I. Prologue


There are only a few papers arguing a solution for the Sorites Paradox based on Wittgenstein’s Philosophical Investigations (“PI”) and related writings. This is understandable for two reasons: 1) Wittgenstein never mentions the Sorites though he must have known about it because he spends considerable effort contemplating vagueness and the problems it poses for constructing any systematic theory of meaning; 2) much of the PI with its emphasis on the meaning of a word being “its use” seems to rule out the possibility of any systematic theory of meaning to instead seemingly reduce the determination of meaning to a relativistic contextual methodology that further seems to make natural language immune from criticism of its normative value for truth which is not a desired result in modern or present analytic philosophy of language. Regarding (1), its absence from the PI seems to be intentional. Wittgenstein’s PI is more concerned with showing vagueness to be more than just an issue of the degree or incremental tolerance of a Sorites, instead it argues all words are in some way vague — vagueness is not limited to quantitative or incremental degrees of tolerance — because the use and usefulness of all words are inter-dependent on the use and usefulness of all other words in the language. Because of (2), the majority of Wittgensteinian solutions to the Sorites Paradox admittedly do not solve it but say they have “dissolved” it. That is, they argue the first premise F(αn) of any Sorites argument is not a sound assumption because it defines its value independently of its context and of its interdependence with all other words in the language. Once this misunderstanding is removed, the argument goes, there is no longer a Sorites. This dissolution further argues this same misunderstanding occurs with all of the further Sorites premises and even for the conclusion: they make an unsound assumption that there exists a non-contextual degree tolerance that does not change the truth value of vague terms. The argument given as to why the Sorites premises appear sound is that we normatively criticize vagueness in language yet all words are in some way vague. Our semantics wants meaning to be context independent but the pragmatics of language wants it context-bound.


I have also come across one argument based on the PI that does not attack the premises but instead argues the reasoning in any degree tolerance Sorites to be invalid. It is argued in A Wittgensteinian Solution to the Sorites: where vagueness is based on incremental or degree tolerance, the validity of a series of valid arguments is not transitive to the validity of the final conclusion. Thus, Sorites is not a paradox but simply invalid argument.


I will argue that a Wittgensteinian dissolution or solution of Sorites does not rule out the possibility of any systematic theory of meaning. Going beyond a surface understanding of the PI, though it is undisputed Wittgenstein saw all wordgames as equal languages, he saw logic and perhaps even mathematics as different in that they were necessary for any and all language wordgames. In particular, he saw logic as a normative criticism of vagueness. Just as the lack of any rule in tennis for how high a ball can go does not make a systematic game of tennis impossible but simply hints at the need for further rules of tennis, the presence of vague words does not make a systematic theory of meaning impossible. For such a system, as Galileo did with his ignoring of friction in his system, philosophers of language would simply ignore vagueness — though a better description from a Wittgensteinian perspective would be that there is nothing to ignore. For the PI, vagueness is not a problem in language only in the philosopher’s need to talk about language and to normatively criticize it to make it less vague. I will begin with a short background history of the work by Frege, Russell, and the early Wittgenstein that led to the PI contemplation on vagueness but only in so far as it would be relevant to the Sorites; then a statement of the Sorites Paradox and the problems it presents; and end with argued Wittgensteinian solutions to it; and then my analysis and conclusion.


II. Historical Background


Wittgenstein’s work was responsive or founded upon the work of Frege and Russell.


A. Gottlob Frege


Frege brought into the philosophy the distinction between sense and reference and a truth-functional view of language. His work had a normative Platonic goal: “the independence of logic from the mind … logic, he claimed, codifies the laws of truth, not of thought, the laws of how we ought to think if we wish to aim at the truth, not of how we do actually think”. Frege’s work had a distinction between objects and concepts (which predicate on objects) as absolute. Objects name and pick out referents. Concepts name and pick out properties of referents. Objects can be described, defined, or predicated about. Concepts cannot. They and their nature can only be hinted at and used but not discussed. In Frege’s terminology, objects are “saturated”: complete unto themselves (such as a proper name). Concepts are “unsaturated”: incomplete (“__ is female”). In standard grammatical terms, objects are subjects, concepts are predicates. An object and a concept together make a proposition although for Frege this is the wrong way to look at it. Rather, given his normative goal, it is never the case that propositions are derived from objects and concepts, but only that objects and concepts are logically derived from propositions and the meaning of a word can only be defined within the context of a proposition. Therefore, for Frege, ordinary language could not function as a means for truth because of the presence of vagueness:

