University of Birmingham School of Computer Science 11 October 2012 Bristol University

1. John loves his wife, and so does peterBjørn JespersenCzech Academy of Sciences, Dept. Logic;Technical University of Ostrava, Dept. Computer Science, Czech Republic University of Birmingham School of Computer Science 11 October 2012 Bristol University Department of Philosophy 12 October 2012

2. ‘Procedural isomorphism, analytic information, and -conversion by value’with Marie Duží (forthcoming)

4. “a has a property, and b has that property” VP ellipsis, anaphora, and ‘sloppy identity’: “John scratched his arm, and so did Mary” “Betsy loves her dog, and Sandy does , too” strict reading: Mary scratched John’s arm; Sandy loves Betsy’sdog sloppy reading: Mary scratched her own arm; Sandy loves her own dog Chomsky cannot accommodate sloppy identity (due to lack of co-indexing), thus missing the ambiguity and picking the less natural reading.

5. “John loves his wife, and so does Peter” Peter has whichever particlar property it is that John has; but which property does John have – loving John’s wife or loving his own wife? • (strict) John loves John’s wife, and Peter loves John’s wife • (sloppy) John loves his own wife, and Peter loves his own wife • (Czech) “Jan miluje jeho ženu a Petr taky” • (Czech) “Jan miluje svou ženu a Petr taky” • (Danish) “Johannes elsker hans kone, og det samme gør Steen” John and Peter both love a third man’s wife, and the reference of ‘hans kone’ is deictic(third reading, not anaphoric). • (Danish) “Johannes elsker sin kone, og det samme gør Steen” John and Peter love their respective wives, and ‘sin kone’ is reflexive; the same woman is possibly the wife both of John and Peter, with John and Peter loving the same woman. Neither Czech nor Danish generate the ambiguity of “John loves his wife”; German (‘seine Frau’) and Dutch (‘zijn vrouw’) do.

6. The question, and a problem Question: How are we to analyze “.... so does ...”-cases of anaphoric reference to a property while distinguishing between strict and sloppy identity in the -calculus? Problem: Two redexes, one contractum. x Fxa  Fa

7. TRANSPARENT INTENSIONAL LOGIC:GROUND AND FUNCTIONAL TYPES IN THE SIMPLE TYPE THEORY • Definition 1 (types of order 1) Let B be a base, where a base is a collection of pair-wise disjoint, non-empty sets. Then: • Every member of B is an elementary type of order 1 over B. • Let α, β1, ..., βm(m > 0) be types of order 1 over B. Then the collection(α β1 ... βm) of all m-ary partial mappings from β1 ... βminto α is a functional type of order 1 over B. • Nothing else is a type of order 1 over B. 

8. TRANSPARENT INTENSIONAL LOGIC:CONSTRUCTIONS Definition 2 (construction) • The variablexis a construction that constructs an object O of the respective type dependently on a valuation v: xv-constructs O. • Trivialization: Where X is an object whatsoever (an extension, an intension or a construction), 0Xis the construction Trivialization. It constructs X without any change. • The Composition [X Y1…Ym] is the following construction. If X v-constructs a function f of type (α β1…βm), and Y1, …, Ym v-construct entities B1, …, Bmof types β1, …, βm, respectively, then the Composition [X Y1…Ym] v-constructs the value (an entity, if any, of type α) of f on the tuple argumentB1, …, Bm. Otherwise the Composition [X Y1…Ym] does not v-construct anything and so is v-improper. • The Closure [λx1…xm Y] is the following construction. Let x1, x2, …, xmbe pair-wise distinct variables v-constructing entities of types β1, …, βmand Y a construction v-constructing an α-entity. Then [λx1 … xm Y] is the construction λ-Closure (or Closure). It v-constructs the following function f of the type (αβ1…βm). Let v(B1/x1,…,Bm/xm) be a valuation identical with v at least up to assigning objects B1/β1, …, Bm/βmto variables x1, …, xm. If Y is v(B1/x1,…,Bm/xm)-improper (see iii), then f is undefined on B1, …, Bm. Otherwise the value of f on B1, …, Bm is the α-entity v(B1/x1,…,Bm/xm)-constructed by Y. • The SingleExecution 1X is the construction that either v-constructs the entity v-constructed by X or, if X is not itself a construction or X is v-improper, 1X is v-improper. • The Double Execution 2X is the following construction. Where X is any entity, the Double Execution 2X is v-improper (yielding nothing relative to v) if X is not itself a construction, or if X does not v-construct a construction, or if X v-constructs a v-improper construction. Otherwise, let X v-construct a construction Y and Yv-construct an entity Z:then 2X v-constructs Z. • Nothing else is a construction. 

9. TRANSPARENT INTENSIONAL LOGIC:RAMIFIED TYPE HIERARCHY Definition 3 (ramified hierarchy of types) • T1(types of order 1). See Definition 1. • Cn(constructions of order n) • Let x be a variable ranging over a type of order n. Then x is a construction of order n over B. • Let X be a member of a type of order n. Then 0X, 1X,2X are constructions of order n over B. • Let X, X1,..., Xm(m > 0) be constructions of order n over B. Then [X X1...Xm] is a construction of order n over B. • Let x1,...xm, X (m > 0) be constructions of order n over B. Then [x1...xmX] is a construction of order n over B. • Nothing is a construction of order n overB unless it so follows from Cn(i)-(iv). • Tn+1(types of order n + 1) Let n be the collection of all constructions of order n over B. Then • n and every type of order n are types of order n + 1. • If 0 < m and , 1,...,m are types of order n + 1 over B, then (1 ... m) (see T1 ii)) is a type of order n + 1 over B. • Nothing else is a type of order n + 1 over B. 

10. TRANSPARENT INTENSIONAL LOGIC:free and bound variables Definition 4 (free and bound variables) Let C be a construction with at least one occurrence of a variable . • Let C be . Then the occurrence ofinC is free. • Let C be 0X. Then every occurrence of  inC is 0bound (‘Trivialization-bound’). • Let C be [x1...xn Y]. Any occurrence of  in Y that is one of xi, 1 in, is -bound in C unless it is 0boundinY. Any occurrence of  inY that is neither 0bound nor-bound in Y is free in C. • Let C be [X X1...Xn]. Any occurrence of  that is free, 0bound, -bound in one of X, X1,...,Xnis, respectively, free, 0bound, -boundinC. • Let C be 2X. Then any occurrence of that is free, -bound in a constituent of C is, respectively, free, -boundinC. If an occurrence of is 0bound in a constituent 0D of C and this occurrence of D is a constituent of X’ v-constructed by X,then if the occurrence of is free, -bound in D it is free, -bound in C. Otherwise, any other occurrence of  in C is 0bound in C. • No other occurrence of  is free, -bound, 0bound in C.

11. “so does peter” “So does Peter” has a fixed, context-invariant meaning that is ‘open’, namely an open construction : wt pwt0Peter

12. “John loves john’s/his own wife” “John loves John’s wife” (1) wt x 0Lovewtx 0Wife_ofwt0John 0John “John loves his own wife” (2) wt x 0Lovewtx 0Wife_ofwtx 0John

13. “john loves john’s/HIS OWN wife, and so does peter” (1.1) wt x 0Lovewtx 0Wife_ofwt0John 0John  pwt0Peter (2.1) wt x 0Lovewtx 0Wife_ofwtx 0John  qwt0Peter But: p, q occur free. Hence (1.1), (2.1) are incomplete analyses of the complete/closed expressions “John loves John’s/his own wife, and so does Peter”

14. Equivalent, but insufficient analyses Let v(p) = loving John’s wife (1.2) wtx 0Lovewtx 0Wife_ofwt0John 0John  x 0Lovewtx 0Wife_ofwt0John 0Peter Let v(q) = loving one’s own wife (2.2) wt x 0Lovewtx 0Wife_ofwtx 0John  x 0Lovewtx 0Wife_ofwtx 0Peter But: (1.2), (2.2) are merely equivalent, and not synonymous (procedurally isomorphic), with the strict/sloppy readings of “John ... Peter”.

15. “so does peter” again The pragmatically identified meaning associated with “So does Peter” in the given situation of utterance/inscription is a closed construction, whereas the context-insensitive meaning of “So does Peter” is an open construction. Upon disambiguation of the source clause, it is determinate which of the two candidate properties a predicate for which must be substituted for ‘so does’ in the target clause.

16. Sub, Tr Sub: substitute C1 for C2 in C3 to obtain D. Example: 0Sub00John 0him 00Wife_ofwt him constructs 0Wife_ofwt0John. Tr: takes object X to 0X. Examples: 0Tr 0John constructs 0John; 0Tr x constructs 0x.

17. Common contractum (!) (1), (2) both yield the contractum (1*2) wt0Lovewt0John 0Wife_ofwt0John This is bad. (1*2) blots out the ambiguity of “John loves his wife, and so does Peter”, and it blots out the anaphoric character of ‘his wife’. Diagnosis: there is a loss of analytic information about which construction constructs the property predicated of John (loving John’s wife/loving his own wife), hence of Peter.

18. ‘HIS OWN WIFE’, SUB, TR (1.3) “John loves John’s wife”, as above. (2.3) “John loves his own wife” wt[wt[x [0Lovewt x 2[0Sub [0Trx]0him 0[0Wife_ofwt him]]]]wt0John] The Double Execution 2[0Sub [0Trx]0him 0[0Wife_ofwt him]] v-constructs the individual who is the wife of the value of him. This construction pre-processesthe meaning of ‘the wife of himself’ (or ‘his own wife’) by substituting the Trivialization of the individual v-constructed by x for him.

19. “john loves his -reducedwife” (1.3) wt 0Lovewt0John 0Wife_ofwt0John (2.3) wt 0Lovewtx 0John 20Sub 0Tr 0John 0him00Wife_ofwthim The subconstructions 0Wife_ofwt0John,20Sub 0Tr 0John 0him00Wife_ofwthim are equivalent, but not procedurally isomorphic (‘synonymous’). Hence (1.3), (2.3) are equivalent, but not procedurally isomorphic. They are different constructions, embedding two different pieces of analytic information about two different ways of constructing one and the same proposition (two structured hyperpropositions, one unstructured proposition).

20. Final analysis of “john...peter” Let PropJohn’s= wt x 0Lovewtx 0Wife_ofwt0John Propown= wt x 0Lovewtx20Sub 0Tr x 0him00Wife_ofwthim Then (1.4) wt PropJohn’swt0John  20Sub0PropoJohn’s0p0pwt0Peter (2.4) wt Propownwt0John  20Sub0Propown0q0qwt0Peter The property John and Peter share in (2.4) is Propown, because the Composition qwt0John  qwt0Peter is v(Propown/q)-congruent with the Composition Propownwt0John  Propownwt0Peter.

21. conclusions • The linguistic phenomenon of VP ellipsis makes the anaphoric reference by means of ‘so does’ to properties ambiguous. Hence strict/sloppy identity. • The ogical problem of ‘two redexes, one contractum’ is solved by means of two TIL constructions that are equivalent, but not procedurally isomorphic (‘synonymous’).