### Might counterfactual and outer modalities

parent 1ca6c762
 ... ... @@ -11,7 +11,27 @@ text \ - sphere: s (Lewis: S, T) \ subsection \Counterfactuals in terms of sphere systems\ subsection \A language of counterfactuals\ datatype ('aa) formula = Falsef (\\\) | Atom 'aa | Impl \'aa formula\ \'aa formula\ (\_ \ _\ 27) | Would \'aa formula\ \'aa formula\ (\_ \\ _\ 25) abbreviation Neg :: \'aa formula \ 'aa formula\ (\~~_\  40) where \~~\ \ \ \ \\ abbreviation Or :: \'aa formula \ 'aa formula \ 'aa formula\ where \Or \ \ \ (~~\) \ \\ abbreviation And :: \'aa formula \ 'aa formula \ 'aa formula\ where \And \ \ \ ~~Or (~~\) (~~\)\ text \The might counterfactual is treated as derived from the would counterfactual. (p. 2 and p. 21)\ definition Might :: \'aa formula \ 'aa formula \ 'aa formula\ (\_ \\ _\ 25) where [simp]: \\\\ \ \ ~~(\ \\ ~~\)\ \\We do not use \abbreviation\ here, as we sometimes want to talk about \\\\ explicitly. For the most time however, we are fine with it being simplified away automatically.\ subsection \Abstract sphere systems\ text \p. 14: “the set \{i}\ having \i\ as its only member belongs to \$\<^sub>i\.”\ definition centered_spheres :: \'world set set \ 'world \ bool\ ... ... @@ -64,18 +84,6 @@ lemma closures_trivial_for_finite_spheres: by (metis Sup_empty Union_in_chain finite_subset subset_chain_def subset_iff, simp add: Inter_in_chain finite_subset subset_chain_def subset_iff) datatype ('aa) formula = Falsef | Atom 'aa | Impl \'aa formula\ \'aa formula\ (\_ \ _\ 27) | Would \'aa formula\ \'aa formula\ (\_ \\ _\ 25) abbreviation Neg :: \'aa formula \ 'aa formula\ (\~~_\  40) where \~~\ \ \ \ Falsef\ abbreviation Or :: \'aa formula \ 'aa formula \ 'aa formula\ where \Or \ \ \ (~~\) \ \\ abbreviation And :: \'aa formula \ 'aa formula \ 'aa formula\ where \And \ \ \ ~~Or (~~\) (~~\)\ locale counterfactuals = fixes S :: \'world \ 'world set set\ and ... ... @@ -84,8 +92,15 @@ locale counterfactuals = sphere_system: \system_of_spheres S\ begin abbreviation possible_worlds :: \'world \ 'world set\ where \possible_worlds w \ \ (S w)\ subsection \Concrete spheres systems\ \\The name ‘outermost sphere’ is introduced on p. 22\ abbreviation outermost_sphere :: \'world \ 'world set\ where \outermost_sphere w \ \ (S w)\ \\“\\$\<^sub>i\ is itself a sphere around \i\ (p. 22)\ lemma \outermost_sphere w \ (S w)\ using sphere_system union_closed_spheres_def by auto abbreviation sphere_order :: \'world \ 'world set rel\ where \sphere_order w \ {(s1, s2). s1 \ S w \ s2 \ S w \ s1 \ s2}\ ... ... @@ -110,8 +125,10 @@ proof - unfolding linear_order_on_def using sphere_ordering_total .. qed subsection \Counterfactual semantics defined in terms of sphere systems\ primrec is_true_at :: \'a formula \ 'world \ bool\ (\\ _ \_\  55) where $$\Falsef\w) = False\ | \(\\\w) = False\ | \(\Atom a\w) = interpretations w a\ | \(\\\\\w) = (\(\\\w) \ \\\w)\| \\Definition of counterfactuals from p. 16\ ... ... @@ -133,6 +150,7 @@ lemma four_counterfactual_cases: ((\~~(\ \\$$\w) \ \\ \\ ~~\\w) \ ((\~~(\ \\ \)\w) \ \~~(\ \\ ~~\)\w)\ using is_true_at.simps by blast end subsection \The But-if-party Example\ ... ... @@ -292,6 +310,76 @@ antecedent-world.” (p. 21) The wording seems a little confusing. Apparently he has a (inverted?) version of the mathematical “eventually” in mind. With a temporal reading of “eventually,” the sentence would be wrong.\ subsection \‘Might’ counterfactuals\ text \Derived truth conditions for ‘might’, p. 21\ lemma (in counterfactuals) might_characterization: $$\\\\\\w) = ( (\s \ S w. \w\ \ s. \\\w$$ \ (\s \ S w. (\w\ \ s. \\\w\) \ (\ws \ s. (\\\ws) \\\\ws)))\ by auto \\This is not quite in line with everyday English: “If spheres were flat, earth might be flat.” and “If spheres were flat, earth would be flat.” seem to include the same statement or non-statement about the possibility of worlds where spheres are flat...\ text \“Under the Limit assumption, we could restate the derived truth conditions for ‘might’...”\ lemma (in counterfactuals_limit_assumption) might_characterization_limited: $$\\\\\\w) = (\wa \ smallest_sphere w \. (\\\wa) \ \\\wa)\ using counterfactual_smallest_sphere_def unfolding Might_def by (metis is_true_at.simps(1,3)) lemma (in counterfactuals) non_vacuously_would_implies_might: assumes \\\ \\ \\w\ \permitting_sphere \ s\ \s \ S w\ shows \\\ \\ \\w\ using assms by (auto, meson in_mono psubsetD sphere_direction) \\This is analogous to the fact that non-empty all-quantification implies existential quanatification.\ lemma (in counterfactuals) neither_would_nor_wouldnt_still_might: assumes \\\\ \\ \\w\ \\\\ \\ ~~\\w\ shows \\\ \\ \\w\ \\\ \\ ~~\\w\ using assms by auto \\“this is the case in which \\\ is true at some of the closest \\\ worlds and \~~\\ is true at others of them.” (p. 21)\ text \Pp. 22f. reinstate the standard modalities (in Kripke style).\ definition Necessary :: \'aa formula \ 'aa formula\ (\\ _\ 20) where [simp]: \\\ \ (~~$$ \\ \\ definition Possibly :: \'aa formula \ 'aa formula\ (\\ _\ 20) where [simp]: \\\ \ \ \\ ~~\\ lemma (in counterfactuals) modal_duality: $$\\\\w) = \~~(\~~$$\w\ $$\\\\w) = \~~(\~~$$\w\ by auto lemma (in counterfactuals) Necessary_ext_def: $$\\\\w) = (\s \ S w. \wo \ s. \\\wo)\ by auto lemma (in counterfactuals) Possibly_ext_def: \(\\\\w) = (\s \ S w. \wo \ s. \\\wo)\ by auto lemma (in counterfactuals) Necessary_outermost_def: \(\\\\w) = (\wo \ outermost_sphere w. \\\wo)\ by auto lemma (in counterfactuals) Possibly_outermost_def: \(\\\\w) = (\wo \ outermost_sphere w. \\\wo)\ by auto \\(That's why Lewis calls them “outer modalities.”)\ text \“the outer strict conditional implies the counterfactual”\ lemma (in counterfactuals) outer_strict_conditional: assumes \\\(\\$$\w\ shows \\\\\\\w\ using assms by auto subsection \Comparative Similarity\ context counterfactuals ... ...
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