Act like you're dancing

The other day, when we were having a fancy dress disco ball with the family, my son interrupted his crazy dance moves to say, "Hey dad, here's another paradox for you: 'act like you're dancing!'". Spot on: you can't pretend to be dancing without, in effect, being engaged in an act of dancing. That would only be possible if you kept perfectly still, but then, that wouldn't count as pretending to be dancing.

Phrasal verbs in the time-away pattern

Remember the verb-particle paradox, which I wrote about last time? To remind you, this paradox concerns combinations such as turn NP [e.g. a paper; a criminal] in, which must be words since they have a special meaning and can (if their meaning is appropriate, of course) occur in patterns where only single words are found (e.g. (1)), but which can't be words, since its parts can appear separated from each-other, as in (2)

(1) 'My gran says that's rubbish,' piped up Neville. (in a Harry Potter novel) (compare '...,' interrupted Neville, which is okay, but *'...,' spoke suddenly Neville, which isn't)

(2) I gave the book back. (compare: *I re- the book turned or *I turned the book re-)

Adele Goldberg, in an article to appear, argues against the word-view of verb-particle combinations on the grounds that we can't find them in the time-away construction:

(3) He vomited the night away.

(4) *He threw up the night away. (Goldberg's judgement) 

I'm not a native speaker of English, but I wonder whether (4) is all that bad. Here are some authentic examples reaped from the world wide web (and checked, as far as possible, for reliability, i.e., the source or user aren't obviously non-native):

(5) Do you sleep in the day away?

(6) soak up the night away in the hot tub

(7) red neck drunks bash up the night away 

(8) Me and my two friends dance off the night away. 

(9) we hung out the night away with greats such as Dave Holland, Max Roach, Roy Haynes, Roy Hargrove, Kenny Garrett, and one of our main piano heroes, Herbie Hancock. 

(10) Designed by Alberto Frias, it's basically a giant egg with a hole where you can crawl up in fetal position and chill out the day away. 

(11) It was great just chilling out the whole weekend away 

(12) We decided to just laze around the morning away before departing for the airport 

(13) Motel Cowboys jerking off The Night Away [apologies for the racy nature of this example]

So, an argument Goldberg uses against a possible morphological analysis of particle verbs can perhaps, if the above examples aren't completely suspect, be turned into an argument for it.

the verb-particle paradox

I'm working hard on an introduction to a text dealing with the verb-particle paradox. On the one hand, particle verbs such as call up behave like single words. There are lots of very solid arguments for a morphological analysis, so that saying that call up is a verb of the form [V V P] makes perfect sense. On the other hand, if they are words, how can we explain that we can split them up and have their parts separated from each other, as in call your mother up, something which you can't do with words?



A solution may have to be sought in considering call up a stored lexical unit which remains unspecified for its grammatical status as either a word or an above-word-level structure. In its discontinuous form (call ... up), the particle manages to trigger the whole form.

When do you say that you're certain?

"You only say you are certain when you are not." (Halliday)

It's true. Despite the fact that they contain an explicit marker of certainty, sentences such as That must be John, That's certainly John, That's definitely John and That's John, no doubt all sound less certain, paradoxically enough, than the modally unadorned sentence That's John.

So, beware of people who say they're certain.

More Wodehouse

'Jeeves,' I said that evening -- and I said it coldly -- 'I shall be obliged if you will pop round to the nearest music-shop and procure me a copy of "Sonny Boy". It will now be necessary for me to learn both verse and refrain. Of the trouble and nervous strain which this will involve, I say nothing.' (op. cit., p. 517)

Some cases of praeteritio are inherently (at least, on a strict reading) self-contradictory. This case is different, as the expression is say nothing of X. There would have been real self-contradiction (again, on a strict reading) if Wooster had said this:

'... I'm not even mentioning the trouble and nervous strain which this will involve.'

More contradiction in Wodehouse

"The rain continued to lash down with what you might call indescribable fury ..." (P.G. Wodehouse. 2008. The World of Jeeves. Arrow Books, p. 454)

Well, yes, you might call it indescribable, but that doesn't describe the fury with which the rain lashed down in any specific way! Of course, it's a play on words, indescribable meaning both 'what cannot be described' and 'extreme'.

It takes someone of the indescribable wit of Wodehouse to turn the self-contradictory word indescribable into an almost indiscernible little language joke.

I'm wondering...

... what I'm wondering about.

Paradox in Jeeves and Wooster?

P.G. Wodehouse, whom I mentioned before, probably had a bit of a chuckle when he wrote the following exchange, between Wooster and his 'manservant' Jeeves:

'Good Lord, Jeeves! Is there anything you don't know?'

'I could not say, sir.'

(P.G. Wodehouse. 2008. The World of Jeeves. Arrow Books, p. 273)

There's a distinct hint of paradox in Jeeves's reply, especially if it's not taken to mean 'My inferior position prevents me from bragging about my knowledge, sir' but 'I have no idea, sir' or 'I don't know, sir, if there's anything I don't know' -- in which case, of course, there IS something he doesn't know!

The exception that proves the rule

How's this for a nice paradox: the exception that proves the rule? Shouldn't an exception be seen as something that contradicts or invalidates the rule, rather than something that proves it?

It isn't true what some people claim about this saying, namely that prove is used here in the sense of 'test'. Rather, the expression goes back to maxim in Roman law, which stated that making an exception proves that the rule exists and is valid in those cases in which you're not making an exception ('Exceptio probat (or firmat) regulam in casibus non exceptis'). Well, that's perhaps rather trivial: an exception does not invalidate the rule, because if it did, we wouldn't call it an exception to that rule. It's because the rule, as formulated, is still seen as generally valid that we can speak of an exception, and by speaking of an exception, we acknowledge the 'otherwise validity' of the rule.

What's less trivial, and considerably more problematic, is that there's a notion in linguistic theory which compels us to accept cases which are at the same time exceptions to the rule but still actually follow the rule, in the sense of being an application of that rule. The notion I'm referring to is that of partial productivity. A rule is partially productive if you can't just apply it generally. Its scope is limited, not open-ended, which means that you have to kind of learn and know the cases which fall under it. Yet, a partially productive rule is not unproductive. New items do on occasion appear, but these are then
1) exceptions to the rule-as-a-list-of-learned-instance: they're not in the list of known cases, hence 'exceptional', special, one-offs.
2) applications of the rule-as-pattern: they are in accordance with the regularity that the rule attempts to capture.

