“The links between logic and games go back a long way. If one thinks of a debate as a kind of game, then Aristotle already made the connection; his writings about syllogism are closely intertwined with his study of the aims and rules of debating. Aristotle’s viewpoint survived into the common medieval name for logic: dialectics. In the mid twentieth century Charles Hamblin revived the link between dialogue and the rules of sound reasoning, soon after Paul Lorenzen had connected dialogue to constructive foundations of logic.” from the Stanford Encyclopedia of Philosophy on Logic and Games
Game Semantics
Usual presentation of game semantics of logic: we have a particular debate / dialogue game associated to a proposition between an Proponent and Opponent and Proponent tries to prove the proposition while the Opponent tries to refute it.
A winning strategy of the Proponent corresponds to a proof of the proposition. A winning strategy of the Opponent corresponds to a proof of the negation of the proposition.
It is often assumed that either the Proponent has a winning strategy in A or the Opponent has a winning strategy in A—a version of excluded middle. At this point our intuitionistic alarm bells should be ringing: we cant just deduce a proof of the negation from the absence of a proof of A. (Absence of evidence is not evidence of absence!)
We could have a situation that neither the Proponent or the Opponent has a winning strategy! In other words neither A or not A is derivable.
Countermodels
One way to substantiate this is by giving an explicit counter model C in which A respectively ¬A don’t hold.
Game-theoretically a counter model C should correspond to some sort of strategy! It is like an “interrogation” /attack strategy that defeats all putative winning strategies. A ‘defeating’ strategy or ‘scorched earth’-strategy if you’d like. A countermodel is an infinite strategy. Some work in this direction has already been done[1]. [2]
Dualities in Dialogue and Logic
This gives an additional symmetry in the system, a syntax-semantic duality distinct to the usual negation duality. In terms of proof turnstile we have the quadruple
⊢A meaning A is provable
⊢¬A meaning $¬A$ is provable
⊣A meaning A is not provable because there is a countermodel C where A doesn’t hold—i.e. classically ¬A is satisfiable.
⊣¬A meaning ¬A is not provable because there is a countermodel C where ¬A doesn’t hold—i.e. classically A is satisfiable.
Obligationes, Positio, Dubitatio
In the medieval Scholastic tradition of logic there were two distinct types of logic games (“Obligationes) - one in which the objective was to defend a proposition against an adversary (“Positio”) the other the objective was to defend the doubtfulness of a proposition (“Dubitatio”).[3]
Winning strategies in the former corresponds to proofs while winning (defeating!) strategies in the latter correspond to countermodels.
Destructive Criticism
If we think of argumentation theory / debate a counter model strategy is like “destructive criticism” it defeats attempts to buttress evidence for a claim but presents no viable alternative.
“The links between logic and games go back a long way. If one thinks of a debate as a kind of game, then Aristotle already made the connection; his writings about syllogism are closely intertwined with his study of the aims and rules of debating. Aristotle’s viewpoint survived into the common medieval name for logic: dialectics. In the mid twentieth century Charles Hamblin revived the link between dialogue and the rules of sound reasoning, soon after Paul Lorenzen had connected dialogue to constructive foundations of logic.” from the Stanford Encyclopedia of Philosophy on Logic and Games
Game Semantics
Usual presentation of game semantics of logic: we have a particular debate / dialogue game associated to a proposition between an Proponent and Opponent and Proponent tries to prove the proposition while the Opponent tries to refute it.
A winning strategy of the Proponent corresponds to a proof of the proposition. A winning strategy of the Opponent corresponds to a proof of the negation of the proposition.
It is often assumed that either the Proponent has a winning strategy in A or the Opponent has a winning strategy in A—a version of excluded middle. At this point our intuitionistic alarm bells should be ringing: we cant just deduce a proof of the negation from the absence of a proof of A. (Absence of evidence is not evidence of absence!)
We could have a situation that neither the Proponent or the Opponent has a winning strategy! In other words neither A or not A is derivable.
Countermodels
One way to substantiate this is by giving an explicit counter model C in which A respectively ¬A don’t hold.
Game-theoretically a counter model C should correspond to some sort of strategy! It is like an “interrogation” /attack strategy that defeats all putative winning strategies. A ‘defeating’ strategy or ‘scorched earth’-strategy if you’d like. A countermodel is an infinite strategy. Some work in this direction has already been done[1]. [2]
Dualities in Dialogue and Logic
This gives an additional symmetry in the system, a syntax-semantic duality distinct to the usual negation duality. In terms of proof turnstile we have the quadruple
⊢A meaning A is provable
⊢¬A meaning $¬A$ is provable
⊣A meaning A is not provable because there is a countermodel C where A doesn’t hold—i.e. classically ¬A is satisfiable.
⊣¬A meaning ¬A is not provable because there is a countermodel C where ¬A doesn’t hold—i.e. classically A is satisfiable.
Obligationes, Positio, Dubitatio
In the medieval Scholastic tradition of logic there were two distinct types of logic games (“Obligationes) - one in which the objective was to defend a proposition against an adversary (“Positio”) the other the objective was to defend the doubtfulness of a proposition (“Dubitatio”).[3]
Winning strategies in the former corresponds to proofs while winning (defeating!) strategies in the latter correspond to countermodels.
Destructive Criticism
If we think of argumentation theory / debate a counter model strategy is like “destructive criticism” it defeats attempts to buttress evidence for a claim but presents no viable alternative.
Ludics & completeness—https://arxiv.org/pdf/1011.1625.pdf
Model construction games, Chap 16 of Logic and Games van Benthem
Dubitatio games in medieval scholastic tradition, 4.3 of https://apcz.umk.pl/LLP/article/view/LLP.2012.020/778