Vagueness and norms

Matja˛ Potrč

University of Ljubljana, Slovenia



Description of the phenomenon of vagueness provides individualistic and collectivistic assessment of the sorites sequence. Vagueness is due to the acknowledgement of boundarylessness, as an effect of impossibility to match the norms steaming from both the individualistic and the collectivistic side. These norms are respectively presented as close to the norms of science and to the norms of common sense.

Boundarylessness is the effect of impossibility to mutually satisfy both individualistic and collectivistic norms. Although these norms do not match, they should be respected.

Transvaluationism is the view of vagueness that is opposed to epistemicism. These views both attempt to account for the phenomenon of vagueness. The difference is in respect to how the treatment of normativity relates to the question of boundarylessness.


-                                             The phenomenon of vagueness: a first description

Vagueness characterizes most of predicates such as they appear in thought and in language. The usual examples figure “being bald”, “being a heap”, and »being poor”. What is vague about such predicates? The answer is that such predicates are vague because one cannot determine the precise point at which they start and where they stop to apply. This is a point determining a difference in numbers “n, n+1”, which mark the upper bound of application concerning a certain predicate, and the first instance of the non-applicability of this predicate. If there would exist such a cut-off point, then predicates would be non-vague.[1] But as the matters stand, many predicates are vague. There is no precise number of hairs, no last determined boundary where a man is not bald, and where he starts to be bald if one of his hairs is pulled out of his scalp.


-                                             Individualistic and collectivistic assessments of the sorites sequence

We already started to discuss vague predicate “being bald”, and this invites us to take a closer look at it. Now, imagine the arrangement of 10.000 men, positioned in a row one besides to the next with an ever-increasing number of hairs on their heads. The first man has one hair, the second one has two hairs, and so on. Ultimately, the 10.000th man has 10.000 hairs appropriately positioned on his scalp. We may picture ourselves these men being lined up in a row, with each successor possessing one more hair as his predecessor. Let us also agree, as most people presumably would, that the man with one hair is bald. Then the following will hold:


If a man with 1 hair is bald, then the man with 2 hairs is bald as well. (General conditional statement).

But a man with 1 hair is bald indeed. (Presupposition based on evidence).

.: So the man with 2 hairs is bald as well.


Now, this reasoning may be reiterated for the ever-adjoining values of hairs as we advance along the sequence. This may be written in such a manner that we take the letter B as representing the predicate “being bald” and numbers 1, 2, …, 9.999, 10.000 as representing the number of hairs. Then we have:


B (1) → B (2)

B (1)

.: B (2)


B (2) → B (3)

B (2)

.: B (3)





B (9.998) → B (9.999)

B (9. 998)

.: B (9.999)


B (9.999) → B (10.000)

B (9.999)

.: B (10.000)


The reasoning using modus ponens form (P → Q, P, .: Q) thus proceeds along the sequence of 1, 2, …, 9.999, 10.000 number of hairs. It extends from the beginning till the end of the sequence by using a pair-wise advancement method. The assignment of the predicate shifts from the lower numbered predicate holder to its immediate neighbor. It starts with the plausible statement of assigning the predicate at the beginning of the sequence. And it stops with the implausible assignment of the predicate at the end of the sequence. We can see that a guy with 10.000 hairs appropriately positioned on his scalp just isn’t bald, despite that this is affirmed in the conclusion of the last instance of the iterated argument form.

            Let us call this manner of viewing the sequence individualistic, for we have repeatedly advanced from one to the next individual hair in the sequence.


            The following will be true as well: We profit from the arrangement of the lineup of the men whose number of hairs increases by one from left to the right. Now we climb on a hill and, using binoculars if necessary, we take a look at the left side of the arrangement. We clearly see bald men there. Then we take a look at the right hand side of the arrangement. Not hard to see that these men have plenty of hair and so that they are not bald. So looking at the whole sequence from a distance will result in clearly seeing cases of baldness on the left hand side and cases of people who are not bald at the right hand side. Somewhere towards the middle of the line we will estimate that we have to do with cases that are undecided between being bald and not being bald. As we are able now to observe the whole arrangement in one brush, our practice may be characterized as taking the line-up in collectivistic manner.

