Commit 4f1a8f42 by benkeks

Impossible antecedents and weakly centered systems

parent 34017c8a
......@@ -11,7 +11,7 @@ text \<open>
- sphere: s (Lewis: S, T)
\<close>
subsection \<open>A language of counterfactuals\<close>
subsection \<open>A language of counterfactuals (\<section> 1.1)\<close>
datatype ('aa) formula =
Falsef (\<open>\<bottom>\<close>) | Atom 'aa |
......@@ -58,15 +58,22 @@ definition intersection_closed_spheres :: \<open>'world set set \<Rightarrow> bo
where
\<open>intersection_closed_spheres S\<^sub>w \<equiv> \<forall>\<S> \<subseteq> S\<^sub>w. \<S> \<noteq> {} \<longrightarrow> \<Inter>\<S> \<in> S\<^sub>w\<close>
text \<open>\<open>centered system of spheres\<close> from p. 14\<close>
text \<open>\<open>system of spheres\<close> (without centering from p. 14\<close>
abbreviation system_of_spheres :: \<open>('world \<Rightarrow> 'world set set) \<Rightarrow> bool\<close>
where
\<open>system_of_spheres S \<equiv> \<forall>w.
centered_spheres (S w) w \<and>
nested_spheres (S w) \<and>
union_closed_spheres (S w) \<and>
intersection_closed_spheres (S w)\<close>
text \<open>\<open>centered system of spheres\<close> from p. 14\<close>
abbreviation centered_system_of_spheres :: \<open>('world \<Rightarrow> 'world set set) \<Rightarrow> bool\<close>
where
\<open>centered_system_of_spheres S \<equiv>
system_of_spheres S \<and>
(\<forall>w. centered_spheres (S w) w)\<close>
text \<open>p. 15 “Note that conditions (2) and (3) of closure under union and intersection are
automatically satisfied when there are only finitely many spheres around \<open>i\<close>, ...”\<close>
lemma closures_trivial_for_finite_spheres:
......@@ -99,9 +106,20 @@ abbreviation outermost_sphere :: \<open>'world \<Rightarrow> 'world set\<close>
\<open>outermost_sphere w \<equiv> \<Union> (S w)\<close>
\<comment>\<open>“\<open>\<Union>$\<^sub>i\<close> is itself a sphere around \<open>i\<close> (p. 22)\<close>
lemma \<open>outermost_sphere w \<in> (S w)\<close>
lemma outermost_sphere_is_sphere: \<open>outermost_sphere w \<in> (S w)\<close>
using sphere_system union_closed_spheres_def by auto
\<comment>\<open>The name ‘innermost sphere’ is introduced on p. 29\<close>
abbreviation innermost_sphere :: \<open>'world \<Rightarrow> 'world set\<close> where
\<open>innermost_sphere w \<equiv> \<Inter> (S w - {{}})\<close>
lemma innermost_sphere_is_sphere_for_nontrivial_systems:
assumes \<open>s0 \<in> S w\<close> \<open>s0 \<noteq> {}\<close>
shows \<open>innermost_sphere w \<in> (S w)\<close>
using assms sphere_system unfolding intersection_closed_spheres_def
by (smt Diff_cancel Diff_eq_empty_iff Diff_insert2 Diff_insert_absorb
insert_Diff intersection_closed_spheres_def singleton_insert_inj_eq sphere_system)
abbreviation sphere_order :: \<open>'world \<Rightarrow> 'world set rel\<close> where
\<open>sphere_order w \<equiv> {(s1, s2). s1 \<in> S w \<and> s2 \<in> S w \<and> s1 \<subseteq> s2}\<close>
......@@ -125,7 +143,7 @@ proof -
unfolding linear_order_on_def using sphere_ordering_total ..
