Epimenides (that sentence…)
(Logic and Language II)
Let
us take another look at Epimenides’ infamous sentence, usually glossed as (or
reduced to): ‘This sentence is false.’ The problem is that this sentence is
contradictory if read as self-referential; that is, it points not to another
sentence (in which case we check the referred-to sentence for truth or
falsity), but to itself (the deictic of ‘this’ is the sentence itself, so we
are asked to check the very sentence that contains ‘this’ for logical or
referential inconsistency). If at this stage we decide that the meaning is that
this sentence is not a ‘real’ sentence (transgressing syntactically,
semantically or logically), then we save ourselves some considerable headache.
If however we decide to follow through the implications raised by the ‘uncanny’
feel of this sentence then we quickly find ourselves in a spiral of
self-contradiction as one step, stage or level contradicts the ones before and
after… If we read the statement of falsity as true, take it at face value, then
it also expresses its truth as false; in which case it can not be true… but
then again… Truth here requires two levels of truth, both must be true for ‘it’
to be true, if one level is false, as is here the case, then the whole is read
as false (echoing the results of basic truth tables). In this case we must
accept that its referent (itself) if true then false (where the brackets just
employed indicate a different level or step) and so on ad infinitum (in terms of an axiomatic logic we have a
transgression of the taboo on self reference).
Otherwise
put (still taken self-referentially), it needs to be true to be false and vice
versa (false to be true). Only if false, (then) true, only if true, (then)
false. Two moments, two stages, statement and inference which when shown
together offer: A = not-A. Contradictory. Illogical. But is this always
catastrophic? What we have in ‘this sentence’ is a clash between statement and
inference, its statement (asserting the unreliability of that particular
sentence, which is fine) and its logical implication… (that ‘is false’, is
true, the predicate as applied back to itself) so between two propositions
(‘this sentence is true’ AND ‘this sentence is false’). The original
subject/predicate and their grammatical relation is unexceptionable (if
existentially empty), as is the procedure of inference. Could it be that there
are two levels involved such that we should not take exception as to the
implied contradiction (as in the case of the solution of the Zeno paradox of
the tortoise and Achilles where the confusion of two very different levels
caused the (apparent) contradiction)? Indeed we could again stop here; the
logical contradiction is after all the confirmation of the sentence’s
unreliability, which it has (honestly) expressed to us (its denial of itself as
a sentence). In other words the infinite regress of the sentence’s logical
inferences only performs what the sentence states; its illogicality, the ground
(if we might use such a term for the bottomless pit onto which it opens) of its
falsity. So if we take heed of its literal message we may call it a halt here.
We have been warned (and the sentence in question, in fact, expresses very
little else).
Should we wish to continue
we might imagine the sentence as described otherwise…’Only if false, (then)
true, only if true, (then) false’. This might be read as: Only if A, (then) B,
only if B, (then) A. Which shows the truth of the two levels in oscillation as
well as perhaps describing ‘things’, events or processes which show oscillation
(flashing lights, seasons, all manner of cyclic phenomena).
Further
pursuing the implications of the two levels, we might ask what if it can be
shown that it is true on one level and not-false on the other, as in the case
of a proposition versus a citation; a proposition and its own citation: read as
two levels of interpretation of the same (of the repetition of the same in
quotation marks), such that in the second case, the ‘citation’ the question of
truth and falsity do not arise – its ‘meaning‘ lies in its relation to its original
(in its quotation marks if you like). “This sentence is false, ‘this sentence
is false’ ”. The implied contradiction is swallowed, or neutralized by the
status of the second level as a citation. So we might suggest that if the
sentence is ‘true’ on these two levels, then the claim to falsity is no longer
self-referential, rather the levels are differentiated according to one being a
citation of the other - so not contradictory; a description of a state of
affairs. (If the cited sentence was read as a proposition it would return to
the alternation of true/false). Also, what does it mean to say ‘this sentence
is true’ (or indeed, ‘false’) in the absence of other material (reference,
context) to be checked for logical or another kind of value? True would be:
‘this sentence is empty’, logically inoffensive and meaningless. We are
returned to our original (and timely) dismissal.
Also the addition of
surrounding quotation marks (surrounding both sentence and quotation, such
that: “sentence A, ‘sentence A’ “), reminds us that the ‘original’ sentence is
not, of course, original at all, but has a history and an origin, that I am in
fact quoting this sentence from somewhere, in this case a received history in
philosophy and logic, and that by omitting the ‘first‘ set of quotation marks I
am up-rooting the sentence from its prior contexts – I will come back to this
soon.
