, which requires a different analysis. As I see it, the pur-pose of this clause is not only to specify a necessary condition for B. Isalso adds a restriction on the sufficiency of A for the obtaining of B. If A holds,and if the additional clause C is satisfied, then the speaker is committed to thetruth of B. Thus, given A, ‘B but only if C’ really means ‘B only if C, and if Cthen B’ – that is, ‘but only if’ amounts to ‘if and only if’, and we can rewrite(*) as(**) If Alf went to the movie, then Beth went to the movie if and only ifshe found a taxi cab,whose symbolization is (somewhat surprisingly):(4) A → (B ↔ C).4In other words, ‘but only if’ has a double effect: it introduces a necessary andsufficient condition, and it imposes a certain parsing of the sentence which tiesthe additional clause to the consequent of the conditional introduced by ‘if’.Student 5. Some prima facie options are:(1) (A → B) ∧ (B → C)(2) (A → B) → C(3) A → (B → C)(4) A → (B ↔ C).However, none of these expressions captures the intuitive truth-conditions ofthe English sentence (*). This is obvious for (1), (2), and (3), all of which fail todo justice to the relevance of ‘Beth went too’ in the evaluation of (*). But thesame applies to (4). For this last expression would be true in a scenario whereAlf did not go to the movie while Beth found a cab and went alone, which is notwhat the sentence says. This is not just a byproduct of the material reading ofthe conditional; it’s the word ‘too’ that is not accounted for. To rule this caseout, we must take C to be a necessary (and, as it turns out, sufficient) condi-tion, not of B, but of the conjunction of A with B. Accordingly, my proposedreading of (*) is(**) If Alf went to the movie, then Alf and Beth went to the movie if andonly if Beth found a taxi cab,which is symbolized as(5) A → ((A ∧ B) ↔ C).Student 6. I am not sure how to do this. I considered a number of options thatcorrespond to plausible parsings of the given English sentence:(1) (A → B) ∧ (B → C)(2) (A → B) → C(3) A → (B → C)(4) A → (B ↔ C)(5) A → ((A ∧ B) ↔ C).However, none of these seems to capture the intuitive truth-conditions of (*),either because the sentential expression can be false regardless of the truth-valueof B (as in (1)), or because it can be true regardless of the truth-value of B (as in5(2) and (3)), or because it fails to capture the contribution made of the word‘too’ (as in (4) and (5)). Indeed, although (5) might seem better, it is tautologi-cally equivalent to (4). So, instead of trying to figure out the answer by directlytrying to parse (*) in some way or other, I think it’s better to turn thingsaround and begin by considering the situations with which the truth of (*) iscompatible. On my reckoning, there are only four such situations:A = T B = T C = TA = T B = F C = FA = F B = F C = TA = F B = F C = FThe other four situations appear to be incompatible with (*):A = T B = T C = F (the ‘only if’ clause is violated)A = T B = F C = T (Beth must go if A and C hold)A = F B = T C = T (Beth cannot go too if Alf does not go)A = F B = T C = F (ditto)So, now, taking these situations to correspond to the rows in a truth table, the(shortest) corresponding sentential expression is(6) B ↔ (A ∧ C)This looks very different in structure form the surface grammar of (*) but, onthe face of it, (6) captures exactly the right truth-conditions.Discussion. Dear students, I have decided not to assign any grade for yester-day’s test. I did not realize that the sentence I assigned was so recalcitrant toregimentation. As it turns out, were it not for the occurrence of the word ‘too’,Student 4 gave the best answer. But even so, I agree that (4) is rather surprisingas a rendering of the logical form of an English sentence in which there is noclear indication of a biconditional. On the other hand, with the word ‘too’ prop-erly assessed, it is Student 6 who gave the answer that I find most appropriatevis-à-vis the intuitive meaning of (*). However, in this case the answer is evenmore surprising. We know that it would be a mistake to suppose that every or-dinary-language sentence wears all of its connectives on its sleeves. But it isvery rare that the underlying logical form is so different from the grammaticalform. And it is peculiar that to figure it out one has to start from scratch and gothrough a complete chart of the possible truth-conditions, ignoring parsing alto-6gether. It is peculiar because, after all, in understanding an English sentence suchas (*) we do not normally do so. Or do we?Student 5 (Addendum). Professor, I realize that my symbolization was mis-taken, and that Student 6’s solution is the only one that fully captures the intui-tive truth-conditions of (*). Still, I think the analysis I offered was basicallycorrect, except that I shouldn’t have said that to account for the meaning of‘too’ we must modify (4) by taking C to be necessary and sufficient for A andB. (That yields an expression that is indeed equivalent to (4).) Instead, I shouldhave simply said that to account for the meaning of ‘too’ we must supplement(4) with the missing bit, namely, with a conjunct that makes it explicit that Bimplies A (Beth went too only if Alf actually went). In other words, this is howI should have phrased my reading of (*):(**) If Alf went to the movie, then Beth went to the movie if and only ifshe found a taxi cab, and she went only if Alf went.This is symbolized, not as (5), but as(7) (A → (B ↔ C)) ∧ (B → A).Now, (7) is tautologically equivalent to (6), so it must be equally correct. I amnot sure that it is closer to the surface grammar of (*) than (6) is, especially ifyou think that (4) is already “surprising”. On the other hand, I don’t think Icame up with this solution by merely surveying the chart of all possible truth-conditions, so I don’t think I ignored parsing altogether. Or did I?Many thanks to Andrea Bonomi, Haim Gaifman, Hsing-Chien Tsai, Philip John-son-Laird, and Alessandro Zucchi for inspiring discussion.

Beth Too, but Only If

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Beth Too, but Only If

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