A definition of a concept (of a possible predicate) must be complete; it must unambiguously determine, as regards any object, whether or not it falls under the concept (whether or not the predicate is truly ascribable to it). Thus there must not be any object as regards which the definition leaves in doubt whether it falls under the concept; though for us men, with our defective knowledge, the question may not always be decidable. We may express this metaphorically as follows: the concept must have a sharp boundary. If we represent concepts in extension by areas on a plane, this is admittedly a picture that may be used only with caution, but here it can do us good service. To a concept without a sharp boundary there would correspond an area that had not a sharp boundary-line all around, but in places just vaguely faded away into the background. This would not really be an area at all; and likewise a concept that is not sharply defined is wrongly termed a concept. Such quasi-conceptual constructions cannot be recognized as concepts by logic; it is impossible to lay down precise laws for them. The law of excluded middle is really just another form of the requirement that the concept should have a sharp boundary. Any object A that you choose to take either falls under the concept or does not fall under it; tertium non datur. E.g., would the sentence ‘any square root of 9 is odd’ have a comprehensible sense at all if square root of 9 were not a concept with a sharp boundary? Has the question ‘Are we still Christians?’ really got a sense, if it is indeterminate whom the predicate ‘Christian’ can truly be ascribed to, and who must be refused it?


Wittgenstein in the PI directly criticizes Frege for his view on vagueness:


… Frege compares a concept to an area and says that an area with vague boundaries cannot be called an area at all. This presumably means that we cannot do anything with it. — But is it senseless to say: “Stand roughly there”? Suppose that I were standing with someone in a city square and said that. As I say it I do not draw any kind of boundary, but perhaps point with my hand — as if I were indicating a particular spot. And this is just how one might explain to someone what a game is. One gives examples and intends them to be taken in a particular way. — I do not, however, mean by this that he is supposed to see in those examples that common thing which I — for some reason — was unable to express; but that he is now to employ those examples in a particular way. Here giving examples is not an indirect means of explaining — in default of a better. For any general definition can be misunderstood too. The point is that this is how we play the game. (I mean the language-game with the word “game”.)

(However, it is interesting to note as I will do later that Frege’s view of the Sorites closely approaches what some argue is Wittgenstein’s approach to it.)


In a letter to Giuseppe Peano and in his personal notes, Frege refers to the Sorites as a fallacy whose mistake in reasoning is in its first premise:

The fallacy known as the ‘Sorites’ depends on something (e.g., a heap) being treated as a concept which cannot be acknowledged as such by logic because it is not properly circumscribed.

The fallacious reasoning is:

… that we treat our vague concepts as if they were sharp without realizing that this is what we are doing, what we take ourselves to be doing, on reflection, is using our vague concepts as vague — hence the unproblematic appearance of the Tolerance Principle —, whereas, in fact, it is necessary for our using them in the way that we want to in practice that they be sharp.


B. Bertrand Russell


Russell’s On Denoting published in 1905 argues against Frege’s sense/reference distinction but ultimately agrees with Frege’s Platonic goal by proposing a Theory of Descriptions arguing that the propositions of ordinary natural language can be rewritten as truth-functional propositions in an ideal language in which the true logical form of propositions represent the simple part of reality and thus the language will ontologically represent the world as it really is. Furthermore, In 1923, he wrote Vagueness in which he argued that natural language as a system, if there is one, and all its component parts are vague with vagueness defined so that it applies to both qualitative and quantitative words including proper names and numerals:

All words describing sensible qualities have the same kind of vagueness which belongs to the word ‘red’. This vagueness exists also, though in a less degree in the quantitative words which science has tried hardest to make precise, such as metre or a second. I am not going to invoke Einstein for the purpose of making these words vague. The metre, for example, is defined as the distance between two marks on a certain rod in Paris, when that rod is at a certain temperature. Now the marks are not points, but patches of a finite size, so that the distance between them is not a precise conception.

It would seem natural to suppose that the name was not attributable before birth; if so, there was doubt, while birth was taking place, whether the name was attributable or not.
If it be said that the name was attributable before birth, the ambiguity is even more obvious, since no one can decide how long before birth the name became attributable.
Death also is a process; even when it is what is called instantaneous, death must occupy a finite time. If you continue to apply the name of the corpse, there must gradually come a stage in decomposition when the name ceases to be attributable, but no one can say precisely when this stage has been reached. The fact is that all words are attributable
without doubt over a certain area, but become questionable within a penumbra, outside of which they are again certainly not attributable.