One
solution using self-citation (Quine) offers, there is a relation such that
[‘a’, a.], = “‘This sentence is false’, this sentence is false” (the second
‘this’ points to the words in quotation marks). However, the deixis of ‘this’
is not so easily resolved. If the second half, or second phrase, is read as
referring to the first, then no problem (in this lies Quine’s ‘solution’): if
only to itself (the second) then redundant. If, however, referring to the whole
sentence (governed by the logic of the full stop), then the propositions are
invalidated. On the other hand, if the second refers to the whole sentence as
one citation and one proposition (as both false in their totality), then the
whole is also invalidated. (Again just as in truth tables, only one of the four
possibilities is ‘true’, here meaning, ‘makes sense’). Note further that Quine
must place the ‘original’ sentence second, giving the implication priority over
the sentence (now a phrase, that ‘originally’ incited it). Not so much a
solution, as a neutralization. One, furthermore, achieved by an inversion of
the logical order of the process. Punctuation is added to ‘solve’ a problem
(the use of quotation marks, abetted by an inversion on temporal order), but
returns (the sentence had become a phrase, a coma is not a full stop) to reopen
the wound… (Gödel anyone…).
Let
us look again at this last problem: … is a sentence quoted within another
sentence really still a sentence, and not ‘just’ a phrase? As a sentence cited,
but as included within another sentence, it must equal a phrase; therefore we
already have a transgression or confusion of levels… Is it this, and not the
‘deeper’ issue, that is the real (key to the) problem? Capital letters and full
stops, if we miss them, or if we loose the quotations marks, irreducible marks
of our thought path or operation, all of these change the material (changes not
only the context-related meaning, but perhaps also the nature of the original
sentence itself – if we may perceive these levels as meaningfully different… a
mark added may be read as a mark of context, not least citation marks…).
Part/whole problems re-echo those of element and set.
Moreover, insofar as ‘This
sentence is false’ refers to its sense or function as a sentence, then grammar,
word order, semantics and if all else fails, pragmatics, are the relevant
meta-languages for analysis…
By contrast, ‘This
proposition is false’ is a logical conundrum (requiring predicate or logical
analysis). Such that by referring to itself and not to any other verifiable
sentence, ‘This proposition is false’, simply rules itself out... according to
the axiom of avoiding contradictory self-reference. Furthermore we have the
choice of deciding whether it is now a false proposition (so still a
proposition) or not (no longer) a proposition… To classify it as no longer a
proposition certainly tidies up the problem. But what of the other alternative;
in deciding that the proposition remains a proposition (a logical relation) but
that it is of a particular kind that we call a false one (negation is a logical
relation) do we repeat the paradox, or do we in fact pose two levels (akin to
that of the solution to the Tortoise and Achilles paradox where a distinction,
or better, confusion of levels invalidates the argument)?So the two levels of
the proposition are to be read as to be kept separate: one is the membership of
the family of logical propositions in general; the second as a a member of a
sub-set of that family, a special kind of proposition marked by negation; with
the two levels not being exchangeable (a part-whole relation, which has been
confused in the fusion or exchange of levels). So the ‘truth’ lies in the
proposition’s existence as a proposition (or a sentence with propositional
value) and the falsity in the existence of its relation of negation. The
inference that its falsity must be read as true is a transgression of levels
(true and false exist on different levels). The crossing of levels allows the
descent into the infinite spiral.
Again,
put another way, there may also be a problem with the idea of an implied next
(or second) step or level, which infers: This sentence is to be read as if true
(we are to believe its claim to falsity). But does not the addition of ‘as if’
change things, we do not read as a ‘true sentence’, ‘this sentence is true’, as
‘as if’ it were so… There is a contextual implication such that we read
sentences ‘as if’ true or literal (to take it on face value until proven
otherwise, as figural or as false, the latter case usually meaning nonsense –
the easy, pre-emptive, solution). Moreover is there really any implication of a
logical relation in this implication? The implication means that we are
encouraged to take the sentence as a sense-making unit, not as a pre-existing
level with logical force, carrying the logical value of ‘true’ as defined by
its opposition to ‘false’ in logical relations. So the second ‘step’ that results
in the chain of contradiction may be a result of over-interpreting the
linguistic habit we have of imputing meaning to utterances according to context
(if we define the context as one foregrounding logical relations does this
legitimise the reading of this (pre) level as logically significant?).