            Whereas the first account of the sorites sequence was individualistic, this second one is collectivistic. So we have individualistic and collectivistic assessments of the sorites sequence. We have taken the predicate “being bald” as our example here, if nothing else because it allows us to illustrate nicely the two perspectives. Many people prefer the example of the heap, where the individualistic assessment starts with one grain, and then it proceeds to the second grain, and so on, shifting the predicate “not being a heap” all the way down to 10.000 appropriately positioned grains. An explicit absurdity follows at the last step. For we can clearly see that there is a heap there, if we take the perspective of observing the whole arrangement in collectivist manner. Of course, it is not excluded that somebody indeed takes the effort of lining up the arrangements of 1, 2, …, 9.999, 10.000 grains. The name “sorites” comes from the Greek expression for the heap. We may perhaps also say that we have a heap of sorites reasonings, one for each of adjacent cases.[2]


            With individualistic and collectivistic assessments of the sorites sequence, and thus of vagueness, the following view on these perspectives offers itself.

Individualistic perspective has counterintuitive consequences, although these consequences arise from following some simple common sense based presuppositions. This is something that the science does. Its starts with the usual and many times it leads us to embrace counterintuitive conclusions, in disagreement with the view of common sense with its refusal to draw such consequences.

            Collectivistic perspective just stays with whatever manifestly reveals itself and does not lead to any additional conclusions that are hidden from what is already displayed. It is a perspective that stays with the common sense. Well, this is as far as it goes. It is true that the collectivist perspective at the sequence of men with increasing number of hairs was obtained on the basis of inspection from the distance on the hill. Perhaps this merits to be reassessed. But let us proceed to the main comparison now.


            Perhaps there is some vicinity between the individualistic and the collectivistic perspectives at the sorites sequence on the one hand, and between scientific and common sense perspectives on the other hand. We can find out by comparing them.

            There is this difference between the scientific and manifest image of the world that people like Sellars pointed to. The main idea in its simplest form is that the common sense compatible manifest image presents chairs and people to us, whereas the refined scientific image of the world will take us beyond these appearances towards a finer structure of atoms, quarks and fields, and eventually of cells and of their constituents. According to this picture, manifest image is nothing but a kind of appearance, behind which a complex and usually a hidden structure may be discovered.

            One consequence of such coming together of manifest and scientific image is that they are not compatible as explanatory schemas. You take one or the other, but you just cannot take both at the same time. But they are again compatible as two perspectives that tend to describe the same reality. Thus it has no real sense to see the world such as it is by itself as inconsistent. There is just incompatibility of the two explanatory schemata that are used to asses the world.


            A natural way to proceed from here is to look at the phenomenon of vagueness as presenting the coming together of two perspectives, of which the first or individualistic one is of scientific inspiration, and where the second collectivistic one is of common sense inspiration. In other words, the difficulty presented by cases of vagueness– difficulty with determining an identification of the cut-off point – comes from joining of the two perspectives inherent to the phenomenon. The striving for achieving the cut-off point, “n, n+1” point of the qualitative difference, where the predicate “being bald” is completely appropriately employed at the first, but not anymore at the next step, may be seen as a kind of scientific manner preciseness. Whereas the common sense perspective, here represented by collectivistic assessment of the sorites sequence, does not really care about the cut-off point. It just wants to have the predicate in question applicable, without any ultimate cut-off point, and it is thus happy about leaving some unresolved inconsistencies along the road.[3]

            Here are a couple of preliminary objections to the above, before the story is continued. First, the cut-off point is perhaps not the ultimate thing to be of concern in the phenomenon of vagueness. Perhaps there is something else, such as incompatibility of the two perspectives. Further, the individualistic perspective is not necessarily scientific, for it starts with whatever is common sensically acceptable: modus ponens conclusion from baldness of one haired man to the baldness of the two haired man is based on the intuitions of common sense. Then, individualistic perspective is not scientific either, if empirical information is required for something to be scientific. Finally, collectivistic perspective from the distance may again be seen as close to some methods employed by science, where a wider view of an area comes into focus. 


-                                             Individualistic assessment of the sorites sequence interpreted as a science inspired enterprise.

The individualistic proceeding in the sorites sequence builds on the modus ponens reasoning which is in force for each individual step. Let us again display the reasoning.


If a man with 1 hair is bald, then the man with 2 hairs is bald as well. (General statement).

