Transparent intensional logic, -rule and Compositionality Marie Duží VSB-Technical University Ostrava

Rules of substition2 Attitude Logic(s)  A reliable test on Compositionality  Attitudes: Notional Propositional  We are dealing with a fine difference between the meanings of sentences like (P1) Charles believes that the Pope is in danger (P2) Charles believes of the Pope that he is in danger  Some authors even claim that (P1) is ambiguous, that it can be also read as (P2).

Rules of substition3 Attitude logics and belief sentences In our opinion it is not so. We can, for instance, reasonably say (it may be true) that Charles believes of the Pope that he is not the Pope, whereas the sentence Charles believes that the Pope is not the Pope cannot be true, unless our Charles is completely irrational. The sentences like (P1) and (P2) have different meanings, and their difference consists in using ‘the Pope’ in the de dicto supposition (P1) vs. the de re supposition (P2). The two sentences are neither equivalent, nor is any of them entailed by the other.

Rules of substition4 Belief sentences in doxastic logics  In the usual notation of doxastic logics the distinction is characterised as the contrast between B Charles D[p] (de dicto) (x) (x = p  B Charles D[x] (de re)  But there are worrisome questions (Hintikka, Sandu 1989): Where does the existential quantifier come from in the de re case? There is no trace of it in the original sentence. How can the two similar sentences be as different in their logical form as they are?  Hintikka, Sandu propose in their (1996) a remedy by means of the Independence Friendly (IF) first-order logic:

Rules of substition5 Belief sentences in doxastic logics  “Independence Friendly (IF) first-order logic deals with a frequent and important feature of natural language semantics. Without the notion of independence, we cannot fully understand the logic of such concepts as belief, knowledge, questions and answers, or the de dicto vs. de re contrast.” Hintikka, Sandu (1989): Informational Independence as a Semantical Phenomenon. In J.E. Fenstad et el (eds.), Logic, Methodology and Philosophy of Science, Elsevier, Amsterodam 1989, pp Hintikka, Sandu (1996): A revolution in Logic? Nordic Journal of Philosophical Logic, Vol.1, No.2, pp

Rules of substition6 Belief sentences and IF semantics  Hinttika, Sandu solve the de dicto case as above, and propose the de re solution with the independence indicator ‘/’: B Charles D[p / B Charles ]  This is certainly a more plausible analysis, closer to the syntactic form of the original sentence, and the independence indicator indicates the essence of the matter:  There are two independent questions: ”Who is the pope” and ”What does Charles think of that person”. Of course, Charles has to have a relation of an ”epistemic intimity” to a certain individual, but he does not have to connect this person with the office of the Pope (only the ascriber must do so). (Chisholm,R.(1976): Knowledge and Belief: ‘De dicto’ and ‘de re’. Philosophical Studies 29 (1976), )

Rules of substition7 Belief sentences and Intensional logics B Charles D(p)(de dicto) x B Charles D(x)(p) (de re) But: x B Charles D(x)(p)  B Charles D(p) ! (applying the rule of -reduction). What then is the difference between de dicto and de re? Why is it “forbidden” here to perform the fundamental rule of -calculi?

Rules of substition8 Solomon Feferman (1995): Logic of Definedness Introduces the axioms (λp) for Partial Lambda Calculus as follows (t↓ means - the term t is defined): i. λx.t ↓ ii. (λx.t(x))y  t(y). The axiom (ii) corresponds to the trivial β-reduction, but the limitation on instantiation in PLC restricts its application to: s↓  (λx.t(x))s  t(s). (but why this restriction?, proof?) Our system (TIL) introduces a generally valid β-reduction for the Partial Higher-Order Hyper- intensional Lambda Calculus.

1/31/2016TIL & beta-rule9 Transparent Intensional Logic Formally:  The language of TIL constructions can be viewed as a hyper-intensional -calculus operating over partial functions.  “hyper-intensional”: -terms are not interpreted as set-theoretical mappings (”modern functions”) but as algorithmically structured procedures (which produce as an output the (partial) mapping).  Procedures, known as TIL constructions, are objects sui generis: they can be not only used but also mentioned within a theory.

1/31/2016TIL & beta-rule10 Suppositio (substitution)  A lot of misunderstanding and many paradoxes arise from confusing different ways in which a meaningful expression can be used.  We are going to show that these different ways consist in using and mentioning entities (by means of an expression)  In which way can an entity be used or mentioned?