Quine’s solution by
citation is the symbolic form of noting the inference that is an everyday code
or habit of language use.
Indeed our first step (or
how we define our first step) is all important. It is not so much a question of
two different levels: such that one = true; the other = false, and that their
unity is asserted (perhaps over and above any description of a process of
alternation as discussed above). But of what are in fact two kinds of starting
point, in which one implies the other, but in a significantly -and
recognisably- different way… Such that by reading the sentence as false we are already taking the step of
pre-supposing the truth of any
sentence we come across, a frame of reference or pre-existing catch-all as
all-encompassing prior habit: however if we read it as true then we must then immediately decide that it is false, and so on… So the first evokes a meta-frame and the
second the self-reference of the descent into the bottomless pit of infinite
regress… self-reference as reference inwards. What we appear to have is the
transgression of levels, the transgression of the two axioms that when
transgressed disable logic of its clarity; the two sources of ‘bad infinity’ ;
‘up’ and ‘down’ , ‘out’ and ’ in’; reference to a meta set and self-reference.
Where ever we start we are sent to the land of paradox. Yet there appear to be
two kinds of paradox (depending upon where we start…). Is this significant?
Only
one starting point is habitual, is ‘default’; the other is its alternative;
there is a question of priority, of directionality, indeed of temporality…but
of cause and effect…? Indeed the assertion of the meta-set does imply a nesting
process … or bad infinity, as we move to ‘higher’ levels. Is the contrary the
case? Indeed the arrow of time does appear to point ‘down’ (as in Chinese from
shang上 to xia下). The question is : is this directionality
reversible (‘ illusory’ as in certain kinds of physics) or irreversible as in
the entropy implied by the second law of thermodynamics? And does the arrow of
time itself have any connection with our basic, intuitive, sense of
temporality; our situation in past, present and future? The past or prior
(level) is implied in the present as memory; we might say that the present
implies a prior state such that certain non-present, or semi-present images are
posited as past (if not this then they are fantasy, or a projection forwards,
the future). The ‘after’ that comes after the present (level), the future, projection
or prediction, is also an implication based upon the past (the reverse seems
untenable, thus the dissymmetry of the two ‘directions’ or semi-presences that
point away from the present). Both of these other levels, or ‘directions’, of
course, may be designated as potentially infinite. An infinity which may be
temporal, reaching back or forwards exponentially, or it may be based upon the
accretion of levels. (The question of a temporal logic, of assuming that some
at least of our intuitions concerning logic come from our fundamental
experience of time, and so deserving of an examination, a ‘thought experiment’
creating a ‘temporal logic’, applicable, to be sure, to a narrow range of uses,
such a speculation will be explored in the following article, ‘0 to
Again,
if ‘S is true (Predicate)’ is no problem, why then is ‘S is false (Predicate)’
a problem? If we allow that the values, ‘true’ and ‘false’ are actually
applicable (which I suspect they are not - again, an ‘easy’ solution), then the
problem must lie in the combination of deixis (pointing or reference) and (or
‘as’) self-reference. True. But this applies equally to the positive or ‘true’
version of the sentence. And if ‘S is true’ is quoted by itself [’a’, a.] there
is no problem, the implication is actualised (again as with the ’false’
sentence, where the double negation may be read as disposing of the
contradiction). Or again we might simply say, ‘this sentence is true’, this
sentence is meaningless.
Also
(further) ‘this sentence is false/true’ and ‘this sentence as (read as if, or
taken as) true/false’, are not the same (as suggested above in the question of
linguistic, semantic or contextual implication). ‘As‘, as less rigorous than
‘is’, plays the role of saying, ‘not the same as’ only in the sense of ‘(not)
exactly the same’; the difference between something totally the same and that
partly the same, as in a set with many elements being identical and being
almost identical, for example in the case where it is imitated by a set with
one element less – the question being of course whether the single element’s
difference constituted a qualitative difference, significant difference, or
just an ‘accidental’, ‘non-essential’ difference, as in the curtailing of a
measurement after the decimal point… This is the realm of metaphor; degrees of
similarity, imitation, identity or isomorphism. The lack of identity (as we
have seen) between the two propositions (or the original proposition and its
implication) such that one may be said to resemble the other, suggests that we
have a proposition and its citation, this combination offers no problem (again
as we have seen). The metaphorical relation, or relation of similitude, which
is not (quite) identity (a citation is more than its original) so offers as solution,
as we read ‘as’ instead of ‘is’. The gap between a metaphorical ‘as’ and an
identical ‘is’ relation being the one between solution and problem
(contradiction).
