Category: logic

The Sorites Paradox – Maybe It’s Not What We Think It is.

It has been asked how, if at all, one might resolve the Sorites paradox. I am not convinced a solution is possible, and in this paper I will explain the responses I have become aware of, and why they fail. In the end, I will conclude that there is no solution to the paradox, but I will offer a few suggestions for a way forward.

The first response might simply be to reject the first premise of the argument. In other words, simply deny that a man with 10 hairs is in fact bald, or that 100 grains of sand is in fact a heap. In essence, this would render vague predicates useless at best, meaningless at worst, since no predicate that allows for a vague border case would be permitted to apply to anything. There is one way in which we might stretch this into plausibility, but I will address the other responses first, before returning to this in the conclusion.

The second response is to set some arbitrary boundary. This means selecting one one among the indefinite number of secondary premises beyond which all others will be false. For example, we might say that thirty-thousand and one grains of sand is the boundary below which we no longer regard a collection of grains to be a heap. At first glance, this approach might seem plausible. After all, we do this frequently in practice: setting the legal drinking age, or the number of credit-hours required to count as a ‘full-time’ student, for example. However, there are two core problems with this. First, from the context of the formal argument, there is no good reason to reject any of the subsequent conditional statements, and there appears to be no means by which we could discover a reason. The implicit modus ponens of the conditional compels us to accept them all. Second, as Wright (Vagueness, 1997) pointed out, vague predicates are inherently coarse by virtue of their intended use. So attempts to impose some sort of specificity would destroy their meaning.

The next approach would be to attempt to define a knowledge gap within some middle range of propositions between the edge false and edge true statements. On one interpretation of the idea, we could use a three-value logic, in evaluating the propositions. At some point, starting with grain one, the proposition ‘this is a heap’ would cease being false, and would instead be valued ‘unknown’ or ‘undefined’. Later, the unknown state would transition to true, once we’ve reached the next threshold. This would make it possible to judge the argument invalid, since any number of its premises were neither true nor false. However, this seems to be attempting to win on a technicality, and it suffers from the same problem as the arbitrary boundary solution, in that we have no real way of determining when the states should change.

The next response might be some form of Edgington’s “degrees” of truth (Vagueness, 1997). But this suffers from it’s own serious flaws. For example, consider the statement ‘it is raining’. That has a ‘degree’ of truth of .5. It’s negation, ‘it is not raining’ will have a degree of truth of .5. The consequence of this, is that the following propositions have exactly the same truth value: ’It is raining, and it is raining’, ’It is raining, and it is not raining’. The same problem exists with our heap of sand. So, again, we’re left with no clear way to determine the truth of the conditionals in the Sorites case.

In the end, there does not seem to be any clear resolution to this paradox. However, I would offer one suggestion. vague predicates, in addition to being inherently coarse, also seem to be describing inherently subjective experiences or judgements. While Sorites arguments seem to want to talk about objective properties of objects. Perhaps there is no solution to the paradox, precisely because “tall” or “heap” or “red” or “bald” is not in fact a property of the object being considered, but a property of the experience the subject has of that object. The paradox is perhaps trying to square subjective interpretation with objective matters of fact. And that’s why it cannot be resolved.

Haack and Dummett on The Justification of Deduction

Susan Haack nicely diagrammed the problem of circularity in her 1976 paper, The Justification of Deduction. In that diagram, she drew a direct parallel to the circularity of the inductive justification of induction, as outlined originally by Hume. Haack argues that justification must mean syntactic justification, and offers an illustrative example argument to show why semantic justification fails – namely, that it is an axiomatic dogmatism: deduction is justified by virtue of the fact that we have defined it to be truth preserving.

Haack goes on to argue on syntactic grounds that justification is a non-starter on at least five other fronts, in addition to being circular. However, Dummett in his 1973 paper by the same name, showed that the only kind of justification that made any sense was semantic justification. First, because syntax necessarily relied on semantics for its meaning, and secondly because the whole point of justification in the first place is confidence in the function of logic as a means of preserving truth values.

Still, Dummett was able to show not only that the justification of deduction was circular, but also that any attempt to do so leads inevitably down one of the horns of Agrippa’s Trilemma. As Haack pointed out, we could simply assert the justification definitionally. But, in attempting to avoid this dogmatic horn, Dummett points out that we have only two other options: the regress horn, or the circularity horn. In the first case, this would mean crafting a set of rules of inference that could be used to independently justify deduction. These new rules would require a language and a theory of soundness and completeness all their own, which in turn, require justification, and then the process would descend yet another level. In the latter case, two different sets of rules of inference might be used to justify each other, in perpetuity. Obviously, none of these options is satisfying.

