Remarks on Kratzer s Ordering- Premise Semantics of Modalities and Conditionals. Norry Ogata Osaka University. Contents

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1 Remarks on ratzer s rdering- Premise Semantics of Modalities and Conditionals Norry gata saka University Contents 1. Definitions of rdering-premise Semantics 2. Their Motivations 3. Some Formal Properties and Flaws 4. Multi-modal Premise Semantics 5. Premise Semantics from the viewpoint of Dynamic Doxastic Logic 1

2 Definitions of rdering-premise Semantics Let W B be a set of possible worlds, functions such that, : W ( ( W )) are called a modal base and ordering source respectively. They gives two types of premise set (set of propositions) to each possible worlds. Let Φ( p) be a set of propositional variables, then formula φ L, >, Φ φ::= p φ φ φ > φ φ is defined by: where φ φ is a material implication which can define 1 φ φ, φ φ, φ with, and φ > φ is a conditional formula

3 Let V : Φ (W) be a valuation of Φ. Then \, >, Φ is called an rdering-premise Model of if p = W, B,, V , >, Φ : L (W) is defined by recursion on φ: = V ( p) ; φ φ = ( W φ ) φ = ; L ; φ u v. where ={ w W B v u & z z v z φ B B u v { p v p} { p u p}. }, 3

4 Lemma 1. (1) is reflexive, transitive, and anti-symmetry, even if =. (2) is reflexive, transitive, and anti-symmetry, even if =. (3) w w even if =. ripke Semantics W : a non-empty set of possible worlds R : a binary relation on W V : Φ ( W ) a valuation of Φ M = W, R, V : a ripke model M, w φ u W. wru M, u φ M M Γ M M, w φ u W. wru &, u φ φ : ϕ Γ w W., w ϕ, w φ 4

5 Lemma 2. (1) R = {( w, z) u B v B z B. z v & v u} = {( w, z) u B z u} is reflexive, transitive, and serial if w B Lemma 1(2) (2) If w W : B( w) =, then W, R is a reflexive, transitive, serial ripke frame R = {( w, z) u W. z u}. (3) R is transitive and serial Lemma 2(1-2). Lemma 3. (1) W, R ϕ ϕ Lemma 2(3), (2) W, R ϕ ϕ Lemma 2(3), (3) W, R? X ( ϕ ϕ) Lemma 2(1-2) 5

6 Problems on rdering-premise Semantics Case1:Some verifiable propositions are not in any premise sets Case2:n vercoming Inconsistencies Case3:n Practical Inference =multimodal inference Case 1:Circumstantial Conversational Backgrounds B is realistic, i.e., w. w B. B( u) = { p, q} ( u) = { p '''}, p q = { u}, u p ' p ''' p '' Then, W, B,, V, u p', p '' even though p' and p '' are not in the conversational background. 6

7 Case 1:Circumstantial Conversational Backgrounds u p ''' q' p ''' p ''' q' Then, W, B,, V, u ( p ''' q'), ( p ''' q') for each q'( u) which is not in the conversational background. Case 2: Judgments Judge A: a is liable ( l); Judge B: a is not liable( l) Judge A=Judge B: the murder is a crime ( c) Judgments = =. Judgments c, c Judgments l, l, for all formula c, l,... 7

8 Case 2: Judgments (ratzer s Solution) w l, l, c Closest to the ideal l, c l, c c More further away than the other two by the ordering source w. B =,i.e., B = W,i.e., R is reflexive, i.e., w is impossible Case 3: Practical Inference B B is circumstantial, i.e., = { gotopub popular} is bouletic, i.e., = { popular, gotopub, hike} = = B 8

9 Case 3: Practical Inference (ratzer s Solution) Closest worlds hike hike gotopub hike gotopub gotopub popular popular popular hike gotopub hike gotopub popular hike gotopub popular should hike, could go, could pop could go, could pop ne wish Case 3: Practical Inference (Frank s Solution) Does this correspond to our intuitions? In one way it does, but in another it doesn t. Frank(1997) She proposes a contraction of p and not-p in the sense of Belief Revision. B B! = ', where ' & ( B ') & X if ' X, then ( X ) =. B For another contraction in DRT, see of Frank(1997). 9

10 Case 3 If ratzer s treatment of practical inference is right, we can add more multi-modal premises: i. The speaker s father s wish: not-hike ii. The law: not-gotopub iii. How many are ordering sources or modal bases needed? r Such a multi-modal situation can be treated? Lemma 4., w ϕ, w ϕ u B v B z B. z v & v u, v ϕ u B v B z B. z v & v u &, v ϕ 10