1/31/2016Use / Mention11 Using / Mentioning Entities Expression usedmentioned to express its meaning : procedure (‘TIL construction’) de dicto / de re mentioned used to produce a function : mentioned used to point at …

Rules of substition12 TIL constructions  Abstract procedures, structured from the algorithmic point of view.  structured meanings: Instructions specifying how to arrive at less-structured entities.  Being abstract, they are reachable only via a verbal definition.  The ‘language of constructions’: a modified version of the typed -calculus, where Montague-like -terms denote, not the functions constructed, but the constructions themselves.  Henk Barendregt (1997): -terms denote functions, yet “... in this interpretation the notion of a function is taken to be (hyper-)intensional, i.e., as an algorithm.”  Operate on input objects (of any type, even constructions) and yield as output objects of any type: they realize functions (mappings)

1/31/2016Constructions13 Kinds of constructions 1.Atomic: do not contain as a used constituent any other construction but themselves (supply objects …)  Variables x, y, p, c, … v-constructing  Trivialisation of X: 0 X 2.Compound.  Composition [X X 1 …X n ]: the instruction to apply a (partial) function f (constructed by X) to an argument A (constructed by X 1,…,X n ) to obtain the value (if any) of f at A. -Closure [x 1 …x n X]: the instruction to abstract over variables in order to obtain a function.  Double execution 2 X: the instruction to use a higher-order construction X twice over as a constituent.

Rules of substition14 TIL Ramified Hierarchy of Types The formal ontology of TIL is bi- dimensional. One dimension is made up of constructions. The other dimension encompasses non- constructions, i.e., partial functions mapping (the Cartesian product of) types to types.

Rules of substition15 TIL Ramified Hierarchy of Types 1 st -order: non-constructionsBase: , , , , partial functions ((())), (()), …, ( 0  1 … n ) 2 nd -order: Base: * 1 constructions of 1 st -order entities, partial functions involving such constructions: ( 0  1 … n ),  i = * 1 3 rd -order: Base: * 2, constructions of 2 nd -order entities, partial functions involving such constructions: ( 0  1 … n ),  i = * 2, or * 1 And so on, ad infinitum

Rules of substition16 -intensions; examples  Functions of type ()  Usually both modal and temporal parameters: (())  Abbreviation:   Propositions /   (individual) offices /   Magnitudes /   Empirical functions (attributes)/()  Attitudes / ( n ) 

1/31/2016Use / Mention17 Definition Used* vs. Mentioned* Let C be a construction and D a sub-construction of C. Then an occurrence of D is used* as a constituent of C iff:  If D is identical to C (i.e., 0 C = 0 D) then the occurrence of D is used* as a constituent of C.  If C is identical to [X 1 X 2 …X m ] and D is identical to one of the constructions X 1, X 2,…,X m, then the occurrence of D is used* as a constituent of C.  If C is identical to [x 1 …x m X] and D is identical to X, then the occurrence of D is used* as a constituent of C.  If C is identical to 1 X or 2 X and D is identical to X, then the occurrence of D is used* as a constituent of C.  If C is identical to 2 X and X v-constructs a construction Y and D is identical to Y, then the occurrence of D is used* as a constituent of C.  If an occurrence of D is used* as a constituent of an occurrence of C’ and this occurrence of C’ is used* as a constituent of C, then the occurrence of D is used * as a constituent of C. If an occurrence of a sub-construction D of C is not used* as a constituent of C, then the occurrence of D is mentioned* in C.

1/31/2016Use / Mention18 Definition Used* vs. Mentioned* Let C be a construction and D a sub- construction of C. Then an occurrence of D is mentioned* in C iff it is not necessary to execute D in order to execute C; Otherwise D is used* as a constituent of C.  Makes a fine individuation possible; finer than just an equivalence.

Rules of substition19 Two kinds of using a construction: de dicto vs. de re supposition. Roughly: C = [… D … ], D  () 1.D occurs in C with de dicto supposition iff D is not composed with a construction A  ;  the respective function / () is just mentioned 2.D occurs in C with de re supposition iff D is composed with a construction A  , and D does not occur as a constituent of a de dicto occurrence D’ (de dicto context is dominant);  the respective function / () is used as a pointer to its actual, current value / 

Rules of substition20 Contexts  suppositio substitution The President of USA knows that John Kerry wanted to become the President of USA. The President of USA is (=) the husband of Laura Bush.  Hence what ? Did John Kerry want to become the husband of Laura Bush?