For example, most instances of the Adj as N pattern, described by the linguist Paul Kay, are lexically listed (fit as a fiddle, cool as a cucumber, happy as a lark, dumb as an ox, etc.). You can't just compare anything to anything else, which means that if you do create a new simile, for example dull as a butter knife (which, to be fair, is not exactly new, as I found it in a corpus), this new application of the rule is, actually, a kind of exception at the same time! We're making an exception in order to apply the rule. Paradox...

Apparent paradox paradox: the slides

Long time no blog. But I haven't been doing nothing. Here are the slides to the presentation on the Apparent Paradox paradox that I gave in Austin, Texas, about a month ago. And last week I gave a talk in Ghent on the partial productivity paradox, which I'll briefly explain in my next post.

Are you experiencing a paradox?

(via Buzzfeed, where you can see a larger image)

#iamparis

To be prepared to preserve peace is one of the most effectual ways of winning this war.
(freely quoted after George Washington)

Could I possibly have cracked the Liar paradox?

It can't possibly be the case, can it, that I have resolved the Liar paradox, this most devilish and demonic of all paradoxes? To remind ourselves, the Liar paradox arises from the fact that a sentence such as

This sentence is not true.

forces us to admit that there are completely okay-looking sentences in natural languages (in this case, in English) which, if we consider them to be true, have to be not true (for that's what the above sentence says about itself), whilst if we assume that they are not true, they have to be true (for if it's not true that the above sentence is true, we have to conclude the opposite of what it says about itself). Either way, it seems like we are confronted with a contradiction: a sentence which is both true and false. And from contradictions, all hell is said to break loose, as per the principle of explosion ("ex contradictione sequitur quodlibet"), which states (to my mind as a way of gross exaggeration, and I'm not the only one) that once we accept contradictions, any conceivable statement can be concluded to be true. And we certainly don't want that. But even if the Liar sentence could somehow be shown not to be a self-contradictory statement, what are we to make of it? Is it false, as it says it is? Is it true nonetheless, in spite of itself? Is it neither true nor false? Is it both true and false? (Well definitely not that -- not if it's not a contradiction.) A liar simply can't be trusted to tell the truth, and if they say they're lying, why would that all of a sudden be true? But if it's a lie, it is true! Oh god...

It would be highly improbable that in the course of just a good week's thinking, I have silenced the Liar, to which have been devoted a 4000-word Wikipedia article, a 9.500-word entry in the Internet Encyclopedia of Philosophy, a 13,000-word entry in the Stanford Encyclopedia of Philosophy, not one, not two, not three, not four, but at least five books published by Oxford University Press (to name but one academic publisher, but without being exhaustive, there's also one book on the Liar which appeared from Yale University Press, another one from Cambridge University Press, and yet another one from the MIT Press), as well as countless scholarly journal articles and more popularizing treatments of the paradox (such as this one). No, I don't think I have really cracked it, and it would be pretentious for me to say I have, as I should first plough through all of the major works written on it. I did take a look at some of the available literature, but obviously not all of it, and most of it is too complicated for someone not versed in advanced philosophy and logic (and philosophical logic) anyway. Anyway, by reading synopses of books, some of the articles that are relatively accessible to the logic layperson, a couple of book reviews and so forth, my own ideas about what seems to be a fruitful approach to the paradox are gradually becoming ripe for some tentative articulation.

This post is long, so let me guide you through its structure. I'll first give 3 sensible ideas about the Liar paradox, all of which have some downsides. I'll then give 3 wrong-headed ideas, though there's also something sensible about them. (I know, I could have merged these two triplets in a single list.) And finally, I'll give my own views on the paradox, realizing that they're perhaps not completely original, given the vast amount of scholarly research on this monster paradox.

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I can't always remember where exactly I've read everything so far, but some rather sensible things have been said about the Liar paradox. I'm giving a very incomplete overview below, with some very incomplete references. It's also clear that each of these points, sensible though they are, are wanting in certain respects.

Sensible idea 1. Ungrounded statements, such as the Liar sentence, are neither true nor false

The Liar sentence given above is sometimes said not to be "grounded". That is, it has no connection to the world of real objects and events. And consequently, it can't be true, nor can it be false. Tim Maudlin (2004) made such a claim. Not being "grounded" is a property that many authors appear to have accused the Liar sentence of, including Saul Kripke, who also points out that the sentence This sentence is true is not grounded either and therefore equally problematic in terms of reference assignment, though it sounds less paradoxical.

But while it sounds intuitively fine to claim, as Maudlin does, that “[c]lassical truth values cannot simply be conjured out of thin air: they must originate always at the boundary of a language, where the language meets the world” (p. 51), this view can be criticized, for example on the grounds (sorry to use that term here) that if we play the ungrounded card, we are compelled to say that a sentence like "All true sentences are true" is not a true sentence, as it's just as ungrounded as the Liar sentence.

Furthermore, if one agrees that This sentence is true has less of a paradoxical flavour than This sentence is not true, this means that that you should also agree that both sentences are supposed to be true, despite being ungrounded. One of the claims to which I will turn later is that there is a tacit assumption that the speaker believes (and wants you to believe) that the statements he or she utters are true. In other words, the Liar sentence and its non-negated counterpart may be ungrounded, but since they have the form of ordinary declarative sentences, they give the impression of being true at first glance. And that is why, at a second glance, This sentence is false can be felt to be paradoxical and This sentence is true can be felt to be tautological (a statement which can't be possibly not be true, whatever the circumstances, like It will rain this afternoon, unless it won't rain).

