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Disparities of Meaning for a Solution to the Liar
James Blackmon
2005
This paper reveals a consequence of a solution to the Liar Paradox given by
Eugene Mills in ‘A Simple Solution to the Liar’. [Mills 1998] The solution is first tested
on Liar pairs, sentences that, without any obvious self-reference, attribute truth value to
each other in a paradoxical way. Mills’ solution proves resilient in passing its test but not
without revealing an interesting disparity between the semantic value of a sentence token
and the semantic value it is systematically taken to have by competent users of the
language. I will briefly characterize this kind of disparity before presenting Mills’
solution and these consequences.
If there is a significant difference between a sentence’s semantically expressed
proposition and the proposition that competent speakers believe it to have for systematic
reasons, then we have a case of what I will call semantic disparity. A case of semantic
disparity may be seen as a special kind of case in which the semantically expressed
proposition and the pragmatically conveyed proposition diverge for a sentence. The
specifying feature is that competent speakers normally and for good reason take the
sentence in question to literally mean one thing when the sentence in fact means
something else.
For example, suppose that a news reporter, upon seeing Charles Lindbergh’s
famous airplane, utters the sentence, ‘The Spirit of St. Louis is a fine airplane’. The
Disparities of Meaning for a Solution to the Liar
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reporter, using the name ‘The Spirit of St. Louis’ in deference to Lindbergh’s dubbing,
intends to express a proposition that attributes the property of being a fine airplane to
Lindbergh’s airplane. Suppose the reporter writes this in a news story read by millions.
Surely there would be many competent speakers who also take the sentence to express a
proposition that attributes the property of being a fine airplane to Lindbergh’s airplane.
But, for all we know, it was the airplane’s propeller that Lindbergh dubbed ‘The Spirit of
St. Louis’, not the entire plane. He may have done this without any malice and without
ever telling anyone about it. Perhaps he never even realized this dubbing ceremony would
mislead others. If, according to some causal theory of names, the sentence semantically
expresses a proposition that attributes the property of being a fine airplane to the
propeller despite the fact that competent speakers (except Lindbergh) take the sentence to
semantically express the proposition that attributes that property to the airplane, then this
is an instance of semantic disparity on this causal theory of names.
On such a theory, the semantics fails to “track the information.” That is, the
proposition semantically expressed by a sentence can differ radically from the
information that competent speakers reasonably take the sentence to have. Sentences such
as ‘The Spirit of St. Louis could not be snuck out the museum’s back door’ and ‘The
Spirit of St. Louis needs a new propeller’ would obviously be systematically used to
successfully inform people about Lindbergh’s airplane but these sentences are,
nevertheless, taken on this causal theory to express falsehoods about a propeller.
1
Cases of semantic disparity should normally be taken as reasons to reject or
modify the theory that generated them. For, a semantical theory is supposed to explain
Disparities of Meaning for a Solution to the Liar
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how a sentence is used to communicate, i.e., how we systematically convey information
to each other. But in these odd cases, the semantic value and the information that is
systematically conveyed differ in a variety of ways. They have different objects, will
normally have different truth conditions, and may easily have different truth values.
Nevertheless, it is reasonable to tolerate rare cases of semantic disparity when the
theory that makes them possible appears to have overcome as costly and tenacious a
problem as the Liar Paradox.
2
This appears to be the case with the solution provided by
Eugene Mills. Let us first consider Mills’ solution.
Mills’ solution
According to Eugene Mills, the Liar sentence does not give rise to intractable
paradox at all. Consider a self-referential use of the sentence L below.
(L) This sentence is false.
Part of Mills’ thesis is that when L is used self-referentially
3
it is simply false due,
as he argues, to its expressing a proposition that entails a contradiction. Since any
proposition that entails a contradiction is false and since any sentence that expresses a
false proposition is derivatively false, L, being a sentence that expresses a proposition
that entails a contradiction, is simply false just as the sentence ‘I am a married bachelor’
is false.
1
I do not purport to be offering a counterexample to causal theories in general. These considerations only
show that this “naïve” causal theory of names is problematic and that any robust theory is better off
avoiding the above consequences.
2
Philosophers have proposed the denial of self-referential sentences, the denial of the Liar sentence’s
meaningfulness, and the denial of bivalence. I take this fact along with the fact that none of these proposals
has been widely accepted to indicate the extent to which the Liar Paradox is both costly and tenacious.
Disparities of Meaning for a Solution to the Liar
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It is tempting to object to Mills’ thesis before investigating his supporting
arguments. The objection would be that, unlike the sentence ‘I am a married bachelor’, L
says of itself that it is false. So, on this objection, from the assumptions that L is false and
that L says that it is false, we derive paradoxically that L is true. The inconstant nature of
L’s presumed falsity appears to be very different than that of the sentence ‘I am a married
bachelor’ which is steadfastly false. On this objection, the paradox has returned if it ever
left.
But Mills’ solution eludes this objection. True, L is false. Also true, if L is false
then any sentence which says and says only that L is false is a true sentence. But,
according to Mills, L is not such a sentence. Here is the second part of Mills’ thesis. L is,
on his view, a sentence which “says of the proposition it expresses not only that it is false
but also that it is true.” [204] L, then, is self-contradictory and for that L is false but
unable to consistently (and so, truthfully) say so. As Mills points out, L is no more
paradoxical than a sentence that expresses that grass both is and is not green. [203] So,
although every self-referential use of L both is false and admittedly does say that much, it
does not say only that much. Instead, it says in self-contradiction that it is both true and
false. The upshot is that L does not mean what we thought it meant for most of us who
thought it meant anything at all.
We have two prominent arguments in the essay: One for the conclusion that L is
false; another for the conclusion that the assumption of L’s falsity does not yield its truth
and so does not result in contradictory truth-value assignments. The first argument
identifies a truth value that L has; the second guards against paradox by ensuring that we
3
L may be used to refer to other sentences as in “Consider the sentence, ‘All beachcombers are wealthy’.
This sentence is false.”
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cannot derive any other truth value. Because my interest is in identifying some results of
Mills’ solution, I will not reproduce his argumentation for the premises and will only
make a few remarks along the way.
Mills explicitly argues for the thesis that L is false by virtue of its entailing a
contradiction from two premises. [202-3]
(1) Every proposition entails its own truth.
(2) The proposition expressed by a self-referential use of L entails its own falsity.
(3) The proposition expressed by a self-referential use of L entails a contradiction
and so is false.
(1) just says that, for any proposition p, p╞ that p is true. For (1) to be false, there
must be some proposition, p, such that it can be the case that p is true even though it is
not the case that that p is true is true. Instead of trying to imagine and defend against
arguments objecting to (1), I accept (1) at face value. For (2) to be false, either a
presumed self-referential use of L must fail to express a proposition or it expresses a
proposition that does not entail its own falsity.
Mills argues for the additional thesis that one cannot derive a contradiction from
the supposition of L’s falsity with what appears to be a strengthened version of the first
argument. This argument, being part of the exposition for the previously cited argument,
is not put explicitly. Its premises, however, are stated explicitly on the pages cited below
and they clearly function to support the argument’s conclusion. [203-5]
(4) Every proposition attributes truth to itself. [205]
(5) The proposition expressed by a self-referential use of L attributes falsity to
itself. [201]
(6) The proposition expressed by a self-referential use of L attributes truth and
falsity to itself.
Disparities of Meaning for a Solution to the Liar
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It is reasonable to expect persistent debate over (4) and I will not try to defend it
here. However, we must note (for reasons that will later become apparent) that Mills
finds support for (4) in an identity attribution which he accepts: for any proposition p, p is
identical to the proposition that p is true.