Now ‘true’ and ‘false’ can only have a meaning when the precise symbols employed – words, perceptions, images, or what not – are themselves precise. We have seen that, in practice, this is not the case. It follows that every proposition that can be framed in practice has a certain degree of vagueness; that is to say, there is not one definite fact
necessary and sufficient for its truth, but a certain region of possible facts, any one of which would make it true. And this region is itself ill defined: we cannot assign to it a definite boundary . . . . Logical words like the rest, when used by human beings share the vagueness of all other words.

… a representation is vague … when the relation of the representing system to the
represented system is not one-one but one-many.

In an accurate language he explains meaning would be a one-one relation; no word would have two meanings, no two words have the same meaning. In actual languages, as we have seen, meaning is one-many. (It happens often that two words have the same meaning, but this is easily avoided, and can be assumed not to happen without injuring the argument.) The fact that meaning is a one-many relation is the precise statement of the fact that all language is more or less vague.



Though I could not find a commentary by Russell on the Sorites, it is clear he would agree with Frege that a logical language cannot have vague predicates. Unlike the “one-many” relation:

Passing from representation in general to the kinds of representation that are specially interesting to the logician, the representing system will consist of words, perceptions, thoughts, or something of the kind, and the would-be one-one relation between the representing system and the represented system will be meaning.


C. Wittgenstein’s Tractatus Logico-Philosophicus


Unlike Frege and Russell’s normative goals, Wittgenstein’s Tractatus (TLP) argues the limits of language for expressing thought. “The book will … draw a limit to thinking, or rather—not to thinking, but to the expression of thoughts …. The limit can … only be drawn in language and what lies on the other side of the limit will be simply nonsense”. In the TLP, language is a picture of states of affairs or objects in the world held together by proper logical form. Thus, for any proposition in language to have meaning, it must have truth values resting on the possibility of its being a representation or picturing of the world that exists outside of language while being held together by proper logical form. For the required logical form, the TLP introduced truth-table definitions for Frege and Russell’s propositional connectives and defined the expression “truth condition” by explicit reference to those truth tables. However this creates a dilemma, since the required logical form for language is itself inside of language, it is not possible to discuss them as if they exist outside of language — despite the fact he had been doing so for the entire TLP. Because the propositions of logic itself do not represent states of affairs and the logical constants do not stand for objects in the world, they cannot have meaning in the world. “My fundamental thought is that the logical constants do not represent. That the logic of the facts cannot be represented”.


The TLP does not contemplate vagueness. Its picture theory of language seems incapable of accepting the existence of vague objects or vague language. How can we have a picture with no border? Therefore like Frege and Russell, the Sorites for the TLP would be put into its fallacy or nonsense category as would any purely logical sentence whose truth or falsehood is recognized “in the symbol alone … and this fact contains in itself the whole philosophy of logic”.


D. Wittgenstein’s Philosophical Investigations


Wittgenstein contemplates vagueness or its opposite absolute sharpness directly in §§38, 65-88, 104-09, and 126-28 of the PI though given the holistic nature of the PI it would be a mistake to consider these in isolation to be its views on vagueness. There is no reference to the Sorites, instead the PI gives and contemplates — like Russell in Vagueness — examples of vagueness that go beyond the degree tolerance vagueness of the Sorites. This seems to be intentional: “A main cause of philosophical disease—a one-sided diet: one nourishes one’s thinking with only one kind of example.” Some examples of vague words contemplated in these sections of the PI include “that”, “game”, “number”, “color”, “pure”, “shape”, “this”, “good”, “Moses” (proper names), “exact”, “inexact”, “exactness”, “vague”, and more. In considering Wittgenstein’s view on vagueness, his view on ‘meaning’ in the PI should be kept in mind. “When I think in words, I don’t have ‘meanings’ in my mind in addition to the verbal expressions, language itself is the vehicle of thought”. “It is no more essential to the understanding of a sentence that one should imagine something in connection with it than that one should make a sketch of it”. Further in the PI, it states:

510. Make the following experiment: say “It’s cold here” and mean “It’s warm here”. Can you do it?—And what are you doing as you do it? And is there only one way of doing it?
511. What does “discovering that an expression doesn’t make sense” mean?—and what does it mean to say: “If I mean something by it, surely it must make sense”? — If I mean something by it? — If I mean what by it?! — One wants to say: a significant sentence is one which one cannot merely say, but also think.
512. It looks as if we could say: “Word-language allows of senseless combinations of words, but the language of imagining does not allow us to imagine anything senseless.” — Hence, too, the language of drawing doesn’t allow of senseless drawings? Suppose they were drawings from which bodies were supposed to be modeled. In this case some drawings make sense, some not. — What if I imagine senseless combinations of words?


Relative to the Sorites, the most relevant PI arguments are §§65, 69, & 88:

65. Here we come up against the great question that lies behind all these considerations. — For someone might object against me: “You take the easy way out! You talk about all sorts of language games, but have nowhere said what the essence of a language-game, and hence of language, is: what is common to all these activities, and what makes them into language or parts of language. So you let yourself off the very part of the investigation that once gave you yourself most headache, the part about the general form of propositions and of language.”
And this is true. — Instead of producing something common to all that we call language, I am saying that these phenomena have no one thing in common which makes us use the same word for all, — but that they are related to one another in many different ways. And it is because of this relationship, or these relationships, that we call them all “language”. I will try to explain this.

69. How should we explain to someone what a game is? I imagine that we should describe games to him, and we might add: “This and similar things are called ‘games’ “. And do we know any more about it ourselves? Is it only other people whom we cannot tell exactly what a game is?—But this is not ignorance. We do not know the boundaries
because none have been drawn. To repeat, we can draw a boundary — for a special purpose. Does it take that to make the concept usable? Not at all (Except for that special purpose.) No more than it took the definition: I pace = 75 cm to make the measure of length ‘one pace’ usable. And if you want to say “But still, before that it wasn’t an exact measure”, then I reply: very well, it was an inexact one. — Though you still owe me a definition of exactness.

88. If I tell someone “Stand roughly here”—may not this explanation work perfectly? And cannot every other one fail too? But isn’t it an inexact explanation?—Yes; why shouldn’t we call it “inexact”? Only let us understand what “inexact” means. For it does not mean “unusable”. And let us consider what we call an “exact” explanation in contrast with this one. Perhaps something like drawing a chalk line round an area? Here it strikes us at once that the line has breadth. So a colour-edge would be more exact. But has this exactness still got a function here: isn’t the engine idling? And remember too that we have not yet defined what is to count as overstepping this exact boundary; how, with what instruments, it is to be established. And so on.
We understand what it means to set a pocket watch to the exact time or to regulate it to be exact. But what if it were asked: is this exactness ideal exactness, or how nearly does it approach the ideal?—Of course, we can speak of measurements of time in which there is a different, and as we should say a greater, exactness than in the measurement of time by a pocket-watch; in which the words “to set the clock to the exact time” have a different, though related meaning, and ‘to tell the time’ is a different process and so on. — Now, if I tell someone: “You should come to dinner more punctually; you know it begins at one o’clock exactly”—is there really no question of exactness here? Because it is possible to say: “Think of the determination of time in the laboratory or the observatory; there you see what ‘exactness’ means”?
“Inexact” is really a reproach, and “exact” is praise. And that is to say that what is inexact attains its goal less perfectly than what is more exact. Thus the point here is what we call “the goal”. Am I inexact when I do not give our distance from the sun to the nearest foot, or tell a joiner the width of a table to the nearest thousandth of an inch?
No single ideal of exactness has been laid down; we do not know what we should be supposed to imagine under this head—unless you yourself lay down what is to be so called. But you will find it difficult to hit upon such a convention; at least any that satisfies you.


III. The Sorites Paradox



The premises in the classic Sorites argument are as follows:

1a: My first premise is that a collection of 1000 grains is a heap. The specific number of grains assumed sufficient for a collection of grains to be a heap is immaterial, we just need to pick a number that in natural language would be undisputedly called a heap.
1b: My second premise is that if a single grain is removed from a heap, the remaining collection of grains is still a heap. By definition, vague expressions are insensitive or tolerant to small differences in degree.
2a: By modus ponens, if a collection of 1000 grains is a heap, a collection of 999 grains is a heap.
2b: If a collection of 999 grains is a heap, by universal instantiation so is a collection of 998 grains.
. . . ****
999b: If a collection of 2 grains is a heap, so is a single grain.
1000b: A single grain is a heap.