Sensible idea 2. (Not) being true is a predicate that should not apply to the subject of the sentence it is a part of; a violation of this condition yields a meaningless sentence rather than a sentence that is either true or false

This idea is closely related to the previous one. The liar sentence only leads to a serious problem if we treat true in the liar sentence above as belonging to the same level as the rest of the sentence. If instead we treat it as a meta-level evaluation, then the paradox vanishes. That's because when we replace "this sentence" in the liar sentence by the full sentence (which we're allowed to do, as the sentence is self-referential), we end up with the following statement, which is not a contradiction:

"This sentence is not true" is not truemeta-level.

Since what's between the quotation marks is rather meaningless drivel (cf. sensible idea 1 above: it's not grounded), it is not even true at the meta-level, just like it would not be true to say that the keyboard slam qsdlkfjq sdlmkjfsj is not true. In other words, what we can conclude here is this:

""This sentence is not true" is not truemeta-level" is not truemeta-meta-level.

Such stratified meta-levels are due to Alfred Tarski. Whether they resolve the liar paradox is not clear, though. For one, introducing a Tarski hierarchy does not explain why the inner sentence is meaningless drivel; it only provides a means of avoiding meaningless drivel, by restricting the use of the truth predicate to objects at a level below that predicate. For another, it does not explain why some self-referential sentences are not problematic, such as The sentence you are now reading is, hopefully, a perfectly grammatical English sentence, since, as far as I can see, it doesn't sin against any rules of grammar (and since 'hopefully', in contrast to what some prescriptivist linguists claim, can be used not just with the meaning of 'in a hopeful way' but also with the meaning of 'as I hope'). What's so special about (not) being true that it can't be predicated of subjects like The sentence you are now reading?

Sensible idea 3. The Liar paradox doesn't require a self-referential Liar sentence

What if we make sure the Liar sentence is not self-referential? Can we then still get a Liar paradox? First of all, as expected, there is no paradox if the first sentence says that the second is true and if the second sentence then says something that could in principle be verified independently, as in the following sequence, taken from the web:

What I'm about to tell you is absolutely true. I was on the 24x from UCSB to downtown one wednesday night in may.

The second sentence, being in the 'truth scope' of the first, is interpreted as true, even more definitely true, if that makes sense, than if it hadn't been preceded by the first sentence. However, what if the second sentence then in turn said something about the first? More specifically, what if it denied that the first was true? The following pair of sentences, neither of which is self-referential, jointly show that, even if we remove self-reference from the Liar sentence, we can still have a Liar paradox. I can't remember who was the first to point this out, but it's a well-known observation.

The following sentence is true.

The previous sentence is false. 

Okay, so self-reference at the sentence level is not required to get a Liar paradox, but what we have here is not much better. It's known as a Liar cycle. Rather like Escher's drawing hands, the two sentences refer to each other and to nothing else.

The liar paradox is apparently parasitical on self-reference, whether within a single sentence or within a pair or larger set of sentences. With all due respect to Yablo's claim that one can construct a liar paradox without self-reference and Sorensen's defence of this view, I tend to agree with Beall that any variant of the liar paradox, one way or another, involves some form of self-reference. Put simply, liar paradoxes as a whole do not say anything about anything except themselves, but we can create the illusion that they do by piling up several sentences none of which is self-referential itself.

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I've also read a few things here and there that do not make any sense at all, although there's also a grain of truth in them.

Wrong-headed idea 1. The Liar sentence is a disguised contradiction and therefore not a paradox

It has been suggested by Arthur Prior, according to the Wikipedia entry on the Liar paradox, that "every statement contains an implicit assertion of its truth". I haven't looked up the source text, so I'll just rely here on what Wikipedia says. If you say "The sun is shining", you actually also convey "It is true that the sun is shining". After all, it would be ludicrous to claim, "The sun is shining, but that is not true". At first sight, that's sensible. So that's not the wrong-headed idea yet. Prior then says, still very sensibly, that the Liar sentence given at the top of this post can be reformulated as follows, with its hidden part made explicit:

It is true that this sentence is false.

Now, for Prior, this sentence amounts to being a simple contradiction, like "p and not-p" (where p stands for any proposition), as It is true that... is equivalent to This sentence is true and... . And a contradiction is just a false statement, not a paradox.

But that is where I would disagree. A paradox is commonly defined as "a statement consisting of two parts that seem to mean the opposite of each other" (e.g. the definition of paradox provided here). What Prior's reformulation does is make the paradoxical nature of the Liar sentence more visible (and therefore less funny -- see further below); it doesn't make its paradoxical nature disappear. Unless I have completely missed the point, Prior's strategy is a bit silly. It aims to show that the Liar paradox doesn't lead to contradiction since it is implicitly a contradiction. That's like saying to your host: "No need to worry any more that that expensive vase in your hall will ever break, 'cause, um, I just broke it."

Wrong-headed idea 2. The Liar sentence never even allows us to evaluate whether it's true or false, because the reference of the subject noun phrase keeps eluding us

It has been claimed (e.g. here, where some further references are given) that the problem of the Liar sentence is that you never even get to evaluate whether it's true or false, as you never manage to pin down the reference of This sentence in the first place. This point is also eloquently made in this video, which asks: "Which sentence exactly?" So, the culprit, on this view, is not the truth predicate but the use of This sentence..., whose reference will embed itself ad infinitum:

This sentence (namely "this sentence (namely "this sentence (namely ...) is false") is false") is false.

This may seem somewhat plausible for the archetypal, so-called 'strengthened' Liar sentence given higher up (This sentence is not true), but the problem (if indeed it is a problem at all) does not obviously arise, it seems to me, with the following version of the Liar sentence:

The sentence you are now reading is not true.

I'm using this formulation because it's close in form to an actually occurring sentence in a book by the linguist John Taylor, which I reproduce here with a bit of context, so that you can interpret it properly:

A narrower approach restricts the term [to wit, 'construction'] to form-meaning pairings which have unit status. 'Construction' thereby becomes synonymous with the symbolic unit in Cognitive Grammar. On this more restricted understanding, the sentence you are now reading is not a construction, since this specific combination of words and phrases has not become established through usage.

What I think we do when we read the latter sentence is this: as soon as we read "is not a construction", we quickly verify this claim by surveying the whole sentence as fast as possible, standing back as it were, breaking it down roughly into some of its syntactic constituents:

On this more restricted understanding / the sentence you are now reading / is not a construction / since this specific ...

