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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 - - 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
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Advantage 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 1
truth-functional propositional operators ¬ (negation) and ∨ (disjunction) as well as
2
the two-place modal operator (conditional”) plus its dual .
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Semantics
Definition 1 FC = W, R is a Chellas frame iff (a) W is a non-empty set of indices and (b) R ⊆ W × W × Pow(W). Definition 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 (ii) w |=M α β iff for all w′: if wRαMw′ then w′ |=M β w |=M α: V(α, w) = 1 w |=M α: not w |=M α for w, w′ ∈ W and X ⊆ W wRXw′: w, w′, X ∈ R αM = {w | w |=M α}
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Proof Theory
LLE if α ↔ β then (α γ) → (β γ) 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
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Advantage 2: Correspondence
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) C∗
Refl
∀w∀w′(wRXw′ → w′ ∈ X) CCM ∀w(∀w′(wRXw′ → w′ ∈ Y) → ∀w′(wRX∩Yw′ → wRXw′)) CRM ∀w(∃w′(wRXw′ ∧ w′ ∈ Y) → ∀w′(wRX∩Yw′ → wRXw′)) COr ∀w∀w′(wRX∪Yw′ → wRXw′ ∨ wRYw′) A lattice of systems available as specified by thirty pairs of conditional logic principles and frame conditions (Unterhuber, 2013; Chellas, 1975; Segerberg, 1989).
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Advantage 2
For example, (α β) → β (T) corresponds to ∀w(wRXw) (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.
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Completeness with a Twist
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)
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Segerberg frames
Definition 3 FS = 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(wRXw′ → w′ ∈ Y)} ∈ P for X, Y ⊆ W Segerberg frames correspond to general frames in normal modal logics. Definition 4 MS = 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: αM ∈ P.
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Completeness: Triviality and Solution
The source of the problem: Theorem 5 For each Chellas model M = W, R, V there exists a Segerberg frame MS = W, R, PV, where PV = {αM | α ∈ L}, so that all formulas valid on M are valid on MS, 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,
- ne referring to single possible worlds and
the other referring to sets of possible worlds (set denoting)
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Completeness: Triviality and Solution
1 No trivial occurrence of subformulæ: No logical equivalent formula which
does not contain that subformula. Expressions such as wRXw′ ∨ ¬wRXw′ 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′((wRXw′ → w′ ∈ Y) → w′ ∈ Y) is excluded, whereas ∀w(wRXw) is not. Important: This requirement generalizes from Kripke model correspondence.
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Many thanks!
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References 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).
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