But man with 1 hair is bald indeed. (Evidence).

.: So the man with 2 hairs is bald as well.


The general statement of this reasoning may be read as a hypothesis that needs to be tested: it comes in the conditional form. Perhaps the general form of the first premise cannot be seen immediately. But modus ponens may be rendered as a general statement indeed, so that it does not depend on a specific number of hairs:


            For any man with n hairs: if this man is bald, so also a man with n+1 hairs will be bald.


More formally:


            (n) (Bn → Bn+1)


Here, „B“ stands for the predicate „being bald“, and „n“ stands for a number of hairs. This is now clearly a general statement, which may also be read as a hypothesis, or as a law with the possibility to be tested in order to figure as a law of empirical science. Because of its conditional form, the first premise is clearly a hypothetical statement.

            Seen from this perspective, the modus ponens reasoning above figuring one haired man and two haired man may be seen just as an instance of the general schema. It provides substitutions for n’s in the generalized form of the modus ponens


(n) (Bn → Bn+1), Bn .: Bn+1


Now, each of the substitutions of Bn, such as “man with 1 hair is bald”, “man with 2 hairs is bald”, may be said to have the role of confirmation of the hypothesis figuring in the first premise. So the second premise then serves as an empirical confirmation instance of the general hypothesis, according to this interpretation. Empirically we take a look at the one haired man and we confirm that he is bald. Then, by consulting the general principle from the first premise, we may draw conclusion that Bn+1, thus that the man with two hairs is bald as well.

            This seems to be fine. But consider the difficulty here that arises at some stage of the confirmation process. At some stage namely, the empirically assessed Bn will not be confirmed if we take number “n” as being big enough. If not sooner, this will be the case about B9.999 substitution for Bn. In other words, the sorites puzzle will not be solved by the empirical hypothesis confirmation. Sorites will stay a puzzle for the individualistic perspective. And if the individualistic perspective at sorites is inspired by science, this means that science inspired approach will not suffice to solve, and perhaps even not to properly articulate the data that are implied into vagueness.

            But here we have a distinction to make in respect to what is confirmed. The second premise, figuring the first instance ob Bn, may be seen as empirically confirmed on the basis of empirical observation. There does not need to be this kind of empirical confirmation at any of the next stages of the modus ponens individualistic reasoning. One may say that the proceeding from the second step on is aprioristic. The reasoning uses just conceptual tools from the second stage of the modus ponens reasoning on. This then explains the force and smoothness with which the (non-intuitive) conclusion is reached, as also the fact that one empirically based observation may lead to the contrary conclusion, embracing whole of the sorites sequence. The empirically evident confirmation in this case is that a 10.000 haired man is not bald, which serves as the second premise in the argument


~B (10.000) → ~B (9.999)

~B (10.000)

.: ~B (9.999).


The non-baldness ascription then extends over the whole of the sequence. Now even more counterintuitive situation figures each of the steps in the sorites sequence as possessing two contradictory semantic values. This comes about in the case where the individualistic proceedings start with both the left hand side and with the right hand side of the sequence, respectively.

            The presupposition of the above is having empirical verification just at the first step, and then proceeding by just conceptual means all till the end of the sequence. One may also go in a different manner here and require empirical verification at each step. This would then make the second premise of the sorites argument to really figure as the premise of empirical evidence in generality, embracing all of the particular substitution instances. This was not the case when the first instance of the whole sequence of reasoning was empirically confirmed. In the new situation though, where we take each instance of the number of hairs to figure as an empirical evidence premise, it will be true that at some point the empirical evidence will disconfirm the general hypothesis. So, B9780 will disconfirm the hypothesis that we have to do with a case of baldness. But the problem to determine the cut-off point will not be solved by this, it will be just differently stated. There will still be the question which number n is the last point of the application of the predicate “being bald”.


            We said that individualistic assessment of sorites sequence might be interpreted as a science inspired enterprise.[4] This may be supported by the hypothesis confirmation nature of the modus ponens reasoning. But it also may be supported by a very simple remark that we start with empirically verifiable and plausible premises at the beginning of the individualistic inspection of the sorites reasoning. While we end up with truths that are not directly supported by observation. As already noticed, this seems to be a characteristic of science. Physics starts by observing the usual and homely (chair), but then it introduces hypotheses allowing it to switch to a finer observation (there must be a finer grain here). And this leads it to postulate something that is not directly confirmed by observation (such as the postulation of quarks). In a similar manner, the individualistic assessment of sorites will end up with postulation that the 10.000 hair man is bald. There is a trouble here though. It is not just that this last fact is not in agreement with the common sense. It is rather simply false.


