1. Uni-Log 1  -reduction by value A plea for  -reduction by value Marie Duží

2. Uni-Log 2 Problem Chomsky’ S. Neale I am going to solve, in a logically rigorous manner, a problem originally advanced as a counter- example to Chomsky’s theory of binding and recently discussed in a 2004 paper by S. Neale. “John loves his wife, and so does Peter.”  Hence John and Peter share a property. (1)Loving John’s wife (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 (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 men are in both cases exemplary husbands).

3. Uni-Log 3 Problem Since “John loves his wife” is ambiguous between attributing (1) or (2) to John, “So does Peter” is also ambiguous between attributing (1) or (2) to Peter. With unrestricted  -reduction, the lambda-term counterparts of the attributions of (1) and (2) to John both  -reduce to (1). Which, intuitively, they should not. With suitably restricted  -conversion, the two redexes do not reduce to the same contractum and can be reconstructed from their respective contracta.

4. Uni-Log 4  -reduction: procedural explication underspecified  -reduction is underspecified by [ x C(x) A] |– C(A/x). The application procedure [ x C(x) A] can be executed in two different ways: ‘by value’ and ‘by name’. by name by name: the procedure A is substituted for x. In this case there are two problems: 1. conversion of this kind is not guaranteed to be an equivalent transformation as soon as partial functions are involved. This is due to the fact that A occurs extensionally as a constituent of the left- hand side construction, whereas when dragged into C its occurrence may become intensional 2. even in those cases when  -reduction is an equivalent transformation, it can yield a loss of analytic information, because when executing  -reduction by name we do not keep track of the function that has been applied by value by value: The idea is simple: execute the procedure A first, and only if A does not fail to produce an argument value on which C should operate, substitute this value for x.

5. Uni-Log 5  -conversion by name: loss of info [ x [x + 1] 3]  [3 + 1] [ y [3 + y] 1] which function has been applied to which argument? No ‘backward path’. Does it matter?

6. Uni-Log 6  -conversion by name: loss of info [ 0 Wife_of wt 0 John] (1) w t [ x [ 0 Love wt x [ 0 Wife_of wt 0 John]] 0 John] [ 0 Wife_of wt x] (2) w t [ x [ 0 Love wt x [ 0 Wife_of wt x]] 0 John] [ 0 Wife_of wt 0 John] (3) w t [ 0 Love wt 0 John [ 0 Wife_of wt 0 John]] It is uncontroversial that the contractum (3) can be equivalently expanded back both to (1) and (2). The problem is, of course, that there is no way to reconstruct which of (1), (2) would be the correct redex

7. Uni-Log 7 “John loves his wife, and so does Peter” The sentence “John loves his wife, and so does Peter” ostensibly shows that the - calculus is too crude an analytical tool for at least one kind of perfectly natural use of indexicals. I will demonstrate that, and how, the - calculus is up for the challenge — provided a form of  -conversion by value is adopted.

8. Uni-Log 8 Contribution  -conversion by value I will detail how to apply the restricted rule of  -conversion by value to contexts containing anaphora such as ‘his’ and ‘so does’. logical contribution  -conversion by value The logical contribution is a generally valid form of  -conversion by value rather than by name. anaphora The philosophical application of  -conversion by value to a context containing anaphora is another contribution.

9. Uni-Log 9

10. 10 Transparent Intensional Logic (TIL) The technical tools of the two disambiguations of the analysandum will be familiar from Montague’s intensional logic, with two important exceptions: variables w ranging over possible worlds and t ranging over times 1. we -bind variables w ranging over possible worlds and t ranging over times. This dual binding is tantamount to explicit intensionalization and temporalization. 2. functional application 2. functional application, symbolized by square brackets, ‘  …  ’, is the logic both of extensionalization of intensions (functions from possible worlds) and of predication. [[Intension w] t]Intension wt Intensions are extensionalized by applying them to worlds and times: [[Intension w] t], or Intension wt for short, is the extension of the Intension at  w, t . Property wt yields a set; Proposition wt yields a truth-value (or no value at all).

11. Uni-Log 11 TIL: procedural semantics functions and functional values  A main feature of the -calculus is its ability to systematically distinguish between functions and functional values. functions and modes of presentation of functions  An additional feature of TIL is its ability to systematically distinguish between functions and modes of presentation of functions. Closure TIL Closure is the very procedure of presenting or constructing a function: x [… x …] Composition TIL Composition is the very procedure of constructing the value (if any) of a function at an argument: [ x [… x …] C] Compositions and Closures operate on input provided by two one-step constructions; variables and Trivializations. Variables Variables x, y, z, p, q, … construct the respective values that an assignment function v has accorded to them; they v-construct. Trivialisation 0 X Trivialisation 0 X constructs X.  The linguistic counterpart of a Trivialization is a constant term always picking out the same object.  An analogy from programming: the Trivialization of an object X (whatever X may be) and its use are comparable to a fixed pointer to X and the dereference of the pointer.  In order to operate on X, X needs to be grabbed, or ‘called’, first. Trivialization is such a grabbing mechanism.

12. Uni-Log 12 TIL: procedural semantics Explicit intensionalization and temporalization encodes constructions of possible-world intensions directly in the logical syntax: w t […w….t…] w t […w….t…] For instance, let 0 Happy construct the property of being happy, and let 0 Pope construct the office of Pope. Then w t [ 0 Happy wt 0 Pope wt ] w t [ 0 Happy wt 0 Pope wt ] is a Closure constructing the possible- world proposition that returns T at those  w, t  -pairs whose incumbent of the office of Pope and whose set of happy individuals are such that the former is an element of the latter; empirical truth-condition P:  w, t   P = df Pope wt  Happy wt.  Whether the pair consisting of the actual world and the present moment is a member of P is beyond logic and semantics, and must be established empirically.

13. Uni-Log 13 TIL: logical core constructionstype hierarchy constructions + type hierarchy (simple and ramified) ramified constructions (types  n ) The ramified type hierarchy organizes all higher- order objects: constructions (types  n ), as well as functions with domain or range in constructions. simple non-constructions The simple type hierarchy organizes first-order objects: non-constructions like extensions (individuals, numbers, sets, etc.), possible-world intensions (functions from possible worlds) and their arguments and values. For the relevant definitions, see References.

14. Uni-Log 14 substitution by value Solution: substitution by value (1) P John (the property of loving John’s wife): w t x [ 0 Love wt x [ 0 Wife_of wt 0 John]] (2) P own (the property of loving wife of himself): w t he [ 0 Love wt he [ 0 Wife_of wt he]]] Strict reading: w t [P John wt 0 John] Sloppy reading: w t [P own wt 0 John] so does Peter “so does Peter”: w t [so1 wt 0 Peter]; w t [so2 wt 0 Peter] so1, so2 are variables ranging over properties

15. Uni-Log 15 substitution by value Solution: substitution by value Now we need to substitute constructions of the properties P John, P own for so1, so2, respectively. To this end we apply the function Sub/(  n  n  n  n ) that operates on constructions: [ 0 Sub C 1 C 2 C 3 ] [ 0 Sub C 1 C 2 C 3 ] constructs the construction D that is the result of (collision-less) substitution of C 1 for C 2 into C 3 [ 0 Sub 00 John 0 he 0 [ 0 Wife_of wt he]]  [ 0 Wife_of wt 0 John] Example. [ 0 Sub 00 John 0 he 0 [ 0 Wife_of wt he]]  [ 0 Wife_of wt 0 John].

16. Uni-Log 16 substitution by value Solution: substitution by value [ 0 Sub 0 0 so1 0 [so1 wt 0 Peter]] [ 0 Sub 0 P John 0 so1 0 [so1 wt 0 Peter]] [ 0 Sub 0 0 so2 0 [so2 wt 0 Peter]] [ 0 Sub 0 P own 0 so2 0 [so2 wt 0 Peter]] “John loves his wife and so does Peter” (1*) w t [[ wt 0 John]  2 [ 0 Sub 0 0 so1 0 [so1 wt 0 Peter]]] (1*) w t [[P John wt 0 John]  2 [ 0 Sub 0 P John 0 so1 0 [so1 wt 0 Peter]]] w t [[ wt 0 John]  2 [ 0 Sub 0 0 so2 0 [so2 wt 0 Peter]]] (2*) w t [[P own wt 0 John]  2 [ 0 Sub 0 P own 0 so2 0 [so2 wt 0 Peter]]] Gloss. Double Execution ( 2 ) is the procedure of executing a construction C twice over. [ 0 Sub 0 0 so1 0 [so1 wt 0 Peter]] constructs the construction [ wt 0 Peter] which is not the right argument of conjunction (  ); we must execute it again to obtain a truth-value. Gloss. Double Execution ( 2 ) is the procedure of executing a construction C twice over. [ 0 Sub 0 P John 0 so1 0 [so1 wt 0 Peter]] constructs the construction [P John wt 0 Peter] which is not the right argument of conjunction (  ); we must execute it again to obtain a truth-value.