All of this happens very fast. We probably don't do a complete parse of the sentence before being convinced that, yes, this particular sentence has, in all likelihood, never been used before, so it can't be stored as a conventional unit in your mind, and so it's not a construction on that definition, and so the author has provided a good example. In other words, it seems to me that it's not correct to claim that the reader interprets the sentence by a full substitution of the whole sentence for the part "the sentence you are now reading", as follows:

... On this more restricted understanding, ["On this restricted understanding, the sentence you are now reading is not a construction, since this specific combination of words and phrases has not become established through usage"] is not a construction, since this specific combination of words and phrases has not become established through usage.

It would, moreover, be rather too far-fetched to claim that on encountering "this specific combination of words and phrases" in the above bracketed sentence subject, the reader then again looks for the reference of this noun phrase and finds it to be, again, the whole sentence, as follows:

... On this more restricted understanding, ["On this restricted understanding, the sentence you are now reading is not a construction, since ["On this restricted understanding, the sentence you are now reading is not a construction, since this specific combination of words and phrases has not become established through usage"] has not become established through usage"] is not a construction, since this specific combination of words and phrases has not become established through usage.

But that still wouldn't do, as these substitutions still contain two instances of "the sentence you are now reading" and "this specific combination of words and phrases", all of which would then have to be further substituted for the whole sentence (and so on and so on), before you even got to the part "is not a construction". Clearly, that is not how a normal person would read Taylor's sentence.

In short, I don't think the Liar paradox can be resolved by pointing out that, as a self-referential sentence, it leads to infinite regress. Other self-referential sentences don't, in practice.

Wrong-headed idea 3. Whenever you think you have resolved the Liar, it will be back with a vengeance

It has been claimed (e.g. in the blurb of this OUP book, ominously called Revenge of the Liar) that a solution to the Liar paradox is simply not possible: the paradox will always manage to take revenge on whoever purports to have found a solution to it. For example, if you think that the solution to the paradox is to say that, in fact, the Liar sentence has no truth value to begin with, then the Liar will rear its ugly head by granting this as a new possibility:

This sentence is either not true or it has no truth value to begin with.

As before, if the sentence is just not true, it is true (because what it says about itself, namely that it may not be true, is then fulfilled). But also if you believe that this sentence, just like the simple Liar sentence (This sentence is not true), quite simply has no truth value to begin with, then again what it says about itself is true, in which case it's not the case that it is either not true or has no truth value to begin with! In either case, the statement comprises a contradiction and, hence, harbours a paradox. And this unfortunate outcome is not different for most, if not all, other suggested solutions to the Liar paradox, such as that it is not a proposition (see somewhere in this book, called Truth, for a demonstration) or that it is meaningless (The YouTube channel Carneades.org shows this in its video on the Liar's Revenge Paradox, challenging all viewers to try and solve it).

This may all sound rather clever, but why would anyone who presents a meta-cognitive, theoretical evaluation of the Liar sentence -- be it that it's not a proposition, that it's meaningless, that it's simultaneously true and false (and that there's nothing problematic about that), that it fails to be definitely true, that it is linguistically ill-formed, what have you -- be prepared to insert that evaluation into the Liar sentence, which is the object of study? Forcing the scholar to do such a thing is utterly perverse. It is a ridiculous mixture of two levels of speaking, one pertaining to the object language and another pertaining to the metalanguage. We typically don't do that, as the awkward example below illustrates, so why would we do that for the Liar sentence?

17 x 24 = 568 or a case of a problem which Daniel Kahneman, in his book Thinking, Fast and Slow, argues has to be handled by System 2, the mental system that we rely on for more complex computations, which require a great deal of concentration.

(By the way, I intentionally provided a wrong solution to the left of or, to make this ridiculously-looking example resemble a Liar's Revenge sentence as closely as possible.)

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Here are some of my own views (which I perhaps have no reason to appropriate as mine and mine only, as they may have been formulated by others already) on how best to approach the Liar paradox.

1. The Liar sentence is not simply both true and false; if anything, it's true and false at different times of the interpretation process

The consensus that the liar sentence is a paradox, because it's true if and only if it’s false and false if and only if it’s true, may rest on a false assumption: its truth and falsity hold simultaneously. What we should not forget in our treatment of the Liar paradox is that we are dealing with language, and that language has a temporal dimension: words are spoken (or written) one after the other, and decoding sentences also takes up time. Works on philosophical logic doesn't seem to be much concerned with this, but it should be. It's only in a popularizing book, Secrets of the Paradox, that I found something written about the time dimension of the Liar sentence, in a section with the nice title "The Life-Cycle of a Proposition". The author writes:

"... think again about the antics of the Olympic athlete name Liar Paradox. Miss Paradox has not yet run in tomorrow's 100 metres final and crossed the finishing line, so she cannot possibly be the gold medallist and therefore she cannot lawfully have the gold medal in her possession. Similarly, the Liar sentence has not yet crossed the verification boundary -- it has not been tested in an appropriate way -- so it cannot possibly have been pronounced false in advance of the event. A gross transgression of the logical and temporal future has occurred. This is malsequence. What should occur subsequently has 'occurred' forcibly several steps before it even could. I repeat: this future does not yet exist; true and false are responses resulting from the verification process and are states which exist solely within the human consciousness, not within propositions.

In the Liar paradox we see that the usual steps: proposition, verification and true/false response, have all been contracted into a single sentence." (pp. 33-34)

In fact, this insight, which rests on a distinction between a proposition and a verification statement, is in line with Tarski's and Quine's view that we can't combine an object-language statement and a meta-language statement involving truth in a single sentence, unless we use quotation marks for the former. By invoking a quite obvious natural sequence of events (first hearing a sentence, then deciding whether or not it's true), it provides a very simple explanation of what we identified as something missing in these philosophers' treatment of the Paradox (or at least in my perhaps ill-informed reading of their treatment): why is it that we can't use the predicate is (not) true after a subject noun phrase like This statement? The reason is now clear: such a predicate comes too early. 

But the temporal dimension of the Liar paradox is actually even larger. To see this, consider the following Liar paradox, which features in the image at the top of this blog post:

It's true, I'm a liar.

