A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Conceptual starting points An end is a condition to be realized. A means is a way of realizing d n e the condition. n a Think possible worlds! Think transitions! Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Conceptual starting points An end is a condition to be realized. A means is a way of realizing d n e the condition. n a Thus: an end is a formula; Think possible worlds! Think transitions! Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Conceptual starting points An end is a condition to be realized. A means is a way of realizing d n e the condition. n a Thus: an end is a formula; a means is an action; Think possible worlds! Think transitions! Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Conceptual starting points An end is a condition to be realized. A means is a way of realizing d n e the condition. n a Thus: an end is a formula; a means is an action; Propositional Dynamic Logic is Think possible worlds! a natural setting. Think transitions! Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL syntax Propositional Dynamic Logic is a logic of actions. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL syntax Propositional Dynamic Logic is a logic of actions. Basic types: a set act of actions , Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL syntax Propositional Dynamic Logic is a logic of actions. Basic types: a set act of actions , Closed under: sequential composition α ; β non-deterministic choice α ∪ β Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL syntax Propositional Dynamic Logic is a logic of actions. Basic types: a set act of actions , Closed under: sequential composition α ; β non-deterministic choice α ∪ β a set prop of propositions . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL syntax Propositional Dynamic Logic is a logic of actions. Basic types: a set act of actions , Closed under: sequential composition α ; β non-deterministic choice α ∪ β a set prop of propositions . Closed under: boolean connectives, dynamic operators [ α ] ϕ , � α � ϕ . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL syntax Propositional Dynamic Logic is a logic of actions. Basic types: a set act of actions , Closed under: sequential composition α ; β non-deterministic choice α ∪ β a set prop of propositions . Closed under: boolean connectives, dynamic operators [ α ] ϕ , � α � ϕ . Intuitions: [ α ] ϕ : after doing α , ϕ will hold. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL syntax Propositional Dynamic Logic is a logic of actions. Basic types: a set act of actions , Closed under: sequential composition α ; β non-deterministic choice α ∪ β a set prop of propositions . Closed under: boolean connectives, dynamic operators [ α ] ϕ , � α � ϕ . Intuitions: [ α ] ϕ : after doing α , ϕ will hold. � α � ϕ : after doing α , ϕ might hold. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL semantics Possible world semantics with transition systems for each action α . α α α α Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL semantics Possible world semantics with transition systems for each action α . α � w ′ means: α w α one can reach w ′ by doing α in w . α α Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL semantics Possible world semantics with transition systems for each action α . α � w ′ means: P α w α one can reach w ′ by doing α in w . α P α ] α [ � w ′ . w ′ | α w | = [ α ] ϕ iff ∀ w = ϕ . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations PDL semantics Possible world semantics with transition systems for each action α . � w ′ means: α w β Q one can reach w ′ by doing α in w . β Q � β β � � w ′ . w ′ | α w | = [ α ] ϕ iff ∀ w = ϕ . � w ′ . w ′ | α w | = � α � ϕ iff ∃ w = ϕ . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Weakly and strongly sufficient means A sufficient means is an action α that can realize one’s end ϕ . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Weakly and strongly sufficient means A sufficient means is an action α that can realize one’s end ϕ . Two interpretations: α α ϕ Weak: α might realize ϕ . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Weakly and strongly sufficient means A sufficient means is an action α that can realize one’s end ϕ . Two interpretations: α α α α ϕ ϕ Weak: α might realize ϕ . Strong: α will realize ϕ . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Weakly and strongly sufficient means A sufficient means is an action α that can realize one’s end ϕ . Two interpretations: α α α α ϕ ϕ Weak: α might realize ϕ . Strong: α will realize ϕ . w | = � α � ϕ w | = [ α ] ϕ ∧ � α �⊤ Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Weakly and strongly sufficient means A sufficient means is an action α that can realize one’s end ϕ . Two interpretations: α α α α ϕ ϕ Weak: α might realize ϕ . Strong: α will realize ϕ . w | = � α � ϕ w | = [ α ] ϕ ∧ � α �⊤ � �� � α can be done. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Weakly and strongly sufficient means A sufficient means is an action α that can realize one’s end ϕ . Two interpretations: α α α α ϕ ϕ Weak: α might realize ϕ . Strong: α will realize ϕ . w | = � α � ϕ w | = [ α ] ϕ ∧ � α �⊤ � �� � α can be done. Caveat: This definition omits relevance. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means-end relations Necessary means seem simpler in practical syllogisms. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means-end relations Necessary means seem simpler in practical syllogisms. The consequence of a necessary means seems well-motivated. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations von Wright’s example I want to make the hut habitable. Unless I heat the hut, it will not be habitable. Therefore I must heat the hut. Expression of an agent’s desire, A necessary means-end relation, Concludes in a necessary action. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means-end relations Necessary means seem simpler in practical syllogisms. The consequence of a necessary means seems well-motivated. But the semantics for necessary means are subtle. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means-end relations Necessary means seem simpler in practical syllogisms. The consequence of a necessary means seems well-motivated. But the semantics for necessary means are subtle. Necessary means (roughly): If α is a necessary means to ϕ , then ϕ can be realized and Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means-end relations Necessary means seem simpler in practical syllogisms. The consequence of a necessary means seems well-motivated. But the semantics for necessary means are subtle. Necessary means (roughly): If α is a necessary means to ϕ , then ϕ can be realized and any weakly sufficient means to ϕ involves doing α . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means and counterexamples Necessary means (roughly): If α is a necessary means to ϕ , then ϕ can be realized and any weakly sufficient means to ϕ involves doing α . Note: Necessary does not imply sufficient. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means and counterexamples Necessary means (roughly): If α is a necessary means to ϕ , then ϕ can be realized and any weakly sufficient means to ϕ involves doing α . Note: Necessary does not imply sufficient. Necessary does not mean immediately necessary. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means and counterexamples Necessary means (roughly): If α is a necessary means to ϕ , then ϕ can be realized and any weakly sufficient means to ϕ involves doing α . Note: Necessary does not imply sufficient. Necessary does not mean immediately necessary. Key unanalyzed term: “involves” Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Involvement Write β � α for: β involves α . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Involvement Write β � α for: β involves α . Loosely: β � α means by doing β , one also “does” α . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Involvement Write β � α for: β involves α . Loosely: β � α means by doing β , one also “does” α . If β � α , then the sufficiency of β does not refute the necessity of α . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Involvement Write β � α for: β involves α . Loosely: β � α means by doing β , one also “does” α . If β � α , then the sufficiency of β does not refute the necessity of α . Basic properties: � is a pre-order. Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Involvement Write β � α for: β involves α . Loosely: β � α means by doing β , one also “does” α . If β � α , then the sufficiency of β does not refute the necessity of α . Basic properties: � is a pre-order. Non-deterministic choice ∪ is the join for � . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Involvement Write β � α for: β involves α . Loosely: β � α means by doing β , one also “does” α . If β � α , then the sufficiency of β does not refute the necessity of α . Basic properties: � is a pre-order. Non-deterministic choice ∪ is the join for � . If β � α , then β ; γ � α ; γ and γ ; β � γ ; α . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Involvement Write β � α for: β involves α . Loosely: β � α means by doing β , one also “does” α . If β � α , then the sufficiency of β does not refute the necessity of α . Basic properties: � is a pre-order. Non-deterministic choice ∪ is the join for � . If β � α , then β ; γ � α ; γ and γ ; β � γ ; α . α ; β � α and α ; β � β . Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means: summarized α is a necessary means to ϕ in w iff ϕ is attainable in w ; Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means: summarized α is a necessary means to ϕ in w iff ϕ is attainable in w ; there is no β such that w | = � β � ϕ , Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means: summarized α is a necessary means to ϕ in w iff ϕ is attainable in w ; there is no β such that w | = � β � ϕ , β � � α and Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means: summarized α is a necessary means to ϕ in w iff ϕ is attainable in w ; there is no β such that w | = � β � ϕ , β � � α and β is ∪ -free Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means: summarized α is a necessary means to ϕ in w iff ϕ is attainable in w ; there is no β such that w | = � β � ϕ , β � � α and (Annoying technical detail) β is ∪ -free Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means: summarized α is a necessary means to ϕ in w iff ϕ is attainable in w ; there is no β such that w | = � β � ϕ , β � � α and (Annoying technical detail) β is ∪ -free Thus, α is necessary iff ϕ is attainable and Hughes A Semantics for Means-End Relations
A brief overview of PDL Means-end relations in PDL Sufficient means-end relations Additional topics Necessary means-end relations Necessary means: summarized α is a necessary means to ϕ in w iff ϕ is attainable in w ; there is no β such that w | = � β � ϕ , β � � α and (Annoying technical detail) β is ∪ -free Thus, α is necessary iff ϕ is attainable and any ( ∪ -free) weakly sufficient means to ϕ involves α . Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Outline Means-end relations in PDL 1 A brief overview of PDL Sufficient means-end relations Necessary means-end relations Additional topics 2 Objects as means Conditional means-end relations Efficacy and fuzzy PDL Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Objects as means A bottle-opener is a means to liquid refreshment. Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Objects as means A bottle-opener is a means to liquid refreshment. But means are actions! Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Objects as means A bottle-opener is a means to liquid refreshment. But means are actions! How to represent objects-as-means in PDL? Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Objects as means A bottle-opener is a means to liquid refreshment. But means are actions! How to represent objects-as-means in PDL? Step 1: Introduce actions “use o ”. Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Objects as means A bottle-opener is a means to liquid refreshment. But means are actions! How to represent objects-as-means in PDL? Step 1: Introduce actions “use o ”. Problem: Keys lock and unlock doors. Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Objects as means A bottle-opener is a means to liquid refreshment. But means are actions! How to represent objects-as-means in PDL? Step 1: Introduce actions “use o ”. Problem: Keys lock and unlock doors. In PDL: [ α ] ϕ ∧ [ α ] ¬ ϕ implies [ α ]( ϕ ∧ ¬ ϕ ). Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Objects as means A bottle-opener is a means to liquid refreshment. But means are actions! How to represent objects-as-means in PDL? Step 1: Introduce actions “use o ”. Problem: Keys lock and unlock doors. In PDL: [ α ] ϕ ∧ [ α ] ¬ ϕ implies [ α ]( ϕ ∧ ¬ ϕ ). Step 2: Move to minimal models. Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Objects as means A bottle-opener is a means to liquid refreshment. But means are actions! How to represent objects-as-means in PDL? Step 1: Introduce actions “use o ”. Problem: Keys lock and unlock doors. In PDL: [ α ] ϕ ∧ [ α ] ¬ ϕ implies [ α ]( ϕ ∧ ¬ ϕ ). Step 2: Move to minimal models. Give up distributivity. Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Objects as means A bottle-opener is a means to liquid refreshment. But means are actions! How to represent objects-as-means in PDL? Step 1: Introduce actions “use o ”. Problem: Keys lock and unlock doors. In PDL: [ α ] ϕ ∧ [ α ] ¬ ϕ implies [ α ]( ϕ ∧ ¬ ϕ ). Step 2: Move to minimal models. Give up distributivity. Gain richer sense of “using” objects. Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL PDL means-end relations are local relations Our definition In w , m is a means to ϕ iff w | = [ m ] ϕ & � m � True . This is a very narrow sense of means-end relation. Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL PDL means-end relations are local relations Our definition In w , m is a means to ϕ iff w | = [ m ] ϕ & � m � True . This is a very narrow sense of means-end relation. Example “Riding the train is a means to reaching Delft.” Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL PDL means-end relations are local relations Our definition In w , m is a means to ϕ iff w | = [ m ] ϕ & � m � True . This is a very narrow sense of means-end relation. Example “Riding the train is a means to reaching Delft.” Do we mean this is true just in this world? Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL PDL means-end relations are local relations Our definition In w , m is a means to ϕ iff w | = [ m ] ϕ & � m � True . This is a very narrow sense of means-end relation. Example “Riding the train is a means to reaching Delft.” Do we mean this is true just in this world? every world? Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL PDL means-end relations are local relations Our definition In w , m is a means to ϕ iff w | = [ m ] ϕ & � m � True . This is a very narrow sense of means-end relation. Example “Riding the train is a means to reaching Delft.” Do we mean this is true just in this world? every world? every world in which we are in Eindhoven? Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL PDL means-end relations are local relations Our definition In w , m is a means to ϕ iff w | = [ m ] ϕ & � m � True . This is a very narrow sense of means-end relation. Example “Riding the train is a means to reaching Delft.” Do we mean this is true just in this world? every world? every world in which we are in Eindhoven? every “normal” world in which we are in Eindhoven? Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Natural means-end relations are conditional Example “Riding the train is a means to reaching Delft.” Natural means-end relations: Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Natural means-end relations are conditional Example “Riding the train is a means to reaching Delft.” Natural means-end relations: are not local more general than just this world Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Natural means-end relations are conditional Example “Riding the train is a means to reaching Delft.” Natural means-end relations: are not local more general than just this world are not global doesn’t express relation about every world Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Natural means-end relations are conditional Example “Riding the train is a means to reaching Delft.” Natural means-end relations: are not local more general than just this world are not global doesn’t express relation about every world are defeasible relation is about normal expectations Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Natural means-end relations are conditional Example “Riding the train is a means to reaching Delft.” Natural means-end relations: are not local more general than just this world are not global doesn’t express relation about every world are defeasible relation is about normal expectations sometimes include preconditions Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Natural means-end relations are conditional Example “Riding the train is a means to reaching Delft.” Natural means-end relations: are not local more general than just this world are not global doesn’t express relation about every world are defeasible relation is about normal expectations sometimes include preconditions Solution: add a non-monotonic conditional operator to PDL. Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Means distinguished by efficacy Different means to a common end have different degrees of reliability. Hughes A Semantics for Means-End Relations
Objects as means Means-end relations in PDL Conditional means-end relations Additional topics Efficacy and fuzzy PDL Means distinguished by efficacy Different means to a common end have different degrees of reliability. End: Get 12 points with one dart. Hughes A Semantics for Means-End Relations
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