Reduction and Elimination in Philosophy and the Sciences

A with distinguished subset Z of terminal histories; (c) P is
a function assigning a player to the elements of H\Z. A
strategy is a function from the preimage of P to a set A. A
winning strategy is a strategy, which leads to a win for a
player irrespective of the opponent’s strategy.
As was mentioned above, game-theoretic characteristics
are equivalent to standard ones. This fact certainly
makes them adequate. However, it induces doubt about
whether game-theoretic conceptualizations introduce any
genuine content into logical theory. The choice of technique
seems to be arbitrary.
This explains the somewhat pessimistic but instructive
remark found in Johan van Benthem's paper: “... all
these games are useful didactic and heuristic tools in this
area, but no significant logical results or insights so far
appear to rest on them exclusively. But again this may
reflect the poverty of the notion of game involved so far in
current literature”. [van Benthem, 1990]
It all depends on what kinds of results are expected.
The games outlined above vindicate what can be called a
strategic viewpoint in logical theory. Consider two graphs
G = and G' = , where V is an
irreflexive and antisymetric relation. Obviously, those
graph structures are non-isomorphic. There is no bijection
between them. Thus I has a winning strategy in game
EFn(G, G'). Actually, I has two winning strategies. He can
force a winning outcome in two or three rounds of
EFn(G, G'). We find an analogous case in semantic games.
It often happens that II or I are in possession (obviously
not simultaneously) of more than one winning strategy.
Hence, winning strategies are more fine-grained semantic
values than truth-values. Although those facts can be
hardly regarded as striking results.
The situation changes when the assumption of perfect
information is abandoned and lack of information
about the opponent’s previous moves is allowed. A winning
strategy is no longer built up on all the possible
moves of the opponent.
A game of imperfect information is the following tuple:
G = . The new element is a set Ii, i∈N
whose elements are called information sets. Information
sets induce partitions (equivalence relation) on a set of
histories such that for all h, h'∈I∈Ι1, A(h) = A(h'), where
A(h) denotes the set of possible actions after history h.
Histories belonging to particular information sets occupy
the same depth on a game tree.
The difference between the semantic game of perfect
as opposed to imperfect information can be seen by
two styles of skolemizing branching-quantifier formulas: (1)
∃f∃gϕ(x, f(x), z, g(x, z)) (2) ∃f∃gϕ(x, f(x), z, g(z)). It is assumed
that Skolem functions correspond to winning
strategies for II. All of this was made explicit by Jaakko
Hintikka who introduced IF FO logic ( being a proper extension
of standard FO logic) where the failure of perfect
information is syntactically expressed by the new element
'/' i.e. (∀x)(∃y/∀x). Allowing imperfect information in a process
of semantic evaluation provides a formalisation of the
idea of dependence/independence relations holding between
quantifiers and attached variables.
Formulas involving slashes introduce equivalence a
relation E(Φ, M) on a particular structure M. Those equivalence
relations correspond to information sets. Whenever
some elements or sequences 3 of elements belong to such
3 Finite ordered sequence is a function f from the finite set of natural numbers
to some finite set A.
On Game-theoretic Conceptualizations in Logic — Maciej Tadeusz Kłeczek
such an equivalence class it is required that a strategy
function f satisfies a uniformity condition: i.e. E(a, b) →
f(a) = f(b).
Determinacy does not survive in this setting. It is
possible that neither player is in possession of a uniform
winning strategy when he is deprived of knowledge about
all previous moves of his opponent. Thus, the inference
from the lack of a winning strategy for one player to the
existence of a winning strategy for the other is no longer
valid. Despite the fact that the game rule for '∼' is the same
as it is in the classical case.
The logic of imperfect information is not closed under
contradictory negation. Consequently the Law of Excluded
Middle does not hold in IF logic and truth-valueless
sentences are allowed. This enables a distinction between
false/ non-true and non-false / true. Game-theoretic negation
under imperfect information satisfies the Strong
Kleene evaluation schema: (a)∼(t) = f (b)∼(f) = t (c)∼(u) = u.
More exactly LEM fails in non-FO fragment of IF logic. A
canonical example of a truth-valueless sentence is:
∀x(∃y/∀x) x ≠ y, where |M| ≥ 2.
The expressive strength of IF logic, which is equal to
the existential fragment of second order logic, opens the
possibility of truth definitions for IF language in the signature
of Peano's arithmetic in the language itself. Gametheoretic
non-determinancy guarantees that the Liar Paradox
will not arise. 4 Applying the standard procedure of
Godel numbering and the Fixed Point Theorem to the Liar
sentence λ we obtain the paradoxical looking sentence
∼Tr[λ] where neither I nor II is in possession of a winning
strategy.
Moreover, IF logic is strictly more expressive than
FO logic 5 . However it validates (a) Downward Skolem-
Lowenheim (Let M be an L-model of infinite cardinality k
and let k' be a cardinal such that k > k' ≥ |L|. Then M has
an elementary submodel of cardinality k') (b) Compactness
(Let W be a set of IF formulas. If every finite subset of W
has a model, then the entire set W has a model). Thus
according to the Lindstrom Theorem its expressive
strength should not exceed those of FO logic. But the
crucial assumption of the Lindstrom theorem is that the
logics he considers are closed under contradictory
negation. Here game-theoric conceptualization shows
once again its force.
In this paper the employment of game-theoretic
apparatus in the core areas of logical theoretizing was
presented. Certainly, the fact that the existence of a
winning strategy corresponds to an assertion that such
and such logical property holds is surprising. At first blush
those concepts might be regarded as categorically
different. All of this introduces an entirely new
methodological perspective, which can, should and is
pursued in a systematic way.
As emphasized earlier, varying purely gametheoretic
properties of a relevant game (affecting the
conditions a strategy to be a winning strategy) entails
change of meaning of the most important logical constant:
negation. As it is self-evident this result is far from trivial
and hardly can be seen as some form of heuristics and/or
didactic enterprise.
4 As it is rightly emphasized by Hintikka the problem of Liar Antinomy does
not concern truth predicate in isolation but the truth predicate in interaction
with negation. Consider Truth-Teller sentence: “(1) This sentence is true”. It
does not give rise to any kind of paradoxical conclusion.
5 Two logics L1 and L2 are equal in the expressive strength if every class of
models definable in L1 is definable in L2, and vice versa.
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