Denying the Liar
The Pennsylvania State University
paradox is standardly supposed to arise from three conditions: classical
bivalent truth value semantics, the Tarskian truth schema, and the formal
constructability of a sentence that says of itself that it is not true.
Standard solutions to the paradox, beginning most notably again with
Tarski, try to forestall the paradox by rejecting or weakening one or more
of these three conditions. It is argued that all efforts to avoid the
liar paradox by watering down any of the three assumptions suffers serious
disadvantages that are at least as undesirable as the liar paradox
itself. Instead, a new solution is proposed that admits that if the liar
sentence is true then it is false, in the first paradox dilemma horn, but
denies that the liar sentence is true, but admits instead that it is
false, and refutes the second paradox dilemma horn according to which it
is supposed to follow that if the liar sentence is false then it is true.
The reasoning for the second paradox dilemma horn is flawed, and that it
is not only not supported by but actually contradicted by the Tarskian
truth schema. We could only infer the second dilemma horn if it were to
classically follow from the assumption that the liar sentence is false and
from the three liar paradox conditions that therefore it is false that the
liar sentence is false. This entire sentence can be shown to be false on
the basis of the standard truth schema, thus blocking the paradox.
Alternative formulations of the liar sentence are discussed, and the
formal proofs and counterproofs for the two paradox dilemma horns, along
with the further philosophical implications of maintaining a resolute
stance that the liar sentence is simply false are considered.