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.