Miscellaneous
Colloquium Mathematical Philosophy, Marie Duzi (Technical University Ostrava) gives a talk at the MCMP Colloquium (15 May, 2014) titled "A plea for beta-reduction by value". Abstract: This paper solves, in a logically rigorous manner, a problem discussed in a 2004 paper by Stephen Neale and originally advanced as a counterexample to Chomsky’s theory of binding. The example I will focus on is the following. John loves his wife. So does Peter. Therefore, John and Peter share a property. Only which one? There are two options. (1) Loving John’s wife. Then John and Peter love the same woman (and there is trouble on the horizon). (2) Loving one’s own wife. Then, unless they are married to the same woman, John loves one woman and Peter loves another woman (and both are exemplary husbands). On the strict reading of “John loves his wife, and so does Peter” property (1) is the one they share. On the sloppy reading, property (2) is the one they share. The dialectics of this contribution is to move from linguistics through logic to semantics. An issue originally bearing on binding in linguistics is used to make a point about -conversion in the typed ß-calculus. Since the properties loving John’s wife and loving one’s own wife as attributed to John are distinct, there is room for oscillation between the sloppy and the strict reading. But once we feed the formal renditions of attribution of these two properties to John into the widespread ß-calculus for logical analysis, a logical problem arises. The problem is this. Their respective ß-redexes are distinct, for sure, but they share the same ß-contractum. This contractum corresponds to the strict reading. So ß-conversion predicts, erroneously, that two properties applied to John ß-reduce to one. The result is that the sloppy reading gets squeezed out. ß-reduction blots out the anaphoric character of ‘his wife’, while the resulting contractum is itself ß-expandable back into both the strict and the sloppy reading. Information is lost in transformation. The information lost when performing ß-reduction on the formal counterparts of “John loves his wife” is whether the property that was applied was (1) or (2), since both can be reconstructed from the contractum, though neither in particular. The sentence “John loves his wife, and so does Peter” ostensibly shows that the ß-calculus is too crude an analytical tool for at least one kind of perfectly natural use of indexicals. The problematic reduction and its solution will both be discussed within the framework of Tichý’s Transparent Intensional Logic. Tichý’s TIL was developed simultaneously with Montague’s Intensional Logic. The technical tools of the two disambiguations of the analysandum will be familiar from Montague’s intensional logic, with two important exceptions. One is that we ß-bind separate variables w1,…,wn ranging over possible worlds and t1,…,tn ranging over times. This dual binding is tantamount to explicit intensionalization and temporalization. The other exception is that functional application is the logic both of extensionalization of intensions (functions from possible worlds) and of predication. I will demonstrates that, and how, the ß-calculus is up for the challenge, provided a rule of ß-conversion by value is adopted. The logical contribution of the paper is a generally valid form of ß-reduction by value rather than by name. The philosophical application of ß-reduction by value to a context containing anaphora is another contribution of this paper. The standard approach to VP ellipsis based on ß-abstracts and variable binding can, thus, be safely upheld. Our solution has the following features. First, unambiguous terms and expressions with a pragmatically incomplete meaning, like ‘his wife’ or “So does Peter”, are analyzed in all contexts as expressing an open construction containing at least one free variable with a fixed domain of quantification. Second, the solution uses ß-conversion by value, rather than conversion by name. The generally valid rule of ß-conversion by value exploits our substitution method and I show that the application of this rule does not yield a loss of analytic information. Third, the substitution method is applied to sentences containing anaphora, like ‘so does’ and ‘his’, in order to pre-process the meaning of the incomplete clause. Our declarative procedural semantics also makes it straightforward to infer that there is a property that John and Peter share.