1/31/ Contexts  suppositio substitution C 1 w t [ 0 = [ w t [ 0 Pres wt 0 USA]] wt [ 0 Husband wt 0 Bush]] extensional context: of using* de re C 2 w t [ 0 W wt 0 K [ w t [ 0 B wt 0 K w t [ 0 Pres 0 USA]]] ] intensional context: of using* de dicto C 3 w t [ 0 Know wt [ w t [ 0 Pres wt 0 USA]] wt 0 [ w t [ 0 W wt 0 K w t [ 0 B wt 0 K w t [ 0 Pres wt 0 USA]]]] ] hyper-intensional context: of mentioning*

Rules of substition22 (The President of the USA …) w t [ 0 Want wt 0 Kerry [ w t [ 0 Become wt 0 Kerry w t [ 0 Pres 0 USA]]]], w t [ 0 = [ w t [ 0 Pres wt 0 USA]] wt 0 Bush] w t [ 0 Know wt [ w t [ 0 Pres wt 0 USA]] wt 0 [ w t [ 0 Want wt 0 Kerry w t [ 0 Become wt 0 Kerry w t [ 0 Pres wt 0 USA]]]]] Types: Want / (     ), Become / (     ), Know / (    1 )  Now we can substitute 0 Bush for [ w t [ 0 Pres wt 0 USA]] wt thus deducing that G.W. Bush knows that John Kerry wanted to become the President of USA, but not that he wanted to become G.W.Bush. The undesirable substitution of 0 Bush for the latter occurrence of the construction w t [ 0 Pres wt 0 USA] is blocked.

Rules of substition23 Using / Mentioning Constructions  Dividing six by three gives two and dividing six by zero is improper. Types: 0, 2, 3, 6 / , Div / (), Improper / ( 1 )the class of v-improper constructions for all v [[[ 0 Div ] = 0 2]  [ 0 Improper 0 [ 0 Div ]]] used* constituents mentioned*

Rules of substition24 Using / Mentioning Constructions  There is a number such that dividing any number by it is improper. Types: Div / (), Improper / ( 1 ), ,/(()), x, y  . Exists x for all y [ 0 Improper 0 [ 0 Div x y]]. But x, y occur in the hyper-intensional context of mention*; they are not free for evaluation or substitution. How to quantify? To this end we use functions Sub and Tr: Sub / ( 1  1  1  1 )the mapping which takes a construction C 1, variable x, and a construction C 2 to the resulting construction C 3, where C 3 is the result of substituting C 1 for x in C 2. Tr / ( 1 )the mapping which takes a number and returns its trivialisation

Rules of substition25 Using / Mentioning Constructions (*) [ 0 y [ 0 x [ 0 Improper [ 0 Sub [ 0 Tr y] 0 y’ [ 0 Sub [ 0 Tr x] 0 x’ 0 [ 0 Div x’ y’ ]]]]]].  Let a valuation v assign 0 to y and 6 to x. Then the sub-construction [ 0 Sub [ 0 Tr y] 0 y’ [ 0 Sub [ 0 Tr x] 0 x’ 0 [ 0 Div x’ y’ ]]]  v-constructs the construction [ 0 Div ], which belongs to the class Improper. This is true for any valuation v’ that differs from v at most by assigning another number to x.  The construction (*) constructs True.

Rules of substition26 De dicto / de re supposition  The temperature in Amsterdam equals the temperature in Prague.  The temperature in Amsterdam is increasing   The temperature in Prague is increasing. Types: Temp(erature in …)/() , Amster(dam), Prague/, Increas(ing)/(  ) .  wt [wt [ 0 Temp wt 0 Amster] wt = (de re) wt [ 0 Temp wt 0 Prague] wt ]  wt [ 0 Increas wt [wt [ 0 Temp wt 0 Amst]] the magnitude is (de dicto) mentioned.

27 Rules of Substitution (logic of partial functions !) “Homogeneous” substitution: no problem  Lebniz’s law Used* de re  extensional context de re Used* de dicto  intensional context de dicto Mentioned* construction  hyperintensional context Used* constructions – constituents: De re (extensional) context: [Cx] = [C’y] co-incidental constructions substitutable De dicto (intensional context): C = C’ equivalent constructions substitutable Mentioned* (hyper-intensional) context: 0 C = 0 C ’ Only identical constructions substitutable

28 Rules of Substitution (logic of partial functions !) Heterogeneous substitutions. Construction of a lower-order into a higher- order context (which is dominant): We must not carelessly draw a construction D occurring in a lower-order context into a higher-order context. Why not? The substitution would not be correct even if there is no collision of variables, due to partiality

Rules of substition29 De re rules The president of CR is (is not) an economist.  de re The president of CR exists. The president of CR is eligible.  de dicto The president of CR may not exist. In the de re case there is an existential presupposition, unlike the de dicto case.