Later in his paper, Dummett attempts to explain how a set of inferential rules might be justified by reference to a theory of meaning for the object language within which it is contained. Essentially, he argues that the soundness and completeness theory of logic provides what a theory of meaning provides for a language: a functional understanding of its use. In other words, if we are to justify logic at all, we must first have a theory of meaning that shows how sentences can carry truth values. But this seems to me to begin the slide back into circularity, because as Dummett goes on to explain, our definitions of true and false themselves determine the means by which we achieve the meanings of sentences judged by those definitions. All we’ve done is to shrink the circle.

Haack and Dummett continue the debate in subsequent papers, but reach no conclusion. I am inclined to wonder, myself, whether any of it matters. The justification problem in induction has been evident for over three hundred years, and the problem of deduction for around seventy-five. Yet somehow, both of these tools of inference continue to be used and taught — and both still seem to be yielding results that most of us find satisfying most of the time.

In a word, yes, some forms of justification are circular (and it seems that no form of justification actually appears to work). But perhaps the problem isn’t what we think it is. Perhaps the process of inference is somehow more fundamental than language. Perhaps it is a feature of consciousness that resides below the level of language, rendering it impervious to notions like justification. Or, perhaps the justification of logic will someday come out of the neurological study of the brain, as an explanation of the evolutionary advantage of a linguistic mind, to a primate that would have otherwise perished on the plains of Africa.

Getting A Handle On The Truth

What is truth?” ~ Pontius Pilate

This is an interesting and surprisingly difficult question. If you look in the OED, what you’ll find there are entirely circular and self-referential explanations: “the quality or state of being true“, ” that which is true or in accordance with fact or reality“, and “a fact or belief that is accepted as true“.

So, the poor souls that rely on the dictionary are left with, essentially, “truth is what’s true”, and “what’s true is what we agree are the facts of reality.” But what if we’re wrong and we still agree? Or worse, what if we disagree, but one of us is right? This can’t be the last word on this topic. What can we say with any confidence about truth, as such? To put it in the words of Bertrand Russell:

“We may believe what is false as well as what is true. We know that on very many subjects different people hold different and incompatible opinions: hence some beliefs must be erroneous. Since erroneous beliefs are often held just as strongly as true beliefs, it becomes a difficult question how they are to be distinguished from true beliefs. How are we to know, in a given case, that our belief is not erroneous? This is a question of the very greatest difficulty, to which no completely satisfactory answer is possible. There is, however, a preliminary question which is rather less difficult, and that is: What do we mean by truth and falsehood?” — The Problems of Philosophy (p. 77)

Thinking on the question a bit, I realized I’m not quite sure what I mean. So, I decided to take a brief look at what what philosophy has had to say on the subject over the centuries, to see if I might find something I’m willing to accede to, at least in the short term.

As Russell is careful to point out in the book I just referenced, any real understanding of truth must start first with understanding what knowledge is. But even this is tricky. I wanted to simply stipulate to the classical definition, in order to shorten this post. But what we find in the traditional definition of knowledge, is yet another circular reference: knowledge is Justified True Belief. In other words, that which is known is that which satisfies all of the following three conditions:

  1. It is believed
  2. That belief is justified
  3. That belief is true

For the sake of brevity, I’ll let the Stanford encyclopedia explain these three conditions in detail, and I’ll set aside common objections to this formulation of knowledge for a later post. Nevertheless, in spite of Stanford’s assertion that “the truth condition is largely uncontroversial“, I think the fact that truth is present in the definition of knowledge is a serious problem for philosophy because it makes the two terms fundamentally dependent upon each other: truth is that which is known is that which is the truth.

As such, I find it hard to blame the dictionary for its circularity when it relies for its definitions on an academic discipline that can’t seem to provide a clear answer to this question. What’s more, I think it’s a little disingenuous for “serious” philosophers to scoff at Ayn Rand for her insistence on unjustified “axioms” like “Existence Exists“, or to laugh at Christians who, facing no real alternative, rely on Jesus’ pronouncement that actually it is he personally who is “…the way, the truth, and the life…” (John 14:6).

To be completely clear, my aim here is not to argue that there is no such thing as truth, or that we cannot know things or cannot justifiably claim to know the truth — or worse, that we should just throw our hands up and simply declare it to be whatever we want it to be. To do so, I’d have to employ the very tools of thought that I’d be condemning. All I am suggesting is that maybe we’re not as sure as we think we are, and that maybe we need to rethink some of these fundamental questions.

What Everyone Else Thinks

As one might expect, given what I have stated above, there are actually numerous philosophical theories of truth. The most popular among them, the “correspondence theory“, offers the greatest appeal to common sense. This theory is probably where the OED gets it’s turn of phrase “in accordance with fact or reality”. The theory states that “a proposition is true provided there exists a fact corresponding to it.” But what does “correspondence” mean? And what, exactly, are facts? Russell makes a lot of hay on this second question, in his own conception of correspondence. In short, this definition “works”, but it’s not entirely satisfying (as Russell notes in the above quote).

Some argue for something called “coherence“, in which each new statement is compared to a complete set of beliefs, and rejected if it does not “fit” within that collection. This theory seems to fail on two grounds: first, that it is not necessary for the collection of beliefs to have any relation to reality, and secondly, as Russell again points out, because of the first problem, there can be many equally “coherent” belief systems existing side by side. How do we know which one to choose? The problems point to a third problem, that I think also plagues the pragmatist, constructivist, and consensus theories of truth. Namely, that they all elevate mere belief to the ontological status of a fact, by virtue of some ex post facto rationale. What’s more, this equivocation seems to go unnoticed (or worse, dishonestly ignored) by the theories’ adherents.

What I think

I find Kant’s idea of the conjunction between the noumenal and phenomenal world somewhat compelling. Although, probably not for reasons Kant would approve. Science shows us that there is a reality that is outside the reach of the senses. Perhaps truth, then, is the extent to which we can apprehend these non-phenomenal parts of reality, and reconcile them with the phenomenal parts. Already, science has provided us with all sorts of tools for doing this (telescopes, microscopes, sensors, meters, etc.). If this is true (somewhat ironically), then the way to the truth is through scientific inquiry. This is certainly a different route to truth via science than the pragmatists propose, but I think the destination may be the same.

On the other hand, although I don’t quite understand his theory, Alfred Tarsky‘s emphasis on semantics got me to wondering.

I have heard truth described by some as a relationship between physical reality and conscious awareness. This is not quite the same thing as correspondence, because the focus here is not on the objects in the relation, but the relation itself. It’s an interesting idea, but I think this isn’t quite complete. Because, if conscious awareness of reality is all that is necessary for a “truth” relation, then beavers and ants and birds would be capable of apprehending the truth. Clearly, then, it must something more.

That difference is language. Truth is as much a semantic concept, as it is a metaphysical one. Like knowledge, the definition of truth is concerned with the objects of mind and reality, and primarily with the nature of the relationship between them. But what is it about the nature of this relation, that makes it truth? I think it is the meaning we assign to that relationship, and the value discovered in the contents of that relationship.

In short, truth is a kind of semantic value judgment of the perception of reality as it is apprehended, by a mind capable of apprehending and valuing. But what does this mean, in practice? Is this just another way of formulating correspondence? Not quite. Is it the same as claiming that the truth is whatever we want it to be? Not quite. Is it pragmatism in another suit of clothes? I don’t think so.

But I’m struggling to find the words necessary to develop the idea any further. And perhaps that’s a clue to the problem with all of these theories. Maybe the problem lies precisely with the fact that our language is woefully lacking, when it comes to the task of describing these sorts of relationships. This is why I am beginning to wonder if we don’t need a new language, or a new way of thinking, or of describing our thoughts, before we can properly answer this question.

Knowledge, Certainty, And Logic

The Epistemic Regress (specifically, the Skeptical variety) is a little out of my depth at the moment, but what is plainly obvious by various presentations of the problem, is that at it’s core lies the Problem of Knowledge. The key question that arises in the examination of major premises in any deductive argument, is “how do you know?” This suggests that something essential about the nature of the premises needs to be discovered, before we are going to solve the riddle.

Perhaps the root of the question actually lies in an unconscious equivocation of analytic and synthetic statements, when we ask it? The latter being knowledge derived from sense perception, the former from “pure reason” (as Kant might have put it). To that end, some suggest that we probably need to revisit the classic problem of Cartesian skepticism yet again. This paper from someone at the University of Alabama discusses a theory called “Foundationalism“, which despite the numerous objections to it, seems somewhat appealing.

However, I think the problem lies precisely in the form of logic itself. It is a tool designed around a positive conception of knowledge; one that presumes that certainty is reasonable and achievable as a standard of knowledge, and requires assertions that are absolute. There’s even a term for it: “Justified True Belief“, in which absolute certainty is the gold standard defining what “knowledge” really is. A view that drove Descartes to his maxim, Cogito Ergo Sum.

But I take my view more from Karl Popper, than from René Descartes: the regress exists, because the tool we’re using and the thing we’re trying to achieve with it are incompatible. We need a new form of critical reasoning, and a new conception of knowledge, that is capable of coping with degrees of uncertainty, and degrees of probability.

Traditional deductive logic (and even some forms of induction) rely too much on a conception of knowledge that demands of its users something that seems, upon very close inspection, to not exist and to not even be possible. We need to get out of the classical playpen of Aristotle and Plato, and grow up a little. What that will look like, is a bit beyond me right now. But maybe someone, somewhere has already beat me to the punch. I hope so. Maybe tentative uncertainty is the most anyone can hope for.