4
Such an identity would account for a thesis
Mills is independently committed to: that an agent’s believing p is a necessary condition
for that agent’s believing that p is true. The penultimate section, §5, of Mills’ essay is
devoted to providing support for (4), making use of the idea that propositions are
individuated by belief. [205-9] Mills has previously argued for (5) in §2. [198-202]
It is from (6) that Mills argues that the sentence L, when expressing a proposition
that attributes truth and falsity to itself, says of itself not simply that it is false but that it is
both true and false. [204] It follows that from the assumption that L is false we cannot use
a premise to the effect that L says it is false to soundly derive L’s truth since L does not
say only that it is false. Indeed, L is false but, despite appearances, it cannot manage to
consistently say so.
Liar pairs and a curious result
At the heart of Mills’ solution is the thesis that every sentence says of the
proposition that it expresses that it is true. This allows L to have a single stable truth
value and thus avoid paradox. But putative solutions to the Liar have shown a tendency to
fall prey to Liar heirs, sentences or propositions that are overlooked or inadvertently
permitted by the putative solution but nevertheless give rise to Liar-like semantic
paradox. Here we will consider one such case, a Liar pair, which is thought to generate
4
Mills takes care to point out that this does not commit him to a redundancy theory. See his footnote 7.
Disparities of Meaning for a Solution to the Liar
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paradox without explicit self-reference. We will see that although Mills’ account avoids
Liar-like paradox in this case, it has some consequences that cannot be ignored.
Consider the pair, a and b.
(a) b is true
(b) a is false
Traditionally, standard Liar pairs are supposed to work as follows. Assume the
truth of one and, according to its content (which concerns the truth value of the other),
derive the truth value of the other. From this derived truth value and the content of the
other, derive the truth value of the original. It has been thought that if there is no
assumption of truth value for any one of the sentences that does not by these means yield
a contradictory truth value for that sentence then this pair of sentences generates paradox.
In this case, we have four assumptions to consider: the truth of a, the falsity of a,
the truth of b, and the falsity of b. First, from the assumed truth of a, we derive that b is
true since that is what a is supposed to say. But b is supposed to say that a is false which
contradicts our assumption that a is true. Second, from the assumed falsity of a, we
derive that b is false since that is what must be the case if what a is supposed to say is
false. By similar reasoning, from the falsity of b, we derive that a is true. But this
contradicts our second assumption, which is that a is false. Third, from the assumed truth
of b, we derive that a is false and from this that b is false—a contradiction. And, fourth,
from the assumed falsity of b, we derive that a is true and from this that b is true—a
contradiction. If all possible assumptions about truth value yield contradictions, we have
a paradox. So, according to this line of thought, the Liar pair results in paradox.
Disparities of Meaning for a Solution to the Liar
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Now let us return to Mills and re-assess this pair while working under his thesis
that every sentence says of the proposition that it expresses that it is true. For this we will
have to be a little more exacting. Take it that ‘a’ and ‘b’ are names for the corresponding
tokens above.
5
Let Ta be the proposition that the proposition expressed by (a suitable use
of) a is true and Fa be the proposition that the proposition expressed by (a suitable use
of) a is false (and similarly for Tb, and Fb).
Then, for Mills, a expresses Tb & Ta. The first conjunct is, as I will call it, a’s
“explicit content”, that b is true. The second conjunct is granted by (4), Mill’s premise
that every proposition attributes truth to itself. For the same reasons, b expresses Fa &
Tb. From this it follows that a’s expressing Tb & Ta is equivalent to its expressing that
(Fa & Tb) is true & Ta. Indeed, the proposition Tb & Ta is identical to the proposition
that (Fa & Tb) is true & Ta.
According to the alleged identity we find in Mills’ §5 (that for all p, p is identical
with the proposition that p is true), the proposition that (Fa & Tb) is true is identical with
Fa & Tb. So that (Fa & Tb) is true & Ta can be simplified to (Fa & Tb) & Ta. But,
clearly now, this thing expressed by a entails the contradiction, ~Ta & Ta. Therefore, a is
false.
b, however, says just this. That is, b expresses Fa & Tb which is equivalent to
~(that (Fa & Tb) is true & Ta) & Tb. This is not a contradiction nor does one follow from
it. Nor does a paradox seem to result from the entailments that hold when a is false and b
is true. If so, then Mills’ account survives this Liar pair test which is failed by some
5
Mills has not explicitly provided an account of Liar sentences that use names to refer to themselves. His L
uses the embedded phrase, ‘This sentence’. I do not see that this matters. If necessary, we may let ‘a’ and
‘b’ correspond to the appropriate embedded phrases.
Disparities of Meaning for a Solution to the Liar
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putative solutions (those that exclude explicit self-reference or simply the self-attribution
of truth value). On Mill’s account, a is simply false and b is simply true.
Now for a curious result. Consider c, which just expresses one of the theses we
have come to hold concerning the duo, a and b.
(c) b is true
The token c is true even though it seems for all purposes type-identical to a which we
have seen is false. c’s truth follows from the fact that it expresses that b is true & Tc.
Imagine things from the point of view of a newcomer who does not know what
‘b’ names and has not spent too much time thinking about the Liar. Such a newcomer
would be confused to learn that a and c, though orthographically indistinguishable tokens
(also tokens of the same language and appropriately fixed in the same contexts), had
different truth values. Presumably this difference in truth value is due to the fact that a
and c manage to get different propositions expressed. But how so? These are tokens of
the same language and the context is fixed. The answer, then, must lie in the one
remaining relevant difference: a is referred to by b but c is not and, for this reason,
different propositions are expressed. Different propositions get expressed because one of
the tokens (the token a) has a name that b uses (in the relevant way) while the other (the
token c) does not. Not a paradox, but unexpected and perhaps problematic.
For suppose the newcomer, S, is no longer so naïve as to think that qualitatively
identical tokens of the same language and in a fixed context have the same truth value. S
happens upon an alley wall on which twenty tokens of the sentence type ‘b is true’ are
scrawled. Suppose S, has now learned what ‘b’ names and has good reason to believe
Disparities of Meaning for a Solution to the Liar
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that there are no other tokens like these twenty before him. Then S can no longer count
all these tokens as true (even if S knows that at least some of them are true) for S has
good reason to believe that one of these tokens is a which is truthfully said by b to be
false.
The curious result here is that some tokens, through no immediate or explicit self-
reference of their own, can fail to consistently say what is in fact the case on a Millsian
view. They can be robbed, as it were, of their ability to do so by “b-like” tokens, tokens
which manage to refer to them in the kind of contradiction-inducing way we have just
investigated.
First Student: “Master’s utterance after the gong rings is true.”
Second Student: “Yes. Master’s utterance after the gong rings is true.”
Gong rings.
Zen Master: “First Student’s utterance is false.”
On Mills’ account, First Student has produced a token that is soon robbed of its
ability to consistently say what is in fact the case by Zen Master’s use of a b-like token.
First Student’s utterance says of Zen Master’s utterance that it is true but, in light of Zen
Master’s utterance, we see that it additionally says of itself that it is both false and true—
a contradiction only to be tolerated by those who have transcended reason. But Second
Student’s utterance succeeds in expressing what is in fact the case: that Zen Master’s
utterance is true while First Student’s utterance is false and, of course, that Second
Student’s utterance is true.
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Conclusion
The case for semantic disparity should by now be clear. Some sentence tokens
under Mills’ solution fail to express what we would naturally (though perhaps naively)
take them to express. Furthermore, we can systematically use them to communicate
effectively even though in a “pure and strict” sense they do not mean what we would use
them to mean.
Normally, this disparity would be good reason to reject a semantic account.
However, this account appears to have resolved the Liar Paradox and some of its ilk and
it appears to have done so without resorting to anything like the previous more extreme
strategies. We are not asked to abandon bivalence or the meaningfulness of the
problematic sentences. Nor are we asked to rule out self-reference or to embrace truth-
value gaps, truth hierarchies, or dialetheism. In light of what Mills’ account appears to
have accomplished and preserved we may find an infrequent semantic disparity to be a
small price to pay.
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References
Mills, Eugene (1998) ‘A simple solution to the Liar’, Philosophical Studies 89: 197-212.