1c: A single grain is not a heap by definition in natural language.

Despite sound premises and valid argument, we have ended with a contradiction. Thus the above argument presents a paradox. To resolve the paradox, we must determine what premises and argument are either unsound, false, or invalid. Further, if there is a resolution, the question remains of whether a given resolution will allow for a systematic theory of meaning.


IV. Wittgensteinian Dissolution or Solution of the Sorites


There are two types of Wittgensteinian resolutions of the Sorites. One would be a dissolution that does not get into its reasoning. The other would attack the reasoning as invalid.


A. Dissolution


The most common Wittgensteinian resolution of the Sorites would begin by attacking the first two premises — an attack that would even apply to the final premise that is the conclusion if one wanted — as not sound and thus never get into the reasoning process thus dissolving the paradox. The argument would be that a “heap” cannot be defined by a propositional account of speaker meaning: a proposition defined as an abstract, mind-and-language-independent entity H(αn) cannot be given a truth value and there are no quantitative degrees of tolerance independent of its use for whatever truth-value there is. The truth value of (1a) is completely context dependent. For Wittgenstein, the meaning of a word is not simply its use as is so often quoted but also its existential value (intentional or pragmatic value for the speaker) or usefulness to that use. “And this can be expressed like this: I use the name “N” without a fixed meaning. (But that detracts as little from its usefulness, as it detracts from that of a table that it stands on four legs instead of three and so sometimes wobbles.)”


As Gricean or speech-act theory brings out, language is not simply a system of symbols that represent things in the world but a process of interaction between speakers that includes and also represents their intentions. Thus, Gricean semantics through concepts such as conversational implicature and then penumbral shift tries to differentiate between the pragmatics of language that is its use but that cannot be given truth-conditions and its strictly semantic meaning that can. The PI agrees with this differentiation conceptually but denies this use and usefulness (pragmatics) can be differentiated for any given proposition or even for any sentence or statement. Like Quine’s holistic fabric for epistemology, the PI sees the meaning of any word to be holistically tied to all other words in any given language. “And it is because of this relationship, or these relationships, that we call them all “language”.”
The meaning of a word is its use. What is it used for: to satisfy our intentions for its use. “When I think in words, I don’t have ‘meanings’ in my mind in addition to the verbal expressions, language itself is the vehicle of thought.” “Pictures and recordings stand for things by possessing certain properties of the original itself. … Sentences are not like this. They do not stand for things in virtue of possessing properties of the original; they do not stand for anything.”


In the PI, a “heap” is vague in the same way all words are vague including the word “vague”, “degrees”, “tolerance”, “exact”, and even “true”: there is no one thing that is their meaning, thus there is no one thing that defines them. (Once the definition of truth-conditions as defined by the TLP is abandoned as it must be in order to link the logical use of truth conditions to anything outside of logic as the TLP was unable to do, what does “true” mean that is different from “accurate”, “right”, “exact”, “clear”, “well-founded”, “certain”, “definite”, and many other words used synonymously for “true”.) For example, if the thousand grains of (1a) were compressed into a ball, would they still be a heap? No. If the thousand grains were spread out all over the floor? All over the floor except for a stack of grains in the middle of the floor? If the grains were all piled one on top of each other so as to be one tall tower of grains one grain wide and each floor one grain high? None of these would be a heap regardless of their still containing 1000 grains. Instead of subtracting, if we add grains, at what point does the heap become a hill of grain and the hill a mountain of grain? At what point does something become a grain so that it can be counted as 1 of 1000 grains? Is a sugar cube a grain or a heap of grains? The same would be true for premises (1b) and the conclusion premise (1c). Subtracting nothing from the heap but only changing its shape or form such as by spreading it out all over the floor would mean that the same 1000 grains would not be a heap without subtracting even one grain. Just one grain that is big enough or having only one grain when many more were expected could mean a heap of trouble — this use of “heap” though arguably has a different meaning not solely because of vagueness but based also on the use of the word. This can go on almost indefinitely.