qed
subsection \<open>Counterfactual semantics defined in terms of sphere systems\<close>
subsection \<open>Counterfactual semantics defined in terms of sphere systems (\<section> 1.3)\<close>
primrec is_true_at :: \<open>'a formula \<Rightarrow> 'world \<Rightarrow> bool\<close> (\<open>\<lbrakk> _ \<rbrakk>_\<close> [20] 55) where
\<open>(\<lbrakk>\<bottom>\<rbrakk>w) = False\<close> |
......@@ -187,7 +205,7 @@ begin
by auto
end
subsection \<open>The Limit Assumption\<close>
subsection \<open>The Limit Assumption (\<section> 1.4)\<close>
text \<open>
In \<section> 1.4, Lewis gives an alternative characterization of counterfactuals under the assumption
......@@ -310,7 +328,7 @@ 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.\<close>
subsection \<open>‘Might’ counterfactuals\<close>
subsection \<open>‘Might’ counterfactuals (\<section> 1.5)\<close>
text \<open>Derived truth conditions for ‘might’, p. 21\<close>
lemma (in counterfactuals) might_characterization:
......@@ -380,11 +398,95 @@ lemma (in counterfactuals) outer_strict_conditional:
shows \<open>\<lbrakk>\<phi>\<box>\<rightarrow>\<psi>\<rbrakk>w\<close>
using assms by auto
subsection \<open>Comparative Similarity\<close>
subsection \<open>Impossible antecedents and strength of counterfactuals (\<section> 1.6)\<close>
text \<open> (p. 26)\<close>
definition Might_weak :: \<open>'aa formula \<Rightarrow> 'aa formula \<Rightarrow> 'aa formula\<close> (\<open>_ \<diamond>\<Rightarrow> _\<close> 25)
where [simp]: \<open>\<phi> \<diamond>\<Rightarrow> \<psi> \<equiv> ((\<phi> \<diamond>\<rightarrow> \<phi>) \<rightarrow> (\<phi> \<diamond>\<rightarrow> \<psi>))\<close>
text \<open>Implicit dual definition (p. 25)\<close>
definition Would_strong :: \<open>'aa formula \<Rightarrow> 'aa formula \<Rightarrow> 'aa formula\<close> (\<open>_ \<box>\<Rightarrow> _\<close> 25)
where [simp]: \<open>\<phi> \<box>\<Rightarrow> \<psi> \<equiv> ~~(\<phi> \<diamond>\<Rightarrow> ~~\<psi>)\<close>
context counterfactuals
begin
lemma might_weak_truth_def:
\<open>(\<lbrakk>\<phi> \<diamond>\<Rightarrow> \<psi>\<rbrakk>w) =
(\<forall>s \<in> S w. (\<exists>w\<phi> \<in> s. \<lbrakk>\<phi>\<rbrakk>w\<phi>) \<longrightarrow> (\<exists>ws \<in> s. (\<lbrakk>\<phi>\<rbrakk>ws) \<and>\<lbrakk>\<psi>\<rbrakk>ws))\<close>
by auto
lemma would_strong_truth_def:
\<open>(\<lbrakk>\<phi> \<box>\<Rightarrow> \<psi>\<rbrakk>w) =
(\<exists>s \<in> S w. (\<exists>w\<phi> \<in> s. \<lbrakk>\<phi>\<rbrakk>w\<phi>) \<and> (\<forall>ws \<in> s. (\<lbrakk>\<phi>\<rbrakk>ws) \<longrightarrow> \<lbrakk>\<psi>\<rbrakk>ws))\<close>
by auto
text \<open> (p. 26)\<close>
lemma would_by_strong_def:
\<open>(\<lbrakk>\<phi> \<box>\<rightarrow> \<psi>\<rbrakk>w) =
(\<lbrakk>(\<phi> \<box>\<Rightarrow> \<phi>) \<rightarrow> (\<phi> \<box>\<Rightarrow> \<psi>)\<rbrakk>w)\<close>
by auto
end
subsection \<open>Actual antecedents (\<section> 1.7)\<close>
locale counterfactuals_centered = counterfactuals +
assumes centered_spheres: \<open>(\<forall>w. centered_spheres (S w) w)\<close>
begin
lemma conditional_collapse_for_actual_antecedent:
assumes \<open>\<lbrakk>\<phi>\<rbrakk>w\<close>
shows
\<open>(\<lbrakk>\<phi> \<box>\<rightarrow> \<psi>\<rbrakk>w) = \<lbrakk>\<psi>\<rbrakk>w\<close>
proof -
from assms have
\<open>\<exists>w\<phi> \<in> {w}. \<lbrakk>\<phi>\<rbrakk>w\<phi>\<close> \<open>(\<forall>ws \<in> {w}. (\<lbrakk>\<phi>\<rbrakk>ws) \<longrightarrow> \<lbrakk>\<psi>\<rbrakk>ws) = \<lbrakk>\<psi>\<rbrakk>w\<close>
by (blast+)
moreover have
\<open>{w} \<in> S w\<close>
using centered_spheres unfolding centered_spheres_def by blast
ultimately show ?thesis
unfolding is_true_at.simps
using singleton_iff subsetD psubsetD sphere_direction
by smt
qed
lemma conditional_collapse_for_actual_antecedent_general:
assumes \<open>\<lbrakk>\<phi>\<rbrakk>w\<close>
shows
\<open>((\<lbrakk>\<phi> \<box>\<Rightarrow> \<psi>\<rbrakk>w) = \<lbrakk>\<psi>\<rbrakk>w) \<and>
((\<lbrakk>\<phi> \<diamond>\<Rightarrow> \<psi>\<rbrakk>w) = \<lbrakk>\<psi>\<rbrakk>w) \<and>
((\<lbrakk>\<phi> \<diamond>\<rightarrow> \<psi>\<rbrakk>w) = \<lbrakk>\<psi>\<rbrakk>w) \<and>
((\<lbrakk>\<phi> \<rightarrow> \<psi>\<rbrakk>w) = \<lbrakk>\<psi>\<rbrakk>w)\<close>
unfolding Might_def Might_weak_def would_strong_truth_def
using conditional_collapse_for_actual_antecedent[OF assms] is_true_at.simps(1,3,4)
by smt
text \<open>“In short: counterfactuals with true antecedents reduce to material conditionals.” (p. 26)\<close>
lemma counterfactuals_are_material_conditional_for_actual_antecedent:
assumes \<open>\<lbrakk>\<phi>\<rbrakk>w\<close>
shows \<open>{\<lbrakk>\<phi> \<box>\<rightarrow> \<psi>\<rbrakk>w, \<lbrakk>\<phi> \<box>\<Rightarrow> \<psi>\<rbrakk>w, \<lbrakk>\<phi> \<diamond>\<Rightarrow> \<psi>\<rbrakk>w, \<lbrakk>\<phi> \<diamond>\<rightarrow> \<psi>\<rbrakk>w} = {\<lbrakk>\<phi>\<rightarrow>\<psi>\<rbrakk>w}\<close>
using assms conditional_collapse_for_actual_antecedent
conditional_collapse_for_actual_antecedent_general by auto
end
text \<open>\<open>weakly centered system of spheres\<close> with assumption (W) from page 29\<close>
locale counterfactuals_weakly_centered = counterfactuals +
assumes weakly_centered_spheres: \<open>\<forall>w. (\<forall>s \<in> (S w). s \<noteq> {} \<longrightarrow> w \<in> s) \<and> (\<exists>s \<in> (S w). s \<noteq> {})\<close>
begin
text \<open>“The world \<open>i\<close> itself is one of the closest worlds to \<open>i\<close>; but there may be others as well” (p. 29)\<close>
lemma innermost_sphere_contains_center:
\<open>w \<in> innermost_sphere w\<close>
using weakly_centered_spheres by blast
end
context counterfactuals
begin
subsection \<open>Comparative Similarity\<close>
\<comment>\<open>p. 48 Section 2.3 Comparative Similarity\<close>
abbreviation at_least_as_similar_as :: \<open>'world \<Rightarrow> 'world \<Rightarrow> 'world \<Rightarrow> bool\<close>
where \<open>at_least_as_similar_as w w1 w2 \<equiv> (\<forall> s \<in> S w. w2 \<in> s \<longrightarrow> w1 \<in> s)\<close>
......@@ -396,10 +498,9 @@ interpretation preorder \<open>at_least_as_similar_as w\<close> \<open>more_simi
end
locale counterfactuals_centered = counterfactuals S
locale counterfactuals_centered2 = counterfactuals S
for S :: \<open>'world \<Rightarrow> 'world set set\<close> +
fixes aw :: \<open>'world\<close>
assumes \<open>centered_spheres (S aw) aw\<close>
begin
abbreviation at_least_as_similar_as_c
......
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