Here we start with a verification statement, and the subject pronoun It cataphorically refers to the statement "I'm a liar", which is itself devoid of a truth predicate. There doesn't seem to be any violation of the normal life-cycle of the proposition here, especially since the pronoun It probably also refers back, anaphorically, to something an invisible addressee had suggested first, something like "You're such a liar". The sentence above then says as much as, "Yes, that's true [= a verification statement of a previous proposition], I'm a liar [= repetition of the previous implicit proposition]". There's nothing that is not well-formed about the sentence (though one could quibble about the use of a comma rather than a full stop or a colon, as the two parts of the sentence are main clauses, syntactically speaking). So why does it still engender the Liar paradox?

If we ignore the complication that It in the sentence above is also an anaphora, then you will agree that the above sentence comes close to the Liar cycle mentioned higher up (The following sentence is true. The previous sentence is false.). Remember that neither of the sentences in this concise two-sentence cycle is self-referential, but that as a whole, the two parts just point at each other and lack any grounding. Here too, we could think: so, if the speaker grants that he's a liar, admitting that it's true, then he must indeed be a liar. But hang on, if he's a liar, and especially if he's a habitual one at that, then we can't trust his "It's true". It's probably not true then. And then it's not true that he's a big liar. But wait, if he's not, then maybe we can trust his "It's true". And so on.

What I have tried to demonstrate by making this train of thoughts explicit is that the it shows it is false to assume that the Liar sentence (or a member of a Liar cycle) is true and false at the same time. Illuminating is how Douglas Hofstadter, in his masterful book Gödel, Escher, Bach: An Eternal Golden Braid (click here at your own peril for a pdf), couches his informally description of the Liar sentence in a temporal unfolding of cognitive events:

"It is a statement which rudely violates the usually assumed dichotomy of statements into true and false, because if you tentatively think it is true, then it immediately backfires on you and makes you think it is false. But once you've decided it is false, a similar backfiring returns you to the idea that it must be true. Try it!" (p. 17)

Note how in this formulation, the Liar sentence flips back and forth from true to false. This is different from saying it's both true and false. To see the difference, let's review a few different ways of dealing with (S), the Liar sentence:

(S) S is not true.

1. S is both true and false.  [purely logical qualification]
2. S is false iff it is true and it is true iff it is false. [again, an atemporal, logic statement]

3. If S is true, it's false, but if it's false, it's true. [atemporal statement equivalent to 2, but without the logical operator iff]
4.  If S is true, then it's false, but if it's false, then it's true. [by using then, the atemporal statement in 3 has become (kind of) temporal]

Suppose we had never heard of the Liar paradox and someone told us, "This statement is not true". What would happen? This is a possible scenario, using Gricean pragmatics:

• Viewed from the perspective of what we typically do when we hear a statement, we tentatively assume it's a true claim.
• Viewed from the perspective of the information we then add to our knowledge base, we accept that it's the case that this statement (whatever that statement may be!) is said not to be true.
• We then realize that we've been tricked, as this statement is the entire sentence, which we assumed to be true.
• We have no new information about anything, so it’s a blatant violation of Grice's first Maxim of Quantity (make your contribution as informative as is required).
• If we still assume the speaker is cooperative, then we must look for what the speaker is trying to do.
•Aha, maybe it was some kind of joke! (cf. next point)

So, it's true that you can't equivocally say, once and for all, whether a Liar sentence is true or false. But it's not true that a Liar sentence is both true and false, thereby implying that these mutually exclusive values hold at the same time. My claim here is that the Liar sentence is true and false at different times of the interpretation process. In that sense, it doesn't really violate the Law of noncontradiction, which states that nothing can be both A and not-A in the same sense and at the same time. I think you will agree with me that the Liar sentence is not a contradiction of the same kind as, say, I was born in Bruges and I was not born in Bruges.

What makes the Liar sentence special is that, on hearing it, it takes some time to figure out that it's self-defeating. Douglas Hofstadter, in his book mentioned above, is right in this respect when he writes: “The difficulty is perhaps underlined when a sentence such as [This sentence is false] is presented to someone naïve about paradoxes and linguistic tricks, such as a child. They might say, “What sentence is false?” and it may take a bit of persistence to get across the idea that the sentence is talking about itself” (GEB, p. 496) We can make the same point, using a non-temporal, visual equivalent: the ouroboros, a snake consuming itself:



Such a concept, or picture, is okay as long as we see different portions of the snake involved in the act of having a bite: the mouth and the tail; it would be a problem if we try to depict physically, or picture mentally, just a mouth eating itself. The time delay is important. Without it, there is true contradiction.

2. Can't we all agree that many variants of the Liar paradox are just comical, nothing more? 

I realize I'm in dangerous logico-philosophical waters when I say that many versions of the Liar paradox truly meaningless (as opposed to "either true or false", "both true and false" or even "neither true nor false"). Their self-contradictory, self-defeating status makes them funny. And that's really their most important property: they're a bit of fun. This is the case, for example, with the Pinocchio paradox:


It's also the case with the Liar question ("Answer truthfully, with yes or no, is your answer to this question going to be "no"?"). My son (8 years old) loves coming up with paradoxes of this kind. Only a couple of days ago, he suggested this one:

One person says to another, "I'll bet you that what you're going to say next is going to be, "I'll bet you it isn't!". 

Other person: "I'll bet you it isn't!"

We just can't resist self-defeating utterances. Here's another one: "I used to believe in reincarnation, but that was in a past life". On Facebook, I recently wrote the following status update:

[dilemma warning] Like if you agree: this status update is way too stupid to like.

Surprisingly enough, it did produce a few likes, four to be exact. (Maybe I should have phrased it as follows: "Like if and only if you agree: ...")


3. A pragmatic approach is needed to deal with apparent self-contradiction

When we are confronted with apparent contradictions in ordinary conversations, we tend not to bother too much. In fact, we may not even notice certain sentences are self-contradictory. This is probably true for such utterances as the following:

Don't listen to what I say. [Well, in fact, the speaker does want the hearer to listen to this, for what would be the point of uttering it if she didn't?]

