P.F. Strawson proposed to translate “No F is G” as ~(∃x)(Fx & Gx) &
(∃x)Fx & (∃x)Gx. (Strawson, 1952, Introduction to Logical Theory,
London, Methuen, pp. 163-179.) Furthermore he proposed to take the term
(∃x)Fx & (∃x)Gx as a PRESUPPOSITION. That is if (∃x)Fx & (∃x)Gx is not
true then ~(∃x)(Fx & Gx) is neither true nor false. So for example the
sentence “All John's children are asleep” is ~(T v F) if John has no
children. How does this help with the Liar paradox? Consider
This sentence is not true (Y)
The argument usually goes as follows.
(1) Assume that Y is true
(2) Then what it says is the case
(3) It says it is not true
(4) Then it is not true
Result: (4) contradicts (1).
Let us now modify the argument and make the assumption (1) part of the
This true sentence is not true (Y')
If we go through the steps (1)-(4) again we find that the sentence is
not true, and therefore there is no such thing as “this true sentence”.
A presupposition of Y' is that its subject term “this true sentence” has
a referent. But it does not! Hence Y' is ~(T v F).
But then the sentence is not true, and what it says is the case!! Well,
in order to resolve the paradox being ~(T v F) is not sufficient - in
addition the sentence must be meaningless. But we in fact do know that Y
(and Y') is meaningless.
But Strawson developed his theory precisely because he thought that the
sentence “All John's children” are asleep” was perfectly meaningful. (We
agree.) He rejected the trichotomy T, F, meaningless. (We agree.)
What follows is my addition. The meaningless sentences are a proper
subset of the sentences that are ~(T v F). In particular when the
non-existence of the subject is necessary [e.g. ~(Ex)(x # x)] then the
sentence is meaningless, when the non-existence of the subject is
merely contingent [e.g. John has no children] then the sentence is
merely ~(T v F).
Gödel's sentence behaves the same way as Y':
~(Ex)(Ey)(Prf(x,y) & This(y)) (G)
where This() is satisfied only by the Gödel number of G. Let us substitute
~(Ex)(Prf(x,<#G#>) & This(<#G#>)) (J)
We observe that (Ex)(Prf(x,<#G#>) is a presupposition of (J), and it is
not true. But then J is not true [it is ~(T v F).] No sound system
should derive either it or G, although there is no reason why it could
Note that (J) and (K) are not the same sentences. The latter talks about
numbers, the former about unicorns. It says that no unicorn is the Gödel
It is possible to specify a semantics such that the latter is true, but
the former is ~(T v F).
If Jack says ‘What I am saying at this very moment is not true’, we can
successfully and truly assert that he did not utter a truth: ‘What Jack
said is not true’. But it is hardly conceivable that Jack’s utterance is
true by virtue of its success in attributing non-truth to itself.
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