Like Frege, this Wittgensteinian dissolution of the Sorites argues “[i]t only looks like there is the paradox of the heap so long as we — unawarely (and absurdly) — hover, by wanting the term ‘heap’ to be both context-bound and context independent in its use.” However, this dissolution by the PI does not see the paradox as a fallacy nor as any realist, anti-realist, quietist, supervaluationist, or contextualist theory for its resolution but as no theory at all:

Vagueness does not imply nonexistence, nor need it imply that the existence of the vague object only comes at the cost of being wholly and objectionably dependent upon us. … But games and languages and heaps and baldness are perfectly real. Thus, I take here neither an ‘epistemic’ view of vagueness nor a ‘metaphysical’ view of it. The idea of such views (or of a spectrum between Realism and Anti-Realism on which we can locate views of vagueness) falls away as an unnecessary and unhelpful abstraction …

So: Nor again have I not put forward a ‘Quietist’ view or theory. For I have not put forward any theory, nor even any view, at all. I have simply, one might say, returned us to our actual practices with words like “vague,” “heap,” “bald,” etc.


Those who argue this dissolution of the Sorites argue that this dissolution is not any type of systematic theory because treating meaning as use and usefulness rules out any descriptive systematic theory of meaning. In the end, the PI is left with the same dilemma as the TLP: it has spent a considerable amount of effort describing in language what it concludes cannot be described by language but for different reasons. This explains its unusual writing style at least. Just as with the TLP, Wittgenstein was trying to write about something he concluded could not be written about, of which he should be silent, but could not because it was important.


B. Resolution


In A Wittgensteinian Solution to the Sorites, Hanoch Ben-Yami makes the only argument that I have found attacking the reasoning of the Sorites based on the PI. He starts out by making a distinction between “degree vagueness” and other types of vagueness such as “combinatory” vagueness. In the former, there exists a borderline “notion of a degree of change too small to make any difference”. In the latter, there exists some other indeterminancy such as Frege’s reference to “Christianity” and religion or to “number” in the PI.


His argument for resolution is that though normally a series of valid arguments is transitive and thus their conclusion would be valid, however where vagueness exists based on incremental or degree tolerance such as in a Sorites in which we know the premises and argument predication at some point “pass vague boundaries of concepts”, they are not transitive to the conclusion. “We thought that this is impossible because our picture of the properties of valid arguments was largely derived – as so much else in philosophy – from mathematics, and mathematical arguments do not employ degree-vague concepts.” Thus, according to Ben-Yami, the Sorites is not a paradox but invalid argument. “We make the mistake in thinking that if there is no fault in any single step, there cannot be any in the one-thousand-steps argument either”. Thus, according to this argument, the validity of a series of valid arguments involving predicates with degree vagueness is not transitive to the validity of the final conclusion.


He argues this resolution does not violate classical logic and its concept of validity: if the premises of an argument are true than by necessity its conclusion is true. For any one argument or short arguments in the Sorites series, the premises are true and the conclusion is true consistent with classical logic. For any long Sorites series and for the entire Sorites, the premises are true and the conclusion false, so it is invalid consistent with classical logic. He does admit and it is obvious his resolution has “some implications for formal validity”. “[I]f we limit ourselves to arguments that do not use degree vague concepts, we can still maintain that all possible arguments of the Sorites form are valid, and therefore formally valid. But we cannot maintain unconditionally that arguments of this form are valid.”


The PI would have no syntax problem with his creation of what is essentially a new rule of logic in order to resolve the Sorites but there is a problem with the semantics. The problem with his resolution is with “middle-sized” Sorites arguments in which the premises are true. Are their conclusions true or false? According to Ben-Yami, in middle-sized Sorites arguments the conclusion is “indeterminate” and thus classical validity is maintained. This seems to be a sleight-of-hand in which “vagueness” is replaced by “indeterminacy”. The middle-sized Sorites are the borderline cases that create vagueness, and we are right back to where we started.


However, even Frege accepted that some propositions may be indeterminate.


V.  Analysis / Conclusion



Ben-Yami’s attempt at resolution of the Sorites is unique. There is no epistemic or metaphysical problem with his argument, and I have seen similar arguments presented to explain why the reasoning of the Gambler’s Fallacy is not always a fallacy. (If a coin comes up heads ten times in a row, a gambler is not justified in believing and it is epistemically irrational to believe it is now more likely to come up tails the eleventh time. However that same reasoning based on a series leads to the justified belief that the coin is loaded to favor heads). Eliminating the vagueness problem of the Sorites by replacing it with an indeterminacy problem seems to transfer the problem of truth-conditional semantics from the proposition to the sentence (as Quine argued should be done). For example, “all penguin-like animals are birds” could be definitely true or false even if “penguin-like” is vague. Worse, he could be abandoning bivalence for three-value logic. I simply do not know enough philosophy of logic to made an analysis of whether this argued resolution of the Sorites resolves any problems the Sorites presents for a systematic theory of meaning but such analysis is ultimately unnecessary since this Sorites solution is not really based on the PI which is the topic of this essay. Ben-Yami claims his resolution is based on the PI, in particular §471 (“It is difficult to begin at the beginning. And not try to go further back.”) in addition to the vagueness contemplations that I previously cited (§§65-71, 76-77, 79-80, 88). However, it seems the PI was more of an inspiration for his solution than a basis for it. He does not need the PI to make his argument. The same argument that the middle-sized Sorites arguments are the source of all the trouble can be and is made by those who argue the dissolution of the Sorites:

In sum: For our purposes, which vary, there will be points at which one member falls under a predicate and the next does not, even if those members are, in isolation, pairwise-indiscernible. (That is, of course, leaving aside what we will often do in such cases: which is simply and happily to speak of a ‘gray area,’ a borderland, a zone where the case in question is neither one thing nor the other.) Thus there will be—unstable, of course—sharp boundaries drawable for most vague predicates, without this in the slightest contradicting the ‘nature’ of vagueness.


Ultimately, for the PI, logic and mathematics are just wordgames that are interdependent as are all other wordgames on each other. For all wordgames, meaning is the use and usefulness of the words including the word “truth” and thus with such a deflationary concept of truth, a descriptive systematic theory of meaning and of language in which meaning is defined by truth-conditions and a propositional account of speaker meaning is not possible.


However, if philosophers were not bothered by having their work be on the same level as other wordgames, the PI does allow for a normative systematic theory of meaning serving to criticize vagueness in order to make language more precise or as precise as possible. “‘Inexact’ is really a reproach, and ‘exact’ is praise. And that is to say that what is inexact attains its goal less perfectly than what is more exact.” — §88 of the PI. I do not mean “normative” in the sense of ultimate meaning, but in the sense of Frege’s original goal I cited in the Prologue of an ideal language: “the independence of logic from the mind … logic, he claimed, codifies the laws of truth, not of thought, the laws of how we ought to think if we wish to aim at the truth, not of how we do actually think” — or at least to avoid the “claptrap” of postmodernism.


And it is worth remarking that Wittgenstein rightly places centrally in his considerations a point about ‘vagueness’ almost universally neglected in the philosophical literature: that “vague” is usually intended, in actual usage, as a term of criticism. — Read, Rupert. A Wittgensteinian Way With Paradoxes. Lexington Books: Lanham, Maryland (2013) p. 131.


Other commentators argue the PI allows for a “critical ‘philosophy of language’” dealing with:

… whether something is a symbol depends on how it is used, that the meaning of an expression is what is explained in explaining correctly how it is to be used, that understanding is an ability not a mental process, that the explanation of a word has the status of a rule or standard of correctness, and that whether a sentence formulates a rule depends on how it is employed. — Hacker, P.M.S.; Baker, G.P. Language, Sense and Nonsense. Blackwell: Oxford (1984) p. 388.


As referenced in my Prologue, the PI notes that the lack of any rule in tennis for how high a ball can go does not make a systematic game of tennis impossible nor does it make it unworkable or not enjoyable. But, ought tennis have a rule as to how high a ball can go? Of course, any such height rule will have its own vagueness problems, but it will make the game more precise — if we want it more precise. Thus, the presence of vague words does not make a systematic theory of meaning impossible but actually gives is it a goal similar to that of science:

The most common misunderstanding about science is that scientists seek and find truth. They don’t – they make and test models, … Building models is very different from proclaiming truths. It’s a never-ending process of discovery and refinement, not a war to win or destination to reach. Uncertainty is intrinsic to the process of finding out what you don’t know, not a weakness to avoid. Bugs are features – violations of expectations are opportunities to refine them. And decisions are made by evaluating what works better, not by invoking received wisdom.
— Neil Gershenfeld, director of the Massachusetts Institute of Technology’s Centre for Bits and Atoms


Like Galileo’s ignoring of friction, vagueness can be ignored until we need to criticize it.



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