In
numbers, perhaps there is no problem, minus one or plus one, both are ok (or
two minuses make a plus, all depending upon how we count, if the different
values are one and zero, not one read as its subtraction, or negation)! The
problem then appears to lie in the nature of propositional value in language.
Or what does false/wrong mean… not well-formed…empty of reference or semantic
content? Given the lack of the latter perhaps we should simply christen the
offending sentence as a non-sentence, semantically speaking, and ignore it.
The
original sentence by Epimenides, however, is ‘All Cretans are liars’. Which,
over and above a perceived lack of politeness, pejorative force, or prejudiced
over-generalization, begins to cause problems only when the fact that
Epimenides himself is Cretan comes to the fore. Whence the alternative title,
‘the Cretan Paradox.’
Adverbial.
‘All Cretans are liars’ (with the contextual knowledge that the subject of
enunciation is a Cretan, such that: and …’I am a Cretan’, is also to be
understood for the paradox to be operational). This sentence does not admit of
self-citation; that is it yields nothing new, no paradox or conflict of levels
results. One (decontexualising, that is pared-down) paraphrase of this is, ‘I
am a liar’, or better, ‘I am always a liar’, the adverb (‘always’ also standing
in for the logical ALL) is required to remove the escape clause of different
instances (‘everyone is sometimes a liar’, is an ALL that asserts as SOME that
is pragmatically true!). Thus ‘all Cretans are liars sometimes’ is (probably) true. Yet the assumption (which is all it
is) that we should imply an ALL here is passed without question. The assumption
is that we have an ALL and ALL-ways, instead of an ALL and SOME-times…
Other adverbial options
include: ‘I am lying’ (now!) clearly an, A=not A, type of paradox that results
in an infinite spiral or alternation of levels… (although in everyday speech
this sentence would be read referring to a prior sentence). To effect a clear
showing of levels (after Quine, for example), that is, through doubling, or the
showing of the citation next to the original (‘a’, a), requires further
paraphrase, that is… more jiggling! (How legitimate is this? To re-jiggle to
reach ‘essence’ or rhetorical paradox is one thing; but to do so to simplify
the solution – that is: to miss out the very things that do not fit your
preferred solution- this is another matter.)
So
bearing the force of adverbials in mind, let us return for a moment to our
original sentence (which was not ‘the’ original sentence, we remember, but an
‘angled’ striped-down paraphrase…). ‘This sentence is false…’ - we have (at
least) three major adverbial options (which are implied in various readings of
‘this sentence’) - ‘now’ or ‘sometimes’ or ‘always’. ‘Now’ also implies that it
is false now; but perhaps not all-ways (so perhaps having the meaning of
‘sometimes’). This sentence is false ‘always’, seems clear; until we remember
that the paradox requires a moment of truth for it to be self-contradictory (so
not strictly-speaking, un-problematically ‘false’.) Again it seems that we may have
a case of ‘sometimes’. In the case of the unequivocal adverbial implication,
‘sometimes’, we have a ‘sometimes’ true; but when is this ‘sometimes’? And is
it a clear ‘sometimes’ or just a case of operating on one level only (as in the
case above)? In the case where ‘it’ is ‘sometimes’ ‘not-false’, we have no
problem. What else might ‘sometimes’ false mean… (‘False’ when the
self-referential, and so contradictory, elements are fore-grounded, false when
the subject ‘this’ is itself and not another…). ‘Sometimes’ (the temporal,
adverbial, value that ‘returns’ in all cases) does rather seem to preempt, and
so describe, the alternation of levels that explication involves.
‘All
Cretans are liars.’ The sentence now has a time and a place, and a population;
in other words, a history, a geography and a culture. The sentence initially
‘quoted’ above is the logical problem distilled; decontextuallised. Logical
essence devoid of any existential or exterior reference. The added factors may
make a difference regarding contradiction as used in ordinary language use, as
resolvable by reference to cultural peculiarities (perhaps including a
reputation for the telling of tall tales) or discovery in the relevant context
of two levels or layers on to which the two contradictory terms may be
projected, divided, and so logically neutralised. Cultural specifics might well
include a form of challenge, with the paradox employed as a challenge to
others, a kind of, ‘look at me, I am cleverer than you!’ In this sense it is an
identity statement (or identity proposition) in which logic is deliberately
sacrificed to deliver a taunt/insult and an assertion of self-worth (and so an
implied social hierarchy). A sacrificial transaction, which supports an
identity exchange, which, in turn, powers the self-assertion. A piece of
rhetoric; a form of ritual.
However
it is on the logical level that the sentence yields more interest by posing a
sequence of questions about meaning, logic and language.
And the uses of self-reference,
which if logically unavoidable, so offering self-contradiction as a essential
(‘making a virtue of necessity’) flavour of our language, then providing the
means of enriching our possibilities of expression.
So in conclusion to return
(briefly) again to the de-contextualised or ‘stripped-down’ (or, if you like,
‘stage-managed’) version: ‘This sentence is false.’ (So placed at the end of a
sentence the two sets of full stops, cited and non-cited, that of the guest and
that of the host, fortuitously coincide – but this does not negate the status
of a part contained within a whole, a phrase within a sentence…)
Does
taking ‘it’ as a citation provide us with a solution? Certainly the
juxtaposition or inclusion of both phrases in one sentence presents the two
moments or levels that we intuit as the nature of the problem. A sentence is
preceded by its citation, such that: ’This sentence is false’ this sentence is
false. The previous sentence should, of course, in turn, be surrounded in
quotations marks. But… this sentence only works as a sentence if the first part
is a phrase – which it declares it isn’t… That is: it (the first phrase)
declares it is a sentence but (as a citation contained in a sentence) it can
only be a phrase. (“‘This sentence is false’…” has no full stop: there is no
full stop until AFTER the following phrase). So, we can pretend (‘it’s a matter
of orthography’ something which disappears with symbolic logic… or does it –
the sense of period remains on the level of proposition and the combination of
propositions into inference, if A then B); but literality (precision) may be
the key to a solution here; suggesting that to construct a paradox may be
harder than this example might make it appear…
Also ‘This sentence…’
cited, gives a subject to the original sentence, which as a phrase it may not
be…
There are other
subject-less, quoted sentences, such that: “’… is not a sentence’ is not a
sentence”. Which is true. (With a similar proviso as to the use of
speech/quotation marks for the whole thing) However, any ‘a something, is a
something’ needs a negative to make it ‘work’, to make it ‘interesting’ (and
again, as noted above, in losing its full stop it also loses its status as a
sentence). ”’… is not a cat’, is not a cat” is meaningful (in a negative sense)
but: “’…is an A’ is an A” and “’...is a two’ is a two” are meaningless without
further definition of ‘A’ or ‘two’ (with the addition of a negative they make
some sense…). Actually this type of structure needs self-reference to make it
interesting. “’…is a sentence‘, is a sentence” is a sentence only because the
quotation provides a subject (the citation is not a sentence, is not even a
phrase, is ill-formed… ungrammatical, incomplete).
‘This sentence…’ is not a
sentence because it is cited; as a citation as part of another sentence it
looses its status as sentence. Thus it is both true and false in a way Quine
probably did not intend: the two levels (true and untrue) are brought out and
shown (performed) by the lack of a full stop which characterizes the embedded
citation. As a citation it is indeed false (because not a sentence). If we read
‘ungrammatical’ (status due to presence or lack of a full stop) as false…
The point here is that it
is the contested nature or questionability of the subject position in the
sentence (such that there are two, first, the grammatical subject, ‘This
sentence...’, and then, second, the whole sentence itself) is what incites the
infinite regress that follows, or the oscillation between levels, such that
A=not A… (If we do not simply read the resulting infinity as a performance of
the literal meaning.)
So, in this reading, and
again as in the case of the ‘paradox’ of the tortoise and Achilles, it is the
collapsing of two distinct levels that causes the apparent paradox. Simply
citing the offending ‘phrase’ (so not a sentence anymore, a sentence cannot
contain a sentence…only a phrase, but what is to be cited is, or was, a
sentence) against itself does not solve the problem, but does point the way.
Yet how fecund this
‘illogical’ relation; how much it tells us about our thinking operations and
our making of meaning in language and logic…
Copyright Peter Nesteruk, 2012.