Reduction and Elimination in Philosophy and the Sciences

On Game-theoretic Conceptualizations in Logic
Maciej Tadeusz Kłeczek, Nottingham, England, UK
Game-theory is a rich mathematical framework formalizing
real-life and intuitive concepts. It comes with a set of
slogans such as: winning, losing, dynamics, interaction,
process, choice. The basic ontology is that of players
acting according to certain definitiary rules of the relevant
game. How one reacts to the merging of game-theory and
logical concepts depends on one's philosophical
assumptions. Some philosophers of logic view with
suspicion the general anthropomorphic flavor and
procedural elements involved.
At least two different levels of analysis are present
in the literature on logic games. Certainly, games are
processes and can be described by process theories, such
as modal logic with some form of bisimulation as the
invariance relation 1 . However my concern in this paper is
more classical and focuses, after preliminary exposition of
paradigmatic logic games, on the interaction of properties
of semantic games with '∼'.
On the standard Tarskian account, truth in a
structure is understood as a certain abstract relation
holding between some particular structure and a formula
(relative to some assignment if the relevant formula is an
open formula). Truth and/or satisfaction conditions (1) M �
Φ[α] are provided in a compositional manner.
The game-theoretic account of truth in a structure is
given as follows:
178
(1') M � + Φ[α] if and only if there is a winning strategy
for the initial Verifier (called II) in a semantic
game G(M, Φ, α). Falsity is defined dually:
(2) M � − Φ[α] iff there is a winning strategy for an initial
Falsifier (called I) in a semantic game G(M, Φ, α).
The definitional rules of the game of semantic evaluation
are given as follows:
(1) If Φ is an atomic formula no action is taken. II
wins iff M � Φ; otherwise I wins.
(2) G(~Φ, M, α) — the game is played as on
G(Φ, M) except that the roles of the players are
transposed.
(3) G(ϕ1 ∧ ϕ2, M, α) — I makes the first choice of a
conjunct from Ω ∈ {1, 2}. The game continues with
the conjunct chosen.
(4) G(ϕ1 V ϕ2, M, α) — II makes the first choice of a
disjunct Ω ∈ {1, 2}. The game continues with the
disjunct chosen.
(5) G(∀xΦ, M, α) — I chooses the witness individual
a from |M|. The game continues
G(Φ, M, α ∪ {x, a}).
(6) G(∃xΦ, M, α) — I makes the first choice of individual
a from |M|. The game continues
G(Φ, M, α ∪ {x, a}).
Assuming the axiom of choice 2 (1) and (1') are equivalent.
Proof proceeds by induction on the complexity of a for-
1 Game trees can be seen as relational models. Lets M = and M' =