Rules of substition30 Charles believes of the president of CR that he is an economist. Types: Ch/ , B/(   ) , Pr(esident of …)/(  ) , CR/ , Ec/(  )  Synthesis (h  , a free variable  the meaning of “he”): He is an economist: w t [ 0 Ec wt h]  v   (anaphora) The President of CR: w t [ 0 Pr wt 0 CR]    a) The President of CR is believed by Charles to be an economist – the passive variant w t [ h [ 0 B wt 0 Ch w t [ 0 Ec wt h]] w t [ 0 Pr wt 0 CR] wt ] Now, can we perform  -reduction ??? Yes, but only the trivial one: w t [ 0 Pr wt 0 CR] wt |  [ 0 Pr wt 0 CR] Collision of variables? Let us rename them:

Rules of substition31 Charles believes of the president of CR that he is an economist.  -reduction “by name” : w t [ h [ 0 B wt 0 Ch w’ t’ [ 0 Ec w’t’ h]] [ 0 Pr wt 0 CR] ] |  ??? w t [ 0 B wt 0 Ch w’ t’ [ 0 Ec w’t’ [ 0 Pr wt 0 CR]]] No collision of variables, But. [ h [ 0 B wt 0 Ch w’ t’ [ 0 Ec w’t’ h]] [ 0 Pr wt 0 CR] ]  [ 0 Exist wt w t [ 0 Pr wt 0 CR]] = [ 0  x [x = [ 0 Pr wt 0 CR]] Unlike the latter. Therefore, don’t perform  -reduction (!?!)

Rules of substition32 Charles believes of the president of CR that he is an economist. b) The direct analysis of the active form, using Tr and Sub.  -reduction “by value”: Now we have to substitute for h the construction of the individual (if any) that actually plays the role of the president: b) w t [ 0 Belive wt 0 Charles 2 [ 0 Sub [ 0 Tr w t [ 0 Pr wt 0 CR] wt ] 0 h (extens.) 0 [ w t [ 0 Ec wt h]] ] ] (intens.)

33 Rules of Substitution (two-phase  -reduction!) We must not carelessly draw a construction D ( w t [ 0 Pr wt 0 CR] wt ) of a lower order into a higher-order context (“calling by name”). Instead, “Calling a sub-procedure by value.” Execute D first, and, if D does not fail (i.e., is not v-improper), then go on to substitute the output of D ([ 0 Tr w t [ 0 Pr wt 0 CR] wt ]). How does it work?

Rules of substition34 2-phase  -reduction: how does it work? w t [ 0 Bel wt 0 Ch 2 [ 0 Sub [ 0 Tr w t [ 0 Pr wt 0 CR] wt ] 0 h 0 [ w t [ 0 Ec wt h]] ] ] 1. Let w t [ 0 Pr wt 0 CR] wt be v-improper (the president does not exist).  Then [ 0 Tr w t [ 0 Pr wt 0 CR] wt ] is v-improper and  The function Sub does not have an argument to operate on:  [ 0 Sub [ 0 Tr w t [ 0 Pr wt 0 CR] wt ] 0 h 0 [ w t [ 0 Ec wt h]] ] v-improper. (And so is the Double execution.)  The so-constructed proposition does not have a truth-value, as it should be (the existential presupposition)

Rules of substition35 Substitution by value (  -reduction) w t [ 0 Bel wt 0 Ch 2 [ 0 Sub [ 0 Tr w t [ 0 Pr wt 0 CR] wt ] 0 h 0 [ w t [ 0 Ec wt h]] ] ] 2. Let w t [ 0 Pr wt 0 CR] wt be v-proper (the president exist). Then  the construction [ 0 Pr wt 0 CR] v-constructs particular individual Y (For instance V. Klaus.) Then  [ 0 Tr w t [ 0 Pr wt 0 CR] wt ] v-constructs 0 Y, and Sub inserts it for the variable h.  the result is the construction: [ w t [ 0 Ec wt 0 Y]] that is executed (Double execution) in order to construct the proposition that is believed by Charles.

36 Substitution by value (  -reduction) Type checking: 2 [Sub [ 0 Tr [ 0 Pr wt 0 CR]] 0 h 0 [ w t [ 0 Ec wt h] ]] (* 1  )  (* 1 * 1 * 1 * 1 ) * 1 * 1 * 1 * 1 (    ) 1. step   2. step (if the 1 st did not fail): 1 [ w t [ 0 Ec wt 0 Y] ]    w t [ 0 Bel wt 0 Ch 20 [ w t [ 0 Ec wt 0 Y] ]   

Rules of substition37  -reduction, another example (*)[ y [ 0 Deg z [ 0 : z y]] 0  x ] (  = square root) (Deg/(  (  ))-a degenerated function) (*n)  -reduced “syntactically-by-name”: [ 0 Deg z [ 0 : z 0  x]]  [[ 0 Exist x] 0  ] ??? NO (*v)  -reduced “by value”: 2 [ 0 Sub [ 0 Tr 0  x] 0 y 0 [ 0 Deg z [ 0 : z y]] ] for:value of (*)of (*n)of (*v) x > 0False x = 0True x < 0UndefinedTrueUndefined

Rules of substition38 Valid rule of  -reduction (2-phase) Let C(y) be a construction with a free variable y, y  , and let D  . Then [[ y C(y)] D]  2 [ 0 Sub [ 0 Tr D] 0 y 0 C(y)] is a valid rule (proof, see above).

Rules of substition39 Rules of inference: Types: y  β, x  , D / (β  ), [Dx]  β, C(y)  α, y C(y)  (αβ), [[ y C(y)] [Dx]]  α. Compositionality: [ 0 Improper 0 [Dx]] |  [ 0 Improper 0 [[ y C(y)] [Dx]]] [ 0 Improper 0 [Dx]] |  [ 0 Improper 02 [ 0 Sub [ 0 Tr [Dx]] 0 y 0 C(y)]] [ 0 Proper 0 [Dx]] |  2 [ 0 Sub [ 0 Tr [Dx]] 0 y 0 C(y)] = [[ y C(y)] [Dx]] = C(y/[Dx]) Special case: Existential presupposition de re Exist / ((  (β  ))  )  the property of a (β  )-function of being defined at a  -argument, [Exist x]  (  (β  ))  [[ 0 Exist x] D] |  [ 0 Improper 0 [[ y C(y)] [Dx]]]  [[ 0 Exist x] D] |  [ 0 Improper 02 [ 0 Sub [ 0 Tr [Dx]] 0 y 0 C(y)]] But not: C(y/[Dx]) |  [[ 0 Exist x] D] …

Rules of substition40 The two “de re principles”: a) existential presupposition Example: [ y [ 0 Deg z [ 0 : z y]] [ 0  x]] |  [[ 0 Exist x] 0  ] 2 [ 0 Sub [ 0 Tr [ 0  x]] 0 y 0 [ 0 Deg z [ 0 : z y]]] |  [[ 0 Exist x] 0  ] Indeed: The square root does not exist for x < 0; for x < 0 the left-hand side is (v-)improper. If the left-hand side is true or false, then the square root exists and x  0. However, the result of the “syntactical” β-reduction does not meet these rules: [ 0 Deg z [ 0 : z 0  x]] and not (for x < 0) [[ 0 Exist x] 0  ].

Rules of substition41 The two “de re principles”: b) inter-substitutivity of co-incidentals [Dx] = [D’ ]  [[ y C(y)] [Dx]] = [[ y C(y)] [D’ ]] = 2 [ 0 Sub [ 0 Tr [D’ ]] 0 y 0 C(y)] Example: The US President is the husband of Laura. The US President is a Republican. Hence: The husband of Laura is a Republican. But not: John Kerry wanted to become the husband of Laura.

Rules of substition42 Substitutions in general Types: c   n, 2 c  , A  , y   a)“by name” (homogeneous substitution): 2 [ 0 Sub 00 A 0 c 0 C(c)] = C(c/ 0 A) 2 [ 0 Sub 0 A 0 y 0 C(y)] = C(y/A) b) “by value” (generally valid, even for heterogeneous substitution): 2 [ 0 Sub [ 0 Tr A] 0 y 0 C(y)] = [ y [C(y)] A]  C(y/A)

Rules of substition43 Conclusions The top-down, fine-grained approach of TIL makes it possible to adequately model structured meanings, and thus:  to formulate meaning-driven (non ad hoc) rules of substitution taking into account the Use/Mention distinction at all levels;  to adhere to Compositionality and anti-contextualism (even in the cases of anaphora, de re attitudes with anaphoric reference, hyper-intensional attitudes, …);  to take into account partiality;  to meet the two de re extensional principles (existential presupposition, inter-substitutivity of co-referentials).