Just ignore what I'm saying. [idem]

I'm not saying anything. [That's not exactly true, is it?]

I'm speechless. [So, where did these words come from?]

We are willing to adjust the strict, literal meaning of these utterances so that they make sense in context. That is why I find it intellectually dishonest of logicians to formulate the Liar sentence in such a way that it can't be contextually coerced -- to produce a maximally "strengthened Liar" -- and then point out that this sentence poses a problem. Well, they make it pose a problem. In everyday conversations, we readily accommodate for utterances that are not fully devoid of self-contradiction. For instance, if someone says, Don't listen to what I say, we take this to mean 'Don't take heed of what I just said' or 'Don't take heed of what I'm about to say' or 'Don't take heed of most of the things I'm saying (except of course what I'm saying now)'. Why do we apply such shifts? The reason is simple: we assume that the other participants in the conversation adheres to what Grice calls the Cooperative Principle, that is, that what they say serves some purpose. We also assume that people engaging in a conversation are trying to deliver coherent speech. If what they say is not really that coherent, we, as hearers, make the necessary adjustments.

For instance, if my son says,

Look, it's not raining... although in fact, it is.

I do not just conclude that he has uttered a contradiction and that he's talking nonsense. I understand that he's first surprised at it having stopped raining and wants me to look up at the sky but that he subsequently feels compelled to modify what he said, as he can still feel or see some rain drops. Similarly, if we get a multiple choice question with one of the four possible answers being "the answer's not given" we interpret this option as "the answer's not given among the three previous options". That's the power of pragmatics.

In particular, a powerful trick that speakers and hearers master is the ability to adopt two perspectives. Accordingly, something can be true from one perspective but false from another. If one were to ask if the protagonists of Escape (Piña Colada song) commit adultery, the answer is Yes, from one perspective (the husband and the wife both go, unbeknownst to one another on a secret date, and the song suggests that the date is successful...), and No, from another perspective (the husband and wife meet each other on the date, see each other in a new light and their passion is rekindled).

How often do we get as a response to a question, "Well, yes and no."?  (Google gives more than 400.00 hits for this combination.) We don't assume, on hearing this, that the respondent is contradicting himself. We know that he means, 'in a sense, yes, but in another sense, no' or 'if you look at it in one way, yes, but if you look at it in a different way, no.' The Proposition Rejection Except Construction also involves two different perspectives: one is (what is portrayed as) the received opinion and the other is the speaker's more informed opinion.

I have suggested here that the Liar sentence is also true and false in different senses, or from different perspectives. If you look at the sentence as a statement just like any other statement directed towards you, then you're prepared to assume it's true, but if you look at it in terms of what you have been told about it (namely that it's not true), then it's false. These are different perspectives, corresponding with, as we said above, different times at which we process the sentence. There's no real contradiction. And if there's no contradiction, the paradox is no more. Ergo, I have resolved the Liar paradox.

Wake me up when I'm dead

This blog has been dormant for a couple of weeks, but it's not dead. Which brings me to this great paradox:

Grampa Simpson: Who has disturbed my tomb? 

Lisa Simpson: Grampa, you're not dead.  

Grampa Simpson: Then wake me up when I am.

the "apparent paradox" paradox

On 23rd October, I'll be giving a talk in Lille on "Conceptual blending: the apparent paradox paradox". Here's an abstract of my talk:

Our use of expressions such as apparent paradox, false teeth, fake orgasm and the like seems to imply that there are objects which are and are not a certain thing. It will be argued that this apparent paradox paradox is only an apparent one. Drawing on insights from mental space theory, polyphony, conversational pragmatics and frame semantics, I will adopt and adapt a recent proposal of how best to deal with privative adjectives like fake, false and fictitious. A central notion in my account is that of perspective-taking, which may help us to solve a wide range of puzzling phenomena, including the problem of negative existentials, as in “Unicorns do not exist”, and possibly even the most fiendish of all paradoxes, the Liar paradox: “This statement is not true”.

I know I promised earlier to give my "solution" (for what it's worth) of the Liar paradox, but I'm still tinkering with it.

I really didn't say everything I said

Here's a nice article by linguist Ben Zimmer about Yogi-isms.

This page intentionally left blank

One of the most common and obvious contradictions in the world of printing is the page which is intentionally left blank... but contains a printed message to that effect.

Cute self-reference

My daughter is in her first year of secondary school, where Latin is on the curriculum. On her daily vocab test today, she was asked what the first person singular present tense form is of the verb errare. The answer should have been erro, but she wrote... error. Which is exactly what it was.

Some sort of solution to the Liar paradox coming up

I've been working on an informal treatment of the Liar paradox (The sentence is not true. If it's true, it's not true, and if it's not true, it is true). I've probed some of the vast literature about it, but since most of the proposed solutions require heavy advanced logic machinery which I don't master (and do not want to master, for that matter), I've finally decided it's best to deal with the paradox more or less my own way.

In a nutshell, my solution involves a realization that the Liar sentence is not both false and true at the same time, that it's just a tantalizing bit of fun and shouldn't be taken too seriously, and that real-life situations of (milder) contradiction suggest a pragmatic treatment may work best. Pragmatic is used here in the everyday sense of 'taking a sensible, not too principled approach' and in the linguistic sense, referring to the work of Paul Grice, whose main claim is that in normal speech, we adhere to what he calls the Cooperative Principle; since the Liar sentence clearly doesn't comply to his conversational Maxims (like being unambiguous, being informative), the only reasonable conclusion is that the speaker addressing us with a silly question like Will your answer to my question be "no"? probably just invites us to have a bit of a laugh -- or to say "Oh, shut up!".

So, stay tuned for a long post, in which I will also highlight some of the sensible and less sensible ideas that have previously been formulated by people attacking the paradox.

If I had more time...

... I would invite linguists to present or update their view on the liar paradox ("This sentence is false"), in the form of succinct contributions to a collective volume titled What Linguists Have to Say to the Liar, or something a little snappier. People to be invited would be Paul Saka, Rennie Gonsalves, Gilles Fauconnier, Leonard Talmy and others.