11 rdering Lemma (1) v u even if = P( u) = = P( v) (2) v u even if P( u) = P( u) = X for all X (3) v u even if P( u) = & P( v) = Propositions on Case 2 = { l, l, c} P( v) = { l, c} P( v) = { l, c} P( u) = { l, c} P( u) = { l, c} P( t) = { l, c} P( t) = { l} P( s) = { l, c} P( s) = { l} B = B = W ( P = { }?) u s; v t; x x;( x w) u, v, t, s, w t, s, w w l, l,? c,?,? w 11

12 rdering Proposition: Case 3 = { pop, go, hike}; w. B = { go pop} P( v) = { hike, go, pop} P( v) = { go} P( u) = { hike, go, pop} P( u) = { pop} P( t) = { hike, go, pop} P( t) = { hike, go} P( s) = { hike, go, pop} P( s) = { hike, pop} Let P = or { go pop} P = Let B = { w} or W t u, v; s u; x x; x w w go, hike, pop, go, pop, ( go pop) = W, B,, V Conditionals in rdering- Premise Semantics φ > φ = { w W w φ }, [ B '/ B ] where wb ' = B { φ }, = { w W u ( B( w) { φ1 }) v ( B { φ1 }) z ( B { φ1 }). z v & v u z φ } 2 (1) material: w B( w) = { w}; w =. (2) strict: wb = ; w =. (3) counterfactual: wb = ; w = { w}. (4) indicative: wb = ; is circumstantial 1 12

13 Counterfactuals in rdering- Premise Semantics A counterfactual is characterized by an empty modal base f and a totally realistic ordering source g. It follows from David Lewis work mentioned above, that this analysis of counterfactuals is equivalent to the one I give in ratzer(1981b). The idea is this: All possible worlds in which the antecedent p is true, are ordered with respect to their being more or less near to what is actually the case in the world under consideration. ratzer(1981a) Standard Semantics of Conditionals (SMC) M= W, f, V ; f : W ( W ) ( W ) (selection function) φ1 > φ2 M = { w W f ( w, φ1 M ) φ2 M}, (1) material: w X w f ( w, X ) & w ( W X ) Y f ( w, X ) Y. (2) strict: f ( w, X ) = X. (3) counterfactual: many versions (e.g., ratzer(1981)=ss;stalnaker(1968)=vcs; Lewis(1973)=VC). 13

14 Counterfactuals in SMC SS=M.P.;RCEA ;RCM;RCE;CC+CA+AC+CS+MP (Pollock 1981) =RCEC;RC;ID+MD+CA+CS+MP (Nute & Cross 2002) CA: f ( w, X Y ) f ( w, X ) f ( w, Y ) AC : f ( w, X ) Y f ( w, X Y ) f ( w, X ) CS : w X f ( w, X ) { w} MP : w X w f ( w, X ) ID : f ( w, X ) X MD : f ( w, W X ) X f ( w, Y ) f ( w, X ) w φ > φ Standardization of rdering- Premise Semantics 1 u ( { X}) v ( { X}) z ( { X}). z v & v u z Y B 2 B u X v X z X. z v & v u z Y, where w. = { w} and X = φ, Y = 1 2 [ B '/ B] f ( w, X ) = { z X u X v X. z v & v u} ( w) B φ. 14

15 Standardization of rdering- Premise Semantics f 's properties ID : obvious MP : yes u X. w w & w u P( u) P ratzer s Samaritan Paradox For each morally accessible world s : No murder occurs ( p) If a murder occurs, the murderer will be given $100.( p q) 1. ( p q) vacluously true 2. p q vacuously true 3. p > q u, z p. v p. z v & v u z q [ B '/ B ] s is excluded but no X must morally occur cannot be said? 15

16 Frank s solution to ratzer s Samaritan Paradox No murder occurs ( p) If a murder occurs, the murderer will be given $100.( p q) F = Cn({ r r Cn({ p, p q})}) H = Cn({ r q r Cn( Cn( Cn( F { p}) { p})! F)} (ignoring variable assignments; In H, p and p are contracted) f satisfies ID in any interpretation, i.e., p > p. The Frank-Zvolensky Problem #If teenagers drink, then they must drink.(deontic) #If I file my taxes, then I must file my taxes.(bouletic) #If a=b, then it is necessarily a=b.(alethic by gata) p > p is generally bad? Frank: X!{ p > p} = Cn( Cn( X { p > p}) { p, p}) p, p are undefined in X!{ p > p} But: : If Dalai Lama is mad, he must do so. (=he must have his reason. Etc by Zvolensky(2002) 16

17 Multi-Modal Premise Semantics (Basic Ideas) Exploiting Belief-Revision Theory, esp., Model Theories of Dynamic Doxastic Logics (Segerberg 2001) etc. Exlpoiting Multi-Modal Logic, esp., BDI- Logics (Rao & Georgeff 1996) etc. Let W be any nonempty set. A topology in W is a family of subsets of W such as: (i) W,, i= (ii) W, Wi W i<n (iii) W. W, is called a topological space. If X, then X i Wi W Model Theory of DDL (1) i is called open. If X is open, W X is called closed. If X is open and closed, it is called clopen. 17

18 Model Theory of DDL (2) A cover C ( ( W ))( ) of a set X W is a family s.t. X C. If every Y C is open, C is called an open cover. If every open cover of is called compact. has a finite subcover, If for any pair of distinc t elements of, one is an element of a clopen set of which the other is not, is called totally separated. If is compact and totally separated, it is a Stone topology. Then W, is called a Stone spa ce. Model Theory of DDL (3) An onion in W, is a nonempty family ( ( W )) satisfying the following conditions: (NESTEDNESS) if X, Y, X Y or Y X, (LIMIT) for any clopen set P, if P, then there is a least element X s.t. P Y. If evey X is closed, is called a closed on ion. 18

19 Model Theory of DDL (4) If an onion satisfies the following conditions, it is called a Lewis onion: (AINT) if C then C ; (AUNI) if C then C ; The belief set B of and the commitment set C of are defined by the conditions: B =, C =. Model Theory of DDL (5) Let W, be a Stone space. An onion frame is a triple W,, D where D : ( W ) ( ( W )), called the onion determiner, is defined by the conditions: (i) for every X dom( D), D( X ) is an onion, (ii) dom( D), (iii) C W.s.t. X dom( D), C = D( X ), (iv) if X dom( D), then, For each clopen set P. C P S P dom( D), where S is the smallest element in D( X ) to overlap with P. C rng( D) is called D-possible belief set. 19

20 Multi-Modal DDL (1) Let A be a set of agents and M a set of modalities. A multimodal index set Ind( A, M ) over A and M is defined by: * ::= ι m( a) ι ; ι ι + ι ι Multi-Modal DDL (2) Let ( p ) Φ be a set of propositional variables ( w ) W a set of possible worlds, ( ι ) Ind ( A, M ) a multimodal index set, and F= W,, D,, Ind( A, M ) a multimodal onion frame. where D : Ind( A, M ) W ( ( W ) ( ( W ))), and D( ι, w) is a onion determiner and : Ind ( A, M ) W ( ( W )), ( ι, w) is a onion for each ι Ind ( A, M ), w W. 20

21 Multi-Modal DDL (3) Let Φ ( p) be a set of propositional variables and IndVar( x) a variable of multimodal indices. L The language ( ϕ) generated IndVar, >, Φ by Φ and Ind ( A, M ) is defined by: ϕ :: = p ϕ ϕ ϕ ϕ ϕ > ϕ 1 2 x 1 x 2 Multi-Modal DDL (4) Let Φ ( p) be a set of propositional variables, Ind( A, M )( ι) a multimodal index set, L Ind ( A, M ), >, Φ ( ϕ) a language W a set of possible worlds, V : Φ ( W ) a valuation, f : W ( W ) ( W ) a selection function, and W,, D,, Ind( A, M ) a multimodal onion frame. The relation W,, f, V, ι, w ϕ is defined by recursion on ϕ : 21

22 Multi-Modal DDL (5) W,, f, V, ι, w p w V ( p) W,, f, V, ι, w ϕ1 ϕ2 W,, f, V, ι, w ϕ1 & D,, f, V, ι, w ϕ2 W,, f, V, ι, w ϕ W,, f, V, ι, w ϕ W,, f, V, ι, w x ϕ ( ι ( x), w) ϕ ϕ = { w W W,, f, V, ι, w ϕ} Multi-Modal DDL (6) Belief-Desire Consistency (cf. BDI-logic) mc( ι) Belief ( a), mc( κ ) Desire( a) ( ϕ ϕ) mc( κ) Belief ( a), mc( ι) Desire( a) ( ϕ ϕ) mc( ι) Belief ( a), mc( κ ) Desire( a) ( ψ > ι ϕ ( ψ > κ ϕ)) mc( κ) Belief ( a), mc( ι) Desire( a) ( ψ > ι ϕ ( ψ > κ ϕ)) ι ι κ κ 22

23 Solution to Many Judges Problem mc( ι ( x)) obligatory( a) x ( ϕ ψ ) ( xϕ xψ ) mc( ι ( x)) obligatory( a) x ϕ x ϕ mc( ι ( x)) obligatory( a) x Judge A: xliable, xmurderisacrime, mc( ι ( x)) obligatory( a) Judge B: liable, murderisacrime, mc( ι ( y)) obligatory( a) No Inconsistency y y Solution to the Practical Inference Problem I want not to go to pub: mc( ι ( y)) Desire( a) & y pub I want to be popular: mc( ι ( y)) Desire( a) & pop It is the case that if I go to pub, then I will be popular: pub > x pop mc( ι ( x)) Belief ( a) ( pub > pop) No Inconsistency Desire( a) y 23

24 Solution to ratzer s Samaritan Paradox No murder occurs: bligatory ( a) p If a murder occurs, the jurors must convene: p > q ( p q) bligatory( a) But #If a murder occurs, the murderer will be given $100. ther index types: bligation( a), Expectation( a), Predict( a),... D,, V, f, ι, w ϕ > ϕ 1 x 2 D, [ D( ι ( x), w)( Z f ( w, ϕ )) /( ι ( x), w)], V, ι, w ϕ 1 2 if Z = min{ S ( ι ( x), w) S f ( w, ϕ ) } D, [ D( ι, w)( ) /( ι ( x), w)], V, ι, w ϕ otherwise means that ϕ > x 2 ϕ is equivalent to [* ϕ ] ϕ where,,... ι A Solution to the Frank- Zvolensky Problem If f does not satisfy (id) f ( w, X ) X, the logic does not satisfy ϕ > ϕ. Lie Segerberg(2001) defines selector function semantics of [* ϕ ] ϕ

25 A Solution to the Frank- Zvolensky Problems W,, f, V, ι, w ϕ1 > ϕ2 f ( w, ( D( ι ( x), w)( Z ϕ1 ))) ϕ2 if Z = min{ S ( ι ( x), w) S ϕ1 } W, [ D( ι ( x), w)( ) /( ι, w)], f, V, ι, w ϕ otherwise and f does not satisfy (id). 2 Loose End I have reviewed ratzer s rdering Premise Semantics of Modalities and Conditionals: but the properties of f has still been clarified. I have proposed multimodal premise semantics based on Dynamic Doxastic Logic, BDI-logics, and Conditional Logics I have provided another solution for each problems which are motivations of rdering- Premise Semantics 25

26 Bibliography Frank, A. (1997). n Context Dependence in Modal Constructions. Ph.D. thesis, University of Stuttgart. ratzer, A. (1981a). The notional category of modality. In H.-J. Eikmeyer and H. Rieser (Eds.), Words, Worlds and Context: New Approaches to Word Semantics, pp Berlin: Walter de Gruyter& Co. ratzer, A. (1981b). Partition and revision: The semantics of counterfactuals.journal of Philosophical Logic 10, ratzer, A. (1991). Modality. In A. von Stechow and D. Wunderlich (Eds.), Semantik: Ein intentionales Handbuch der zeitgenoess-ischen Forscung, pp Berlin: Walter de Gruyter & Co. Nute, D. and C. B. Cross (2002). Conditional logic. In D. Gabbay and F. Guenthner (Eds.), Handbook of Philosophical Logic, 2nd Edition, Volume 4, pp Dordrecht: luwer Academic Pub. Rao, A. and M. P. Georgeff. (1998). Decision Procedures for BDI Logics. Journal of Logic and Computation 8:(3) Segerberg,. (2001). The Basic Dynamic Doxastic Logic of AGM. In M.-A. Williams and H. Rott (eds.) Frontiers in Belief Revision, pp Dordrecht: luwer Academic Pub. Zvolenszky, Z. (2002). Is a possible-worlds semantics of modality possible?:a problem for ratzer s semantics. In Proceedings from Semantics And Linguistic Theory, Volume