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Informal Formal Fuzzy A Semantics for Means-End Ascriptions Jesse Hughes Technical University of Eindhoven September 21, 2004 Jesse Hughes A Semantics for Means-End Ascriptions Informal Formal Fuzzy Outline Means and ends, informally 1

  1. Informal Norms in Knowledge Formal Initial analysis Fuzzy Some examples of functional ascriptions “The function of the heart is to pump blood.” “That switch mutes the television.” “The subroutine ensures that the user is authorized.” “The magician’s assistant is for distracting the audience.” We ascribe functions to biological stuff, artifacts, algorithms, personal roles. . . Jesse Hughes A Semantics for Means-End Ascriptions

  2. Informal Norms in Knowledge Formal Initial analysis Fuzzy Some examples of functional ascriptions “The function of the heart is to pump blood.” “That switch mutes the television.” “The subroutine ensures that the user is authorized.” “The magician’s assistant is for distracting the audience.” We ascribe functions to biological stuff, artifacts, algorithms, personal roles. . . Jesse Hughes A Semantics for Means-End Ascriptions

  3. Informal Norms in Knowledge Formal Initial analysis Fuzzy How functions relate to means and ends “That switch mutes the television.” ⇓ One can use the switch to mute the television. ⇓ There is an action involving the switch that will cause the television to be muted. Functions imply means-end relations. Step one: Provide a semantics for means-end relations. Jesse Hughes A Semantics for Means-End Ascriptions

  4. Informal Norms in Knowledge Formal Initial analysis Fuzzy How functions relate to means and ends “That switch mutes the television.” ⇓ One can use the switch to mute the television. ⇓ There is an action involving the switch that will cause the television to be muted. Functions imply means-end relations. Step one: Provide a semantics for means-end relations. Jesse Hughes A Semantics for Means-End Ascriptions

  5. Informal Norms in Knowledge Formal Initial analysis Fuzzy How functions relate to means and ends “That switch mutes the television.” ⇓ One can use the switch to mute the television. ⇓ There is an action involving the switch that will cause the television to be muted. Functions imply means-end relations. Step one: Provide a semantics for means-end relations. Jesse Hughes A Semantics for Means-End Ascriptions

  6. Informal Norms in Knowledge Formal Initial analysis Fuzzy How functions relate to means and ends “That switch mutes the television.” ⇓ One can use the switch to mute the television. ⇓ There is an action involving the switch that will cause the television to be muted. Functions imply means-end relations. Step one: Provide a semantics for means-end relations. Jesse Hughes A Semantics for Means-End Ascriptions

  7. Informal Norms in Knowledge Formal Initial analysis Fuzzy How functions relate to means and ends “That switch mutes the television.” ⇓ One can use the switch to mute the television. ⇓ There is an action involving the switch that will cause the television to be muted. Functions imply means-end relations. Step one: Provide a semantics for means-end relations. Jesse Hughes A Semantics for Means-End Ascriptions

  8. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  9. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  10. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  11. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  12. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  13. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? No. Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  14. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? No. Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  15. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? No. Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  16. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? No. Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  17. Informal Norms in Knowledge Formal Initial analysis Fuzzy Why use formal semantics? Why not? Formal semantics are artificial. Too simple or bloody complicated. Or both. You’ve convinced me. . . Can we go? No. Formal semantics provide precise claims. Consequences are clear, indisputable. Yield rules of inference and (importantly) . . . In our project description. Jesse Hughes A Semantics for Means-End Ascriptions

  18. Informal Norms in Knowledge Formal Initial analysis Fuzzy What is an end? a mean? An end is some desirable condition. A means is a way of making the end true. Means change things: means are actions . Some controversies. Ends-in-themselves? Objects as means? Jesse Hughes A Semantics for Means-End Ascriptions

  19. Informal Norms in Knowledge Formal Initial analysis Fuzzy What is an end? a mean? An end is some desirable condition. A means is a way of making the end true. Means change things: means are actions . Some controversies. Ends-in-themselves? Objects as means? Jesse Hughes A Semantics for Means-End Ascriptions