Paul Saka: to hell with the theory of truth

In his outstandingly clear and convincingly argued paper "Beyond truth", Paul Saka ends with a passage that attempts to give the final stab to truth-conditional semantics:

"It is unlikely for any expert semanticist in either linguistics or philosophy not to have heard of the liar paradox, and not to know that it seems to prove that the theory of truth is illegitimate; most use the theory of truth in the foundations of their research, and yet practically none at all even acknowledge the inconsistency of their position. It is nothing less than scandalous."

Paul Saka: fierce theory of truth buster

Our established practice of science (and by extension, the teaching programs that help propagate it) does not come out unscathed either:

"It also proves, in case an object lesson were needed, that research programs do not rise and fall according to intellectual merit alone. When a theory is logically refutable, and everyone knows it, yet the theory enjoys orthodoxy nonetheless, then obviously non-rational forces are at work, be they sociological or psychological."

I'll definitely read more of Paul Saka.

Update: maybe I should have used quotation marks in the title of this post, as it now seems as though I want to say something to Paul Saka, namely: "To hell with [your] theory of truth". That's most definitely not what I meant, as it's Saka himself who is highly dismissive of this theory. I didn't use quote marks for the simple reason that he didn't use these actual words. Incidentally, Paul Saka has written extensively on quotation, e.g. this paper.

Worth watching

Over at Carneades.org, a YouTube channel, October last year was The Month of Paradox. Its presentation of paradoxes, one a day, is framed in terms of Quine's general classification of paradoxes into veridical ones, falsidical ones and antinomies, which the creator of the Carneades videos calls "the most fiendish of all paradoxes". Check it out!

Visual illusion

I've had a bit of fun today being totally puzzled by the vanishing leprechaun visual illusion, dating from the 1960s. If you cut the top half of the picture below into three pieces and swap the two upper parts, such that you get the lower half of the picture, you'll find that one leprechaun appears to have disappeared.

An explanation of the illusion is provided, with some building of suspense, on this website.
The Guardian recently had a piece on a variations of this puzzle, including one with sixties pin-up girls.

I've actually also been looking at more serious stuff today, namely MIT professor Agustín Rayo's plea for localism, which I hope I'll be able to come back to soon in a separate post, since it provides an original account of the Liar paradox. I have a hunch that what connects the pictorial illusion and Rayo's ideas is that context is everything.

Praeteritio at Prattenburg

In an episode of Zo heurt het eigenlijk, a Dutch TV programme on high society lifestyle, one can hear Lady Vicky van Asch van Wijck née Freifrau von Papen utter the following praeteritio to let us know that the grounds of her nice country estate Prattenburg do not belong just to her husband and herself but are open to visitors, so to all of us:

"ik wil niet zo ver gaan om te zeggen "Het is van ons allemaal", maar in feite is het toch van ons allemaal."

'I don't want to go as far as to say "It belongs to all of us", but in fact is does belong to all of us.'

Maybe unbeknownst to her, this is a contradiction, as she clearly does go so far as to say what she's saying. Perhaps she meant the first part of the sentence as a hedge: 'I normally wouldn't want to declare this so openly (because it might sound as if we're bragging about our services to the public), but since you're asking me...'.

There's no shame in being a pariah

Just heard in an episode from The Simpsons

Lisa: I guess we better get used to being pariahs.
Marge: There's no shame in being a pariah.

Marge had better look up the meaning of pariah again.

Strange loop in Google circles

I just noticed that I seem to have included myself in my own Google circle, thereby boosting the number of my followers with, oh, a whopping 34%, to round the percentage a bit in my favour. (There were 3, and now there are 4, me included.)
What's strange about this is that, apparently, you can also not be in your own circle.
Even stranger is that, now that I'm in my circle, my page profile includes a thumbnail of itself. When I click it, I think the page just refreshes itself.
Being your own follower is certainly a paradoxical concept. Try doing it physically! Only some dogs manage to do so, chasing their own tail.

"I have said about that what I have said about that"

Belgian federal State Secretary for Asylum Policy and Migration Theo Francken reacted today curtly to suggestions that our country might accept more than 250 (mainly Syrian) asylum seekers a day:

Ik heb daarover gezegd wat ik daarover heb gezegd.

'I have said about that what I have said about that' 

This is a clear tautology, unlike Wooster's "When we Wooster's are adamant, we are -- well, adamant". We interpret tautologies not just as trivial truisms. As they are so obviously true and hence provide nothing in terms of information that we could not already have known by ourselves, our task as listeners is to find out why they are uttered at all. In this case, it is: 'I stand by what I said' -- precisely the tautological interpretation of Wooster's utterance (which, let me repeat, I don't think was initially intended to come out as such a tautology).

Tautologies are not paradoxes but they are linked to them in a straightforward way: we only have to negate them to end up with a paradox, or definitely a contradiction (I still don't know whether I should make a distinction between paradox and contradiction):

Ik heb daarover niet gezegd wat ik daarover gezegd heb.

'I haven't said about that what I have said about that' 

This is why Wooster's utterance is not a tautology. We can negate it without the result being a paradox:

When we Woosters are adamant, we are not adamant.

With a small amount of effort, we can make sense of this as conveying the following:

'When we Woosters are adamant, it is possible for us to change our mind about being adamant'

This is not pure nonsense. You can be adamant about things and know that you'll never change your mind, but you can also be adamant about things temporarily -- say for a week or so -- and then evaluate your comportment to see whether you can relax your being adamant about them.

Awkward adamance

What ho! Have you ever read anything by P.G. Wodehouse? You haven't? By Jove, you should! It's immensely funny. Here's a typical exchange between Bertie Wooster, one of London's idle rich, and his intellectually superior manservant Jeeves, who tends to disapprove of Wooster's vestimentary taste:

‘… Soft silk shirts with evening costume are not worn, sir.’

