Brandon Moores
Good afternoon. In the time I have, I would like to focus on a particular problem discussed in Dr. Müller's paper, one familiar, I'm sure, to almost everyone here: the liar paradox. As you all know, Dr. Müller suggests that the paradox is unable to communicate meaning in the sense of something "trustable" about our perceived reality. I would like to dispute this claim, and defend the idea that, despite the fact that the statement is logically false, we may still be able to think our way around it, and in the end use it to uncover the laws on which our reality - whether an invention of the mind or independent of it - is based.
The paradox is so often discussed that I at first passed over it, and might have left it entirely, but for a whimsical turn of my memory. I recalled, as I was reading the paper over, a line from Milton's Paradise Lost (I later looked it up, and found it in Book I, lines 335-336): "Nor did they not perceive the evil plight/ In which they were…" This line was taken up most famously by Stanley Fish in his book Surprising Sin, where he uses it as a cornerstone of his attempt to build a reader-response model of literature. I will be working in another direction, but I will start from the same general observation as Fish: that a double negative is not the same thing as an affirmation, though the two may share the same logical result.
Milton's line suggests something we already intuitively recognize: that the concepts of affirmation and negation point to something more than a simple judgement of truth or falsity. The language of logic is not a precise analog to the language of these concepts. I would like to emphasize that not only are the words "truth" and "lie" or "falsity"* used differently in language and in logic; they refer to quite distinct things. We have already seen that in logic, the signs can cancel each other out quite efficiently; this suggests that each sign can "eclipse" the other perfectly, that they are, as it were, two sides of the logical coin. (Though, interestingly, there are problems even with this basic metaphor: positive and negative are not exact opposites of each other; otherwise, how could we explain the fact that a double negative makes a positive, but a double positive does not make a negative? Perhaps a more fruitful method is to think of a positive statement as a stable one, and of a negation as a sign of change; but even here, can we not imagine a system of logic where the negative is conceived as a "ground state", and it is the positive that brings about change? Is the entirety of logic simply a system for the application of these two apparently interchangeable predicates? In which case, we return to our first problem). This is obviously not the case in language. There, true and false exist side by side; they do not cancel each other out; indeed, a truth can be used to support a lie, and a lie a truth. They are, on the surface, identical statements, with the simple distinction that one of them happens to be the case, and the other does not. "Truth" and "lie", used in this original sense, are less opposites than brothers. It is precisely because they are so closely related that we are capable of enjoying fiction, and find it so easy to suspend our disbelief. (One may even go so far as to say that "truth" and "lie" are foreign or peripheral to regular linguistic statements; though, for practical reasons, they are often considered "true").
When applied to the liar paradox, we find that treating the problem simply in terms of logic unnecessarily limits the debate. Many things are suggested by the statement "This sentence is a lie": the problem of containment (can a sentence contain/refer to itself this way? What rules govern such interactions? How analogous is it to the problem of sets?); the problem of satisfaction or completion (should this be interpreted as a "closed" or an "open" statement? It seems complete in itself, yet also seems to demand countless iterations); finally, the problem of oscillation between two perspectives, each of which directs us to the other; together, the two seem to make a bell-like hum that, despite its inability to rest in any one state, nonetheless suggests a certain meaning.
We have seen from Milton's line that a double negation can communicate more than a positive state of affairs. Likewise, a lie can have positive meaning: think of the famous puzzle where one is presented with two personages, one of whom always tells the truth, and one of whom always lies, and asked to deduce a truth from a single question - without knowing which is the liar and which is the truth-teller. The answer, of course, is to involve the liar in the question by referring to both, and then conclude the opposite of the answer given. Note that the answer is always a lie; yet it communicates a truth. "This sentence is a lie" is, indeed, a lie no matter which way one turns it; but that does not make it meaningless, or reduce prima facie its ability to communicate.
Even, and perhaps especially, within a working-ontology system, then - one where statements have value or meaning depending on their "trustability", their ability to comment on our experience - the liar paradox may contain a meaning, albeit a well-defended one. Logically, it is barren; conceptually, it is rich. We are fortunate that we can approach it through (at least) these two avenues; were we confined to one or the other, we could indeed make little headway. By approaching it creatively, rather than deductively, we may well be able to squeeze from it important insights about the structure of our world. It is this hope that has kept the paradox alive for 2500 years; to understand something that always hovers at the edge of reason: the relation between language and logic.
* These latter two words, of course, imply something subtly different from simple negation, which can be and often is true (i.e. it is true that I am not currently on the moon). But my point is precisely that they are not commensurable with logic; that the action of saying "I am lying" and the logical operation of negating a statement do not correspond so closely as they seem.