"I always lie," says the liar. If he's telling the truth, then he is indeed lying. But if he's lying, then he's telling the truth.
This sentence is false. If it's true, then it is indeed false. If it's false, then it is indeed true.
P = P is false
These sentences, and others like them, seem to propose an insoluble problem for our logic and language. They appear to be both true and false, or neither. Throughout the history of logic and philosophy, many solutions have been proposed for these paradoxes (labelled together as the Liar paradox), yet their inconceivable force lingers on. Hence the question mark in the title: while I believe Michael Polanyi has sketched an interesting solution to the paradox, it contains a large assumption based on his epistemology, and thus would collapse along with his epistemological framework. But of course this is the case with any purported 'solution', and unlike some of the more radical solutions (e.g. dialetheism's acceptance of the existence of true contradictions, or fuzzy logic's non-binary approach to truth), Polanyi's is intuitively appealing, at least to me.
Before outlining his solution, it's worth considering how the paradoxical sentence is typically evaluated. At first glance, it seems to be the same as any sentence: a correspondence between the meaning of the words and the facts. 'The birds outside are chirping' is evaluated as true or false through an empirical verification of the sentence's meaning, that is, by listening to the sounds outside and verifying if the sound of chirping birds is present. Similarly, 'this sentence is six words long' is evaluated by counting the number of words in the sentence referred to. Finally, 'this sentence is false', seems to involve a similar procedure, with one big difference. We look at the sentence referred to, namely, 'this sentence is false', and use the information therein to determine its truth or falsity. But unlike the previous two examples, 'this sentence is false' cannot be verified by listening or counting, or any other empirical verification. Instead, its verification lies in the meaning of the sentence. The sentence is about its own falsity, and is closer in form to a meta-sentence such as: 'the sentence 'the birds outside are chirping' is false'. It is an assertion of truth or falsity, rather than a mere assertion. 'p is false' or 'p is true' rather than simply 'p'.
Yet it stands to reason that if 'p' is true, then 'p is true' is true and 'p is false' is false. One would think that the truth-value of an assertion, and an assertion about that assertion, are the same. Only when p is itself a meta-assertion about its own falsity do we encounter paradox; if p is itself 'p is false'. This consideration should give us pause, as this is a highly specific kind of sentence, one not encountered in ordinary life. Could there be something meaningless about the sentence that allows us to dismiss the paradox? If we rely on the correspondence method of verifying sentences described above, then there is no room to dismiss it, as its correspondence with the truth is established through its meaning, and its meaning is inherently paradoxical.
Polanyi offers an alternative picture of sentences and their meaning. According to his epistemology as laid out in Personal Knowledge, we cannot apply the same method of establishing the truth of a sentence for assertions and meta-assertions. This is because 'p' is a sentence asserting a fact, while 'p is true' or 'p is false' are not actually sentences, but expressions of commitment. These expressions do not refer to facts; they are acts of endorsements. Polanyi claims that every sentence with a truth value contains two elements: the words conveying meaning, and the speaker's affirmation or dissent of that meaning (whether implied or explicit). So any sentence 'p' tacitly contains 'p is true' (or 'p is false' if p is said sarcastically, as a lie, etc.). The mistake we make when analyzing sentences is that we often forget about this tacit component - we exclusively analyze the first element, the meaning of the words, to see if it corresponds with the facts. This is natural, and even a helpful way of establishing truth, as we don't always want the speaker's personal commitments to impinge on our judgements. But Polanyi wants us to understand that there can be no sentence without this second, assertive element: "An unasserted sentence is no better than an unsigned cheque; just paper and ink without power or meaning" (PK 27).
Drawing on the work of Frege, he points out that if we were to formalize language, each statement should contain an assertion sign '⊢' to signify the personal commitment of the speaker. So ⊢ P would mean: 'I believe P is true'. And just as it would seem incomplete to hear a sentence that was just 'I believe', so too is it incomplete to just hear 'P': "Written down by itself the signpost symbol ⊢ conveys as little meaning as would a solitary question mark or exclamation mark, which are its nearest analogues among existing symbols. This incompleteness of the symbol has an important and perhaps not so readily acceptable correlate. It suggests that a declaratory sentence is by itself also an incomplete symbol" (27). This mimics how we interpret things in our ordinary life; we do not hear disembodied declarations coming out of the ether, but specific people saying things in a certain way, with a certain tone, nuance, or emphasis. Even in such an abstract endeavor as symbolic logic, like in Frege's Begriffsschrift or Russell and Whitehead's Principia Mathematica, the assertion sign ⊢ is used to flag that it is their own judgements being used. Even though they aim to convey universal logical truths, they still have to assert them, and they still might be wrong.
Thus for Polanyi, the Liar paradox is predicated on treating an incomplete symbol as a complete one. It is like evaluating an exclamation or question as true or false. Returning to the cheque metaphor, he claims, "we cannot assert the expression 'p is true', any more than we can endorse our own signature; only a sentence can be asserted, not an action" (254). From this reasoning Polanyi rejects the meaningfulness of the Liar paradox: "if ‘p is false’ merely declares that the speaker denies acceptance to p, then ‘p is false’ is not a sentence and the paradox does not arise" (255). For a sentence to be true or false, it must contain the two elements mentioned above: the objective meaning of the words, and a subjective commitment to them. There is no assertive sentence possible without this combination, and hence the Liar paradox is a product of an artificial creation; the transformation of an act of commitment into an objective statement. A meaningless string of words that gives the illusion of having meaning.
One might reasonably be troubled that a subjective element is necessarily involved in the determination of truth here. Polanyi's project in Personal Knowledge is to allay that objectivist impulse, and to show that having a purely objective standard for truth is an impossible ideal (254):
"Unless an assertion of fact is accompanied by some heuristic or persuasive feeling, it is a mere form of words saying nothing. Any attempt to eliminate this personal coefficient, by laying down precise rules for making or testing assertions of fact, is condemned to futility from the start. For we can derive rules of observation and verification only from examples of factual statements that we have accepted as true before we knew these rules; and in the end the application of our rules will necessarily fall back once more on factual observations, the acceptance of which is an act of personal judgment, unguided by any explicit rules. And besides, the application of such rules must rely all the time on the guidance of our own personal judgment. This argument formally confirms the participation of the speaker in any sincere statement of fact."
The paradox is a result of attempting to depersonalize all sentences, even assertions. There is no 'it is asserted', there is only 'I assert', 'they assert, or 'Polanyi asserts': "an ‘impersonal allegation’ is a contradiction in terms—just as an ‘anonymous cheque’ would be" (256). This does not imply a thoroughgoing relativism, however, as Polanyi's project ultimately aims to show that the personal participation we have in our knowing is not separable from the objective requirements we have to fulfill in order to claim having knowledge. I will not try to spell this out completely here, but I hope that this purported solution to the Liar paradox provides some inspiration to dig deeper into the epistemology that makes it possible.