‘Jeeves,’ I said, looking the blighter diametrically in the centre of the eyeball, ‘they’re dashed well going to be. I may as well tell you now that I have ordered a dozen of those shirtings from Peabody and Simms, and it’s no good looking like that, because I am jolly well adamant.’
‘If I might-’
‘No, Jeeves,’ I said, raising my hand, ‘argument is useless. Nobody has a greater respect than I have for your judgement in socks, in ties, and – I will go farther – in spats; but when it comes to evening shirts your nerve seems to fail you. You have no vision. You are prejudiced and reactionary. Hidebound is the word that suggests itself. It may interest you to learn that when I was at Le Touquet the Prince of Wales buzzed into the Casino one night with soft silk shirt complete.’
‘His Royal Highness, sir, may permit himself a certain licence which in your own case-’
‘No, Jeeves,’ I said firmly, ‘it’s no use. When we Woosters are adamant, we are – well, adamant, if you know what I mean.’
‘Very good, sir.’

(Quotation quoted from https://pechorinsjournal.wordpress.com/2009/08/26/, where incidentally I also spotted another instance of that 'complete denial' use of except that I discussed earlier:

And that’s it, with that template and the odd minor variation you could in theory write most of the Jeeves and Wooster stories, except you couldn’t at all because none of that matters in the slightest.)

Just like when I just wrote that I quoted a quotation, in the above dialogue Wooster makes use of this wonderful mechanism known as recursion to say that he's adamant about being adamant. Only, in Wooster's case, it's not clear whether there's any usefulness in adding a meta-level of adamance. What does it mean for someone to be unshakable in their determination to be unshakable in their determination? Wooster seems to be aware of how little his utterance makes sense, or least of how awkward his formulation is, witness his addition of "if you know what I mean". Perhaps, he intends it as a tautology, that is, a truism like "War is war" or "Boys will be boys" (about which the linguist Wierzbicka has written extensively). What he means to say then is this: "If we Woosters say no, we say no" or simply "No is no". I doubt that this is the interpretation Wooster had in mind when he started the sentence. I think the humorous effect lies in the reader seeing how Wooster probably wanted to utter a sentence of the following general structure:

When we Woosters [verb phrase], we are adamant. (Meaning: 'When we Woosters [verb phrase], we are determined not to change our mind about it')

e.g. When we Woosters have decided to take a course of action, we are adamant. ('When we Woosters have decided to take a course of action, we will carry it through')

e.g. When we Woosters have principles, we are adamant. ('When we Woosters have principles, we are unyielding and faithful to them')

The structure above, with adamant at the end, could have functioned as an effective rhetorical means to say, 'I won't change my mind about [my opinion or course of action mentioned in the preceding discourse], so that's the end of the discussion.' Note how Wooster, earlier in the dialogue, said "it's no good looking like that, because I am jolly well adamant", again with the word adamant at the end. Wooster ruins the potential firmness of the above structure by using are adamant too early in the sentence, making the second occurrence of it sound silly and superfluous.

By the way, have you noticed how adamant is one of those adjectives that necessarily takes a complement? Even when it's not used with about X, we know that it applies to some opinion, principle or purpose that we have to retrieve from the context. It's one of those words that requires contextual 'saturation', to use a term by the French philosopher Fançois Recanati. So, you can't just say, out of the blue, I'm adamant, without expanding on it, while it's perfectly okay to say I'm hungry.

Afterthought: So, what's the point of this post? How is it relevant to paradox? There's nothing paradoxical about Wooster's sentence, is there? No, there isn't, but it's undeniably the case that many paradoxes in the family of Liar Paradoxes involve the use of a meta-level -- or perhaps, more precisely, the failure to use one. If someone says, "This sentence is a lie," there is no stable ground, no external level from which you can judge the truth of something else. Understanding paradox is understanding how meta-levels work and why some sentences lead to strange loops (e.g., This statement is false) and others don't (e.g., This sentence contains five words).

Except it's not

I just read this sentence in a Slate article on Wittgenstein:

"So, language is quicksand—except it’s not."

On the face of it, this doesn't make much sense. To use except, one should normally use a list first, and a single statement hardly qualifies as a list. The use of except can be made acceptable if we see the preceding statement instead as a kind of container of truth. The quoted sentence then says: 'So, that language is quicksand is completely true -- except...' The first part is then scalar and this also allows for an exception, in the same way that we can say "I liked everything about the show, except the finale".

So far so good, but what follows except in the quote above is not really an exception to the totality of truth of what preceded. It's a complete denial of what preceded. It may seem somewhat pointless to state something and then deny it right after, but the [Statement -- except pronoun + {contracted verb + negator / verb + contracted negator}] pattern is very common. Here are ten examples retrieved from the Corpus of Contemporary American English (COCA):

1. I mean reaching back to what EJ was just saying and people think he's a Columbia Harvard guy for whom everything's always been sort of easy... GEORGE-STEPHANOPOU# (Off-camera) And Hyde Park. EJ-DIONNE-1 " THE-W# Hyde Park.DEE-DEE-MYERS-1 " V# Yeah. GEORGE-WILL-1-ABC# (Off-camera) Because he is. DEE-DEE-MYERS-1 " V# Well, except that he's not.
2. Dj vu all over again. Except that it wasn't.
3. It was almost funny. Except that it wasn't.
4. Well maybe the whole crime decline was a myth. Except that it wasn't.
5. Neuroscientists told Schmorrow that it was impossible. Except that it wasn't.
6. Finally, it seemed everything was going to be great. Except that it wasn't.
7. Until he found himself on a winning streak, with a small crowd cheering him on, and the only thing to do was keep betting. Except that he didn't.
8. ... and she knew he would turn at once to watch and make sure she didn't brush the moisture off her eyelashes except that he didn't.
9. You can't keep going like this, Jon, you'll burn out. He heard it so many times that he almost believed it. He must be exhausted. Except that he wasn't.
10. He cruised through his workouts for NBA scouts and believed he had a good chance of being taken No. 1 in the draft by the Chicago Bulls. Except that he wasn't.

There's something of a conventionalized ironic understatement in this cliché: 'yes, everything you (or they, or even I) have been saying makes perfect sense -- apart from the small tiny problem that it's just not true'.

Update: I just realize that I searched for the pattern with that after except. Removing it from my search query shows the that-less version is rather more frequent than the that-containing version. Here's just one bonus example from COCA of this more succinct form:

There was no appeal, and that was that. Except it wasn't.