  20. Informal Norms in Knowledge Formal Initial analysis Fuzzy What is an end? a mean? An end is some desirable condition. A means is a way of making the end true. Means change things: means are actions . Some controversies. Ends-in-themselves? Objects as means? Jesse Hughes A Semantics for Means-End Ascriptions

  21. Informal Norms in Knowledge Formal Initial analysis Fuzzy What is an end? a mean? An end is some desirable condition. A means is a way of making the end true. Means change things: means are actions . Some controversies. Ends-in-themselves? Objects as means? Jesse Hughes A Semantics for Means-End Ascriptions

  22. Informal Norms in Knowledge Formal Initial analysis Fuzzy What is an end? a mean? An end is some desirable condition. A means is a way of making the end true. Means change things: means are actions . Some controversies. Ends-in-themselves? Objects as means? Jesse Hughes A Semantics for Means-End Ascriptions

  23. Informal Norms in Knowledge Formal Initial analysis Fuzzy What is an end? a mean? An end is some desirable condition. A means is a way of making the end true. Means change things: means are actions . Some controversies. Ends-in-themselves? Objects as means? Jesse Hughes A Semantics for Means-End Ascriptions

  24. Informal Norms in Knowledge Formal Initial analysis Fuzzy Possible worlds and making propositions come true Ends are propositions we want to make true. But actions don’t change the meaning of propositions. Think of a set of possible worlds. At each time, one world is the actual world. And at each world, every proposition is true or false. Jesse Hughes A Semantics for Means-End Ascriptions

  25. Informal Norms in Knowledge Formal Initial analysis Fuzzy Possible worlds and making propositions come true Ends are propositions we want to make true. But actions don’t change the meaning of propositions. Think of a set of possible worlds. At each time, one world is the actual world. And at each world, every proposition is true or false. Jesse Hughes A Semantics for Means-End Ascriptions

  26. Informal Norms in Knowledge Formal Initial analysis Fuzzy Possible worlds and making propositions come true Ends are propositions we want to make true. But actions don’t change the meaning of propositions. Think of a set of possible worlds. At each time, one world is the actual world. And at each world, every proposition is true or false. Jesse Hughes A Semantics for Means-End Ascriptions

  27. Informal Norms in Knowledge Formal Initial analysis Fuzzy Possible worlds and making propositions come true Ends are propositions we want to make true. But actions don’t change the meaning of propositions. Think of a set of possible worlds. At each time, one world is the actual world. And at each world, every proposition is true or false. Jesse Hughes A Semantics for Means-End Ascriptions

  28. Informal Norms in Knowledge Formal Initial analysis Fuzzy Possible worlds and making propositions come true Ends are propositions we want to make true. But actions don’t change the meaning of propositions. Think of a set of possible worlds. At each time, one world is the actual world. And at each world, every proposition is true or false. Jesse Hughes A Semantics for Means-End Ascriptions

  29. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Outline Means and ends, informally 1 Norms in Knowledge Initial analysis Means and ends, formally 2 Propositional Dynamic Logic Brown’s Logic of Ability Means and ends, fuzzily 3 Why go fuzzy? Fuzzy sets Jesse Hughes A Semantics for Means-End Ascriptions

  30. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy A simple example of possible worlds Our Universe A set of worlds involving a footrace and starter pistol. Two basic properties: Footrace started? Pistol loaded? Jesse Hughes A Semantics for Means-End Ascriptions

  31. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy A simple example of possible worlds Our Universe A set of worlds involving a footrace and starter pistol. Two basic properties: Footrace started? Started Pistol loaded? Jesse Hughes A Semantics for Means-End Ascriptions

  32. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy A simple example of possible worlds Our Universe A set of worlds involving a footrace and starter pistol. Two basic properties: Footrace started? Pistol loaded? Loaded Jesse Hughes A Semantics for Means-End Ascriptions

  33. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy A simple example of possible worlds Our Universe A set of worlds involving a footrace and starter pistol. Two basic properties: Footrace started? Started Pistol loaded? Loaded Jesse Hughes A Semantics for Means-End Ascriptions

  34. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy A simple example of possible worlds Our Universe Two basic actions: Loading the pistol Firing the pistol Started Loaded Load Jesse Hughes A Semantics for Means-End Ascriptions

  35. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy A simple example of possible worlds Our Universe Two basic actions: Loading the pistol Firing the pistol Started Loaded Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  36. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy A simple example of possible worlds Our Universe Two basic actions: Loading the pistol Firing the pistol Started Loaded Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  37. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy A simple example of possible worlds Our Universe Started Loaded In Saturn, loading and firing is Load Fire a means to starting the race. Jesse Hughes A Semantics for Means-End Ascriptions

  38. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy A simple example of possible worlds Our Universe Started Loaded In Saturn, loading and firing is Load Fire a means to starting the race. Jesse Hughes A Semantics for Means-End Ascriptions

  39. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe Started Loaded In Saturn, loading and firing is Load Fire a means to starting the race. Jesse Hughes A Semantics for Means-End Ascriptions

  40. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe Started Loaded In Saturn, loading and firing is Load Fire a means to starting the race. Jesse Hughes A Semantics for Means-End Ascriptions

  41. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Introducing the formal language PDL Propositional dynamic logic Examples Basic ingredients: A set act of actions A set prop of propositions act = { load , fire } Action constructions: prop = { Loaded , For building complex actions Started } Composition, iteration, choice, test Formula constructions: Boolean connectives [ m ] ϕ — strong dynamic operator � m � ϕ — weak dynamic operator Jesse Hughes A Semantics for Means-End Ascriptions

  42. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Introducing the formal language PDL Propositional dynamic logic Examples Basic ingredients: A set act of actions A set prop of propositions act = { load , fire } Action constructions: prop = { Loaded , For building complex actions Started } Composition, iteration, choice, test Formula constructions: Boolean connectives [ m ] ϕ — strong dynamic operator � m � ϕ — weak dynamic operator Jesse Hughes A Semantics for Means-End Ascriptions

  43. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Introducing the formal language PDL Propositional dynamic logic Examples Basic ingredients: A set act of actions A set prop of propositions act = { load , fire } Action constructions: prop = { Loaded , For building complex actions Started } Composition, iteration, choice, test Formula constructions: Boolean connectives [ m ] ϕ — strong dynamic operator � m � ϕ — weak dynamic operator Jesse Hughes A Semantics for Means-End Ascriptions

  44. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Introducing the formal language PDL Propositional dynamic logic Examples Basic ingredients: A set act of actions A set prop of propositions act = { load , fire } Action constructions: prop = { Loaded , For building complex actions Started } Composition, iteration, choice, test Formula constructions: Boolean connectives [ m ] ϕ — strong dynamic operator � m � ϕ — weak dynamic operator Jesse Hughes A Semantics for Means-End Ascriptions

  45. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Introducing the formal language PDL Propositional dynamic logic Examples Basic ingredients: A set act of actions A set prop of propositions act = { load , fire } Action constructions: prop = { Loaded , For building complex actions Started } Composition, iteration, choice, test Formula constructions: Boolean connectives [ m ] ϕ — strong dynamic operator � m � ϕ — weak dynamic operator Jesse Hughes A Semantics for Means-End Ascriptions

  46. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Introducing the formal language PDL Propositional dynamic logic Dynamic operators Basic ingredients: A set act of actions A set prop of propositions [ m ] ϕ : Action constructions: m will bring about ϕ . � m � ϕ : For building complex actions Composition, iteration, choice, m can bring about ϕ . test Formula constructions: Boolean connectives [ m ] ϕ — strong dynamic operator � m � ϕ — weak dynamic operator Jesse Hughes A Semantics for Means-End Ascriptions

