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?
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.