-                                             Collectivistic assessment of the sorites sequence interpreted as the common sense inspired enterprise.

Sorites sequence may be also assessed from the collectivist perspective. There is no special hypothesis needed here, and no inferential reasoning supported by generalization. There is no observational datum here either to support the hypothesis. We have a direct observation of what is given. We do not mesh with details, but take an overall look at the sequence in its entirety. This is why we talk about the collectivist assessment.

            Now, such a view may be interpreted as a perspective from the point of view of common sense. It is just a matter of what one immediately observes, without any hypotheses, and what one reports on this basis.

            It is true that in the case of increasing hair alignment, we had to shift our viewing perspective to the distance from which we can observe sequence in its entirety. But this shift of perspective does not influence the direct nature of the immediate observation.

            Common sense nature of the collectivist perspective is recognized by acceptance of the result steaming from the immediate observation. It is just obviously and common sensically clear that there are these cases of baldness on the left hand side of the sequence, that there are these cases of non-baldness at the right hand side of the sequence, and that there are these cases of indecision somewhere in between both of these.

            Although collectivistic view is rather direct and has a tendency of not being complicated, it does not contribute to the solution of the sorites problem. It gives us good approximate account about judging the cases to belong to baldness, to non-baldness and to the area of indecision about these that arises between these two. But it does not deliver solution to the problem concerning points of semantic transition: where exactly is the number of hairs such that this number determines the last case of the bald, and so that its successor will be the first case of the non-bald?

            Collectivistic assessment of the sorites sequence thus may be understood as the common sense inspired approach. But this approach just isn’t able to solve or even to appropriately tackle the problem posed by boundarylessness as the main problem determining vagueness.


-                                             The normativity of science.

In order to understand slightly better what is involved into comparison between individualistic and collectivistic account of sorites on the one hand and between science and common sense on the other hand, it is worthwhile to look at the requirements that are usually posed to the science and to the common sense. Let us look in a very general way at the requirements that science should satisfy.

            Main desiderata for proceedings of science, as compared to the common sense approach, seem to be in a stricter way of treating the matters at hand. Whereas in common sense one takes for granted or at the face value whatever is simply there, this is not the case for the science. In the science, there is the methodology, there is the hypothesis, and there is the observation. Methodology consists in that the hypothesis is tested in view of data provided by the observation. So observational data are not simply sitting there in the case of the science. They are the data interpreted by the hypothesis.

            One aim of science is to provide methods for selection of the data, for their measurement, for their statistical comparing and estimation, for their quantitative description. As already remarked, the requirements for something to count as a datum of science are rather demanding. In any way they are much stricter than this is the case for the common sense. For something to be able to count as a datum appropriate for scientific investigation, the context imposes several quite strict requirements that need to be satisfied. First, these are requirements of methodology, which are themselves guided by their proper normativity.


-                                             The normativity of common sense.

The normativity for something to be an appropriate datum in the view of common sense is quite different. It imposes fewer requirements as does the scientific context, and accordingly common sense is less strict than this is the case with the science.

            This is easily illustrated by very simple observations about what makes one satisfied so that these are appropriate observations from the perspective of the common sense. Is there a table here? Yes, there it is, that’s obvious. I can see it.

            There is certainly a wisdom of common sense. And this wisdom has to be respected. But it is not necessarily a wisdom that complies to the requirements imposed by the normativity of generalizations, especially not of exceptionless generalizations. Generalizations in common sense do not come in the form of exceptionless rules. Many times they contradict themselves, such as this is the case with proverbs: “Many cooks spoil the broth” and “Several people know the things better than just one does”.[5] As normativity implied in the common sense needs not really to be strict, such contradictions will be viable.


-                                             Normativities of science and of common sense do not match.