  47. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Introducing the formal language PDL Propositional dynamic logic Dynamic operators Basic ingredients: A set act of actions A set prop of propositions [ m ] ϕ : Action constructions: m will bring about ϕ . � m � ϕ : For building complex actions Composition, iteration, choice, m can bring about ϕ . test Formula constructions: Boolean connectives [ m ] ϕ — strong dynamic operator � m � ϕ — weak dynamic operator Jesse Hughes A Semantics for Means-End Ascriptions

  48. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe [fire] Started fire will make Started true. Started True: Loaded False: Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  49. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe [fire] Started fire will make Started true. Started True: Loaded False: Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  50. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe [fire] Started fire will make Started true. Started True: Loaded False: Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  51. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe [fire] Started fire will make Started true. Started True: Loaded False: Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  52. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe [fire] Started fire will make Started true. Started True: Loaded False: Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  53. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe [fire] Started fire will make Started true. Started True: Loaded False: Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  54. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe � load � Loaded load can make Loaded true. Started True: Loaded False: But every world satisfies [load] Loaded ! Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  55. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe � load � Loaded load can make Loaded true. Started True: Loaded False: But every world satisfies [load] Loaded ! Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  56. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe � load � Loaded load can make Loaded true. Started True: Loaded False: But every world satisfies [load] Loaded ! Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  57. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Some PDL examples Our Universe � load � Loaded load can make Loaded true. Started True: Loaded False: But every world satisfies [load] Loaded ! Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  58. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy But the world isn’t deterministic, is it? Actions may have uncertain outcomes. Randomness Uncertain conditions Actions may require skill Malfunctioning artifacts Our models should support non-determinism. Jesse Hughes A Semantics for Means-End Ascriptions

  59. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy But the world isn’t deterministic, is it? Actions may have uncertain outcomes. Randomness Uncertain conditions Actions may require skill Malfunctioning artifacts Our models should support non-determinism. Jesse Hughes A Semantics for Means-End Ascriptions

  60. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy But the world isn’t deterministic, is it? Actions may have uncertain outcomes. Randomness Uncertain conditions Actions may require skill Malfunctioning artifacts Our models should support non-determinism. Jesse Hughes A Semantics for Means-End Ascriptions

  61. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy But the world isn’t deterministic, is it? Actions may have uncertain outcomes. Randomness Uncertain conditions Actions may require skill Malfunctioning artifacts Our models should support non-determinism. Jesse Hughes A Semantics for Means-End Ascriptions

  62. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy But the world isn’t deterministic, is it? Actions may have uncertain outcomes. Randomness Uncertain conditions Actions may require skill Malfunctioning artifacts Our models should support non-determinism. Jesse Hughes A Semantics for Means-End Ascriptions

  63. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy But the world isn’t deterministic, is it? Actions may have uncertain outcomes. Randomness Uncertain conditions Actions may require skill Malfunctioning artifacts Our models should support non-determinism. Jesse Hughes A Semantics for Means-End Ascriptions

  64. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Adding non-determinism to our model Our Universe The pistol has a weak spring. Sometimes, bullet doesn’t fire, world doesn’t change. Started � � � � � fire � = . Loaded , Actions take a world to a set of worlds! � fire � : W → PW Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  65. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Adding non-determinism to our model Our Universe The pistol has a weak spring. Sometimes, bullet doesn’t fire, world doesn’t change. Started � � � � � fire � = . Loaded , Actions take a world to a set of worlds! � fire � : W → PW Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  66. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Adding non-determinism to our model Our Universe The pistol has a weak spring. Sometimes, bullet doesn’t fire, world doesn’t change. Started � � � � � fire � = . Loaded , Actions take a world to a set of worlds! � fire � : W → PW Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  67. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Adding non-determinism to our model Our Universe The pistol has a weak spring. Sometimes, bullet doesn’t fire, world doesn’t change. Started � � � � � fire � = . Loaded , Actions take a world to a set of worlds! � fire � : W → PW Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  68. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Adding non-determinism to our model Our Universe The pistol has a weak spring. Sometimes, bullet doesn’t fire, world doesn’t change. Started � � � � � fire � = . Loaded , Actions take a world to a set of worlds! � fire � : W → PW Load Fire Jesse Hughes A Semantics for Means-End Ascriptions

  69. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Ability and modal logic: Kenny’s analysis Ability is closely related to Means-end ascriptions. Modal logic cannot represent ability (Kenny). 1 �| = ϕ → Can ϕ I hit the bull, but I am not able to hit the bull. 2 �| = Can ( ϕ ∨ ψ ) → (Can ϕ ∨ Can ψ ) I can hit bottom or top, but NOT (I can hit bottom -or- I can hit top). Jesse Hughes A Semantics for Means-End Ascriptions

  70. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Ability and modal logic: Kenny’s analysis Ability is closely related to Means-end ascriptions. Modal logic cannot represent ability (Kenny). 1 �| = ϕ → Can ϕ I hit the bull, but I am not able to hit the bull. 2 �| = Can ( ϕ ∨ ψ ) → (Can ϕ ∨ Can ψ ) I can hit bottom or top, but NOT (I can hit bottom -or- I can hit top). Jesse Hughes A Semantics for Means-End Ascriptions

  71. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Ability and modal logic: Kenny’s analysis Ability is closely related to Means-end ascriptions. Modal logic cannot represent ability (Kenny). 1 �| = ϕ → Can ϕ I hit the bull, but I am not able to hit the bull. 2 �| = Can ( ϕ ∨ ψ ) → (Can ϕ ∨ Can ψ ) I can hit bottom or top, but NOT (I can hit bottom -or- I can hit top). Jesse Hughes A Semantics for Means-End Ascriptions

  72. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Ability and modal logic: Kenny’s analysis Ability is closely related to Means-end ascriptions. Modal logic cannot represent ability (Kenny). 1 �| = ϕ → Can ϕ I hit the bull, but I am not able to hit the bull. 2 �| = Can ( ϕ ∨ ψ ) → (Can ϕ ∨ Can ψ ) I can hit bottom or top, but NOT (I can hit bottom -or- I can hit top). Jesse Hughes A Semantics for Means-End Ascriptions

  73. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Ability and modal logic: Kenny’s analysis Ability is closely related to Means-end ascriptions. Modal logic cannot represent ability (Kenny). 1 �| = ϕ → Can ϕ I hit the bull, but I am not able to hit the bull. 2 �| = Can ( ϕ ∨ ψ ) → (Can ϕ ∨ Can ψ ) I can hit bottom or top, but NOT (I can hit bottom -or- I can hit top). Jesse Hughes A Semantics for Means-End Ascriptions

  74. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Ability and modal logic: Kenny’s analysis Ability is closely related to Means-end ascriptions. Modal logic cannot represent ability (Kenny). 1 �| = ϕ → Can ϕ I hit the bull, but I am not able to hit the bull. 2 �| = Can ( ϕ ∨ ψ ) → (Can ϕ ∨ Can ψ ) I can hit bottom or top, but NOT (I can hit bottom -or- I can hit top). Jesse Hughes A Semantics for Means-End Ascriptions

  75. Informal Propositional Dynamic Logic Formal Brown’s Logic of Ability Fuzzy Ability and modal logic: Kenny’s analysis Ability is closely related to Means-end ascriptions. Modal logic cannot represent ability (Kenny). 1 �| = ϕ → Can ϕ I hit the bull, but I am not able to hit the bull. 2 �| = Can ( ϕ ∨ ψ ) → (Can ϕ ∨ Can ψ ) I can hit bottom or top, but NOT (I can hit bottom -or- I can hit top). (1) rules out strong modal logics. Jesse Hughes A Semantics for Means-End Ascriptions

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