17. Uni-Log 17 Conversion by value The procedure of pre-processing shows why  -conversion by value makes for a superior analysis when compared to conversion by name. If P John is substituted for so1 in (1*), the result is that Peter loves John's wife. If P own is substituted for so2 in (2*), the result is that Peter loves his own wife. If conversion by name is deployed, the result is inevitably that Peter loves John's wife.

18. Uni-Log 18 Conversion by value share a common property Both (1*) and (2*) entail that John and Peter share a common property, though not the same in both cases. Here are the two (structurally identical) proofs. For any  w, t  -pair the following proof steps are truth-preserving: (1) [[P John wt 0 John]  2 [ 0 Sub 0 P John 0 so1 0 [so1 wt 0 Peter]]]  (2) [[P John wt 0 John]  [P John wt 0 Peter]]Sub, 1 (3)  p [[p wt 0 John]  [p wt 0 Peter]]EG, 2 (1’)[[P own wt 0 John]  2 [ 0 Sub 0 P own 0 so1 0 [so1 wt 0 Peter]]]  (2’)[[P own wt 0 John]  [P own wt 0 Peter]Sub, 1’ (3’)  q [[q wt 0 John]  [q wt 0 Peter]]EG, 2’ The sloppy reading, which we prefer, entails that the shared property is P own, because the Composition [[q wt 0 John]  [q wt 0 Peter]] is v(P own /q)-congruent with the Composition [[P own wt 0 John]  [P own wt 0 Peter].

19. Uni-Log 19 Conversion by value: general rule For simplicity’s sake, I introduce the rule in its simplified version for unary functions (generalization to n-ary functions is obvious): [[ x Y] C] |– 2 [ 0 Sub [ 0 Tr C] 0 x 0 Y] Note Double Execution. C v-improper  the attempt to substitute fails. C is v-proper, it v-constructs an entity, say e  the result of the first step is the construction Y( 0 e/x). The resulting construction must then be executed in order to obtain the value of the function v-constructed by [ x Y] at the argument e. loss of analytic information In this manner compositionality is preserved, the above rule of  - conversion by value is always valid even when C is v-improper, and no loss of analytic information.

20. Uni-Log 20 Concluding remarks On too coarse an analysis, the respective redexes of the sloppy and the strict reading reduce to the same contractum, which corresponds to the strict reading. The unpleasant consequences are that the anaphoric character of ‘his wife’ is lost in conversion and that two properties – loving John’s wife and loving one’s own wife – are predicted, wrongly, to be equivalent. Solution  A generally valid rule of  -reduction by value (that exploits our substitution method) whenever there is a clause hosting a pragmatically incomplete meaning. one and the same open construction dynamically pre-process We assign one and the same open construction to such a clause in every context. The substitution method makes it possible to dynamically pre-process the open construction by supplying values for the free anaphoric variables. Thus no analytic information is lost.

21. Uni-Log 21 Concluding remarks functional programming language TIL-Script In general, all  -reductions can be restricted to ‘call by value’. My research team have been developing a computational variant of TIL, namely the functional programming language TIL-Script. In this interpreted formalism only  -reduction by value is used. hyperpropositional attitudesattitudes de re quantifying into hyperintensional contexts The substitution technique was originally developed by Duží in order to properly analyze sentences containing anaphoric reference. Later the technique turned out to be very useful in cracking many other hard nuts in formal semantics, e.g. substitution inside hyperpropositional attitudes, attitudes de re, and quantifying into hyperintensional contexts. For details, see Duží et al. (2010, Ch. 5).

22. Uni-Log 22 TIL References Procedural isomorphism, analytic information and  -conversion by value Duží, M., Jespersen, B. (2013): Procedural isomorphism, analytic information and  -conversion by value, Logic Journal of the IGPL, vol. 21, pp. 291-308; DOI: 10.1093/jigpal/jzs044. Procedural Semantics for Hyperintensional Logic Duží M., Jespersen B. and Materna P. (2010): Procedural Semantics for Hyperintensional Logic. Foundations and Applications of Transparent Intensional Logic. Berlin: Springer, Logic, Epistemology, and the Unity of Science, vol. 17.