It is commonly acknowledged that normativities of science and of common sense are not of equal footing. As remarked just now, the requirements applied to the science are stricter than are the requirements applied to the common sense. Some people have disagreed with this. So Churchlands claim that the generalizations of common sense, such as


            If you desire x, and if you believe that x is in your reach, so you will go for x


may be understood as exceptionless general statements. They are generalizations, just the kind of generalizations to be found in scientific laws. But, contrary to the case of real science, these generalizations, though exceptionless, are not behaving appropriately. This now shows that common sense is an inappropriate kind of science, which deserves to be eliminated and to be substituted with the real scientific talk. It may take a while, but common sense talk will be gradually eventually substituted by the scientific talk.

            If the above suggestion would be right, then there would be just the normativity of science, for normativity of common sense would match with the normativity of science. But this does not seem to be right. In the reasoning above, once the normativity of common sense is recognized to be on the same footing with the normativity of science, it is then quickly stated that facts pertaining to the common sense do not match with scientific requirements. But this may also be interpreted as the recognition of the difference between the normativity of science and between the normativity of the common sense. I would propose to stay with this last option.


-                                             Sorites sequence shows lack of matching between individualistic and collectivistic norms that follow scientific and common sense normativities.

Now let us stick with the difference between scientific and common sense normativities, by claiming simply that the requirements of the first are stricter than the normativities of the second one. And let us apply this to the case of sorites sequence.

            It seems clear enough that the requirements of normativity proper to individualistic and collectivistic assessments do not match. The conjecture is here that the mismatch is caused by the incompatibility between stricter normativity implied in the individualistically and science inspired assessment of the sequence, and between the collectivistically or common sense inspired assessment of the sorites sequence.

            Individualistic view of the sequence is guided by the norms which are stricter in that they take each step individually and apply modus ponens reasoning to it, thus a general hypothesis as the confirmable generalization. Collectivistic view of the sequence follows less strict and thus looser norms. Each kind of these norms has its own normativity. These normativities do not match. And in a similar sense, the results of those normativities also lead to different conclusions. So, individualistic approach results in the same semantic value being extended throughout all of the sequence, in two opposing ways. And collectivistic approach results in the recognition of different values at different ends of the sequence. Neither of these though does recognize the cut-off point.


-                                             Boundarylessness: an effect of impossibility to match two kinds of the norms involved.

There is this normativity requirement proper to the individualistic assessment of the sorites sequence: “Never change the semantic value of any immediate successor in the sorites sequence”. This means that, whatever value one starts with, it has to extend over all of the rest of the sequence. The manner of the value to advance is pair wise: it always shifts from a case in point to its immediate successor.

            Normativity requirement implied in the collectivistic assessment of the sorites sequence may be formulated as follows: “Determine just the overall areas of semantic values that you immediately notice, without caring about where exactly to position the cut-off point”.

            It may be recognized that the main concern of the first requirement based on the individualistically inspired methodology is propelled by the chasing of the boundary or of the cut-off point. Each adjoining pair of values is confronted by the question: “Is this the cut-off point?” And the answer is always that it is not the cut-off point. Take this man with one hair, and take this other man with two hairs. Is one of them bald while another one is not bald? If this would be so, there is a cut-off point here. The answer is given through the conditional modus ponens that actually denies the presupposition of the cut-off point that one chases for: “If the man with one hair is bald, then the two haired man will be bald as well. But the first one is bald. So, the second one is bald as well.” So, there is no cut-off point here. There is no cut-off point at all for any of the two comparing pairs all along the sorites sequence.

            The collectivistic assessment of the sorites sequence, to the contrary, practically ignores the possibility of any cut-off point. The collectivistic normative requirement is simply satisfied with any obvious assignment of the semantic value to each point at the sequence, from the overall perspective. So, boundary is no concern for collectivism. This proves that collectivism is common sensically inspired.

            Now, this shows that individualistic assessment of the sorites keeps on chasing for the boundary. While collectivistic assessment of the sorites sequence does not care about the boundary at all. The mismatch between these norms is obvious. It is this mismatch that produces boundarylessness, the real constitutive impossibility to find a cut-off point on the sorites sequence. Thus it would be wrong to think that boundarylessness is just the effect of the collectivistic assessment, because it does not normatively recognize the boundary. And neither is it the result of the individualistic assessment because it performs a vain chase for the boundary all along the sorites sequence. Boundarylessness is really the product of impossibility for both of these concerns to come together. Individualistic and collectