should the result be BE, then the row player would have known it, and he would have instead picked A rather than B
I think this thing that gets predictably falsified by accepted policy shouldn’t be called knowledge, it’s merely a meaningful proposition, in this case a false proposition.
It is indeed correct that “the result be BE” is a false proposition in the real world. In fact, this is the reason why they are called counterfactuals and why the subjunctive tense (“would have”) is used.
Nashian game theory is based on the indicative tense, for example common knowledge is all based on the indicative tense (A knows that B knows that A knows etc). Semantically, knowledge can be modelled with set inclusion in Kripke semantics: A knows P if the set of accessible worlds (i.e., compatible with A’s actual knowledge) is included in the set of possible worlds in which P is true (and we can canonically identify this set with P, i.e., conflate a logical proposition with the set of worlds in which it is true).
What is important to understand is that counterfactuals can be rigorously captured and anchored into the actual world. This has been researched in particular by Lewis and Stalnaker in the 1960s and 1970s. A statement such as “should the result be BE, then the row player would have known it,” is mathematically modelled as a counterfactual implication P>Q with P=”the result is BE”, Q=”the row player knows that the result is BE” where P and Q are predicates on all possible worlds (including counterfactual worlds) that can be understood as subsets of the set of all possible worlds.
Q=Krow(P) is itself a compound predicate, because it is a knowledge statement: Q is true in a specific world ω if the set of worlds accessible to the row player from ω is included in P.
Counterfactuals imply some sort of distance between possible worlds: the more different they are, the farther apart they are. Given P and Q, the predicate P>Q is defined as true in a world ω if, in the closest world to ω in which P is true, denoted fP(ω), Q is also true.
So all in all, the statement is formally modelled as the predicate P>Krow(P) being true in the actual world and is to be distinguished from the logical implication P⟹Krow(P), which is trivially true in the actual world if P is false. But the counterfactual implication P>Krow(P) on the other hand is not trivial and can potentially be false in the actual world even when P is false.
I hope it helps clarify! For more details, the 1972 book by Lewis “Counterfactuals” is a very interesting read.
Counterfactuals imply some sort of distance between possible worlds [...] predicate P>Q is defined as true in a world ω if, in the closest world to ω in which P is true, Q is also true
What does “closest world to ω in which P is true” mean? Is this still data extracted from a Kripke frame, a set of worlds plus accessibility, or does this need more data (“some sort of distance”)? What sort of distance is this, what if there are multiple worlds in P at the same distance from ω, possibly with different truth of Q?
Keeping to the example at hand, what are the possible worlds/counterfactuals here, just the (row, column) pairs? Their combination with possibly-false-in-that-world beliefs? Something else intractably informal that can’t be divided by equivalence for irrelevant distinctions to give an explicit description? What is the accessibility in the Kripke frame? What are the distances? Is “the result is BE” just the one-world proposition true in the world BE? If some of the assumptions in my questions are wrong (as I expect them to be), what are the worlds where “the result is BE” holds? What does it mean to enact row A in the situation where the result is BE (or believed to be BE)? Or is “he would have instead picked A rather than B” referring to something that shouldn’t be thought of as enactment?
It is indeed correct that “the result be BE” is a false proposition in the real world
(I meant it’s false in the world where it’s believed, where row A would be taken instead as a result of that belief, so that in fact in that world row A is taken rather than B, so that the belief that row B is taken in that world is false. I didn’t mean to imply that I’m talking about the real world.)
I think this thing that gets predictably falsified by accepted policy shouldn’t be called knowledge, it’s merely a meaningful proposition, in this case a false proposition.
Thank you for your comment, Vladimir_Nesov.
It is indeed correct that “the result be BE” is a false proposition in the real world. In fact, this is the reason why they are called counterfactuals and why the subjunctive tense (“would have”) is used.
Nashian game theory is based on the indicative tense, for example common knowledge is all based on the indicative tense (A knows that B knows that A knows etc). Semantically, knowledge can be modelled with set inclusion in Kripke semantics: A knows P if the set of accessible worlds (i.e., compatible with A’s actual knowledge) is included in the set of possible worlds in which P is true (and we can canonically identify this set with P, i.e., conflate a logical proposition with the set of worlds in which it is true).
What is important to understand is that counterfactuals can be rigorously captured and anchored into the actual world. This has been researched in particular by Lewis and Stalnaker in the 1960s and 1970s. A statement such as “should the result be BE, then the row player would have known it,” is mathematically modelled as a counterfactual implication P>Q with P=”the result is BE”, Q=”the row player knows that the result is BE” where P and Q are predicates on all possible worlds (including counterfactual worlds) that can be understood as subsets of the set of all possible worlds.
Q=Krow(P) is itself a compound predicate, because it is a knowledge statement: Q is true in a specific world ω if the set of worlds accessible to the row player from ω is included in P.
Counterfactuals imply some sort of distance between possible worlds: the more different they are, the farther apart they are. Given P and Q, the predicate P>Q is defined as true in a world ω if, in the closest world to ω in which P is true, denoted fP(ω), Q is also true.
So all in all, the statement is formally modelled as the predicate P>Krow(P) being true in the actual world and is to be distinguished from the logical implication P⟹Krow(P), which is trivially true in the actual world if P is false. But the counterfactual implication P>Krow(P) on the other hand is not trivial and can potentially be false in the actual world even when P is false.
I hope it helps clarify! For more details, the 1972 book by Lewis “Counterfactuals” is a very interesting read.
What does “closest world to ω in which P is true” mean? Is this still data extracted from a Kripke frame, a set of worlds plus accessibility, or does this need more data (“some sort of distance”)? What sort of distance is this, what if there are multiple worlds in P at the same distance from ω, possibly with different truth of Q?
Keeping to the example at hand, what are the possible worlds/counterfactuals here, just the (row, column) pairs? Their combination with possibly-false-in-that-world beliefs? Something else intractably informal that can’t be divided by equivalence for irrelevant distinctions to give an explicit description? What is the accessibility in the Kripke frame? What are the distances? Is “the result is BE” just the one-world proposition true in the world BE? If some of the assumptions in my questions are wrong (as I expect them to be), what are the worlds where “the result is BE” holds? What does it mean to enact row A in the situation where the result is BE (or believed to be BE)? Or is “he would have instead picked A rather than B” referring to something that shouldn’t be thought of as enactment?
(I meant it’s false in the world where it’s believed, where row A would be taken instead as a result of that belief, so that in fact in that world row A is taken rather than B, so that the belief that row B is taken in that world is false. I didn’t mean to imply that I’m talking about the real world.)