P ossible W orlds S emantics for C onditionals : T he C ase of C - PowerPoint PPT Presentation

P ossible W orlds S emantics for C onditionals : T he C ase of C hellas -S egerberg S emantics Matthias Unterhuber Munic Center for Mathematical Philosophy matthias.unterhuber@lrz.uni-muenchen.de D agstuhl S eminar 15221 May 25-29, 2015

I ntroduction Chellas-Segerberg Semantics: A Possible-Semantics for Conditionals Goes back to Chellas (1975) and Segerberg (1989) Modal base logic for conditional logic A number of advantages Correspondence and completeness results for a lattice of systems based on thirty principles and frame conditions (Unterhuber, 2013; Unterhuber & Schurz, 2014) Can, for example, model the systems of Kraus, Lehmann, and Magidor (1990), Lewis (1973), and Adams (1966). Today: Outline of the correspondence and completeness result

A dvantage 1 Full language, in contrast to the probabilistic semantics of Adams (1966, 1977) without triviality à la Lewis (1976). Full language L : 1 contains the set of atomic variables AV = { p , p ′ , . . . } and 2 is closed under truth-functional propositional operators ¬ (negation) and ∨ (disjunction) as 1 well as the two-place modal operator � (conditional”) plus its dual � . 2

S emantics D efinition 1 F C = � W , R � is a Chellas frame iff (a) W is a non-empty set of indices and (b) R ⊆ W × W × Pow ( W ) . D efinition 2 Let F = � W , R � be a Chellas frame. Then, M = � W , R , V � is a Chellas model iff V is a valuation function from AV × W to { 0 , 1 } and for all formulas α , β ∈ L and w ∈ W it holds: (i) w | = M ¬ α iff w �| = M α (ii) w | = M α ∨ β iff w | = M α or w | = M β for all w ′ : if wR � α � M w ′ then w ′ | = M β (ii) w | = M α � β iff for w , w ′ ∈ W and X ⊆ W w | = M α : V ( α, w ) = 1 w �| = M α : not w | = M α � α � M = { w | w | = M α } wR X w ′ : � w , w ′ , X � ∈ R

P roof T heory if α ↔ β then ( α � γ ) → ( β � γ ) LLE RW if α → β then ( γ � α ) → ( γ � β ) AND ( α � β ) ∧ ( α � γ ) → ( α � β ∧ γ ) LT α � ⊤ More interesting: What it is missing

A dvantage 2: C orrespondence In CS Semantics structural (frame-based) conditions can be specified. Note: Such structural conditions cannot be specified when R ⊆ W × W × L . Examples: (Refl) α � α (CM) ( α � γ ) ∧ ( α � β ) → ( α ∧ β � γ ) (RM) ( α � γ ) ∧ ( α � β ) → ( α ∧ β � γ ) (Or) ( α � γ ) ∧ ( β � γ ) → ( α ∨ β � γ ) ∀ w ∀ w ′ ( wR X w ′ → w ′ ∈ X ) C ∗ Refl ∀ w ( ∀ w ′ ( wR X w ′ → w ′ ∈ Y ) → ∀ w ′ ( wR X ∩ Y w ′ → wR X w ′ )) C CM ∀ w ( ∃ w ′ ( wR X w ′ ∧ w ′ ∈ Y ) → ∀ w ′ ( wR X ∩ Y w ′ → wR X w ′ )) C RM ∀ w ∀ w ′ ( wR X ∪ Y w ′ → wR X w ′ ∨ wR Y w ′ ) C Or A lattice of systems available as specified by thirty pairs of conditional logic principles and frame conditions (Unterhuber, 2013; Chellas, 1975; Segerberg, 1989).

A dvantage 2 For example, ( α � β ) → β (T) corresponds to ∀ w ( wR X w ) ( C ) in the following sense: (T) is valid on a Chellas frame � W , R � iff C holds for all X ⊆ W , where valid on frame � W , R � is being true at all worlds w ∈ W for all models � W , R , V � . For a given Chellas frame � W , R � , (T) is true at all worlds w ∈ W for all models � W , R , V � (short: valid on frame � W , R � ) iff C holds for all X ⊆ W . This is not true for Chellas models . Consider (T) and consider the model M = � W , R , V � , where W = { w , w ′ } , where w and w ′ can see each other with respect to ∅ and W and where V ( α, w ) = V ( α, w ′ ) for all α ∈ L .

C ompleteness with a T wist Road map: Completeness proof with respect to classes of Chellas frames does not go through (as in Chellas, 1975). Problem: There are sets of possible worlds in Chellas models for which there is no formula α such that α is true in all worlds of that set. Completeness with respect to classes of Segerberg frames (to be defined), as in Segerberg (1989), is trivial: Any such logic system would be complete. Solution: Use a narrower class of Segerberg frames (standard Segerberg frames)

S egerberg frames D efinition 3 F S = � W , R , P � is a Segerberg frame iff (a) � W , R � is a Chellas frame and (b) R ⊆ W × W × P for P ⊆ Pow ( W ) such that (i) ∅ ∈ P , (ii) if X ∈ P then − X ∈ P , (iii) if X , Y ∈ P then X ∪ Y ∈ P , and (iv) if X , Y ∈ P then X � ∗ Y ∈ P . X � ∗ Y = { w ∈ W | ∀ w ′ ∈ W ( wR X w ′ → w ′ ∈ Y ) } ∈ P for X , Y ⊆ W Segerberg frames correspond to general frames in normal modal logics. D efinition 4 M S = � W , R , P , V � is a Segerberg model iff (a) � W , R , P � is a Segerberg frame and (b) � W , R , V � is a Chellas model such that for all α ∈ L : � α � M ∈ P .

C ompleteness : T riviality and S olution The source of the problem: T heorem 5 For each Chellas model M = � W , R , V � there exists a Segerberg frame M S = � W , R , P V � , where P V = {� α � M | α ∈ L} , so that all formulas valid on M are valid on M S , and vice versa. In particular, Segerberg frame completeness does not exclude non-standard models. Solution: Restrict the class of Segerberg frames which qualify for the extension of the base system (excluding non-standard Segerberg frames). Language of frame conditions: Multi-sorted language with two types of terms, one referring to single possible worlds and the other referring to sets of possible worlds (set denoting)

C ompleteness : T riviality and S olution 1 No trivial occurrence of subformulæ: No logical equivalent formula which does not contain that subformula. Expressions such as wR X w ′ ∨ ¬ wR X w ′ are excluded. 2 No trivial occurrence of set variables: No set denoting term is equivalent in with an additional set variable occurs (Boolean equivalence + X). Expressions such as Y in X ∩ ( Y ∪ ¬ Y ) . Essential: No set variable occurs in a frame condition satisfying of (1) and (2) unless it occurs also in the third argument place of some occurrence of R . This gives us exactly what we wanted. For (T) ∀ w ∀ w ′ (( wR X w ′ → w ′ ∈ Y ) → w ′ ∈ Y ) is excluded, whereas ∀ w ( wR X w ) is not. Important: This requirement generalizes from Kripke model correspondence.

Many thanks!

R eferences I Adams, E. (1966). Probability and the Logic of Conditionals. In J. Hintikka & P . Suppes (Eds.), Aspects of Inductive Logic (pp. 265–316). Amsterdam: North-Holland Publishing Company. Adams, E. (1977). A Note on Comparing Probabilistic and Modal Logics of Conditionals. Theoria , 18 , 186–194. Chellas, B. F. (1975). Basic Conditional Logic. Journal of Philosophical Logic , 4 , 133–153. Kraus, S., Lehmann, D., & Magidor, M. (1990). Nonmonotonic Reasoning, Preferential Models and Cumulative Logics. Artificial Intelligence , 44 , 167–207. Lewis, D. (1973). Counterfactuals . Blackwell. Lewis, D. (1976). Probabilities of Conditionals and Conditional Probabilities. The Philosophical Review , 85 , 297–315. Segerberg, K. (1989). Notes on Conditional Logic. Studia Logica , 48 , 157–168. Unterhuber, M. (2013). Possible Worlds Semantics for Indicative and Counterfactual Conditionals? A Formal Philosophical Inquiry into Chellas-Segerberg Semantics . Frankfurt am Main: Ontos Verlag (Logos Series).

R eferences II Unterhuber, M., & Schurz, G. (2014). Completeness and correspondence in Chellas-Segerberg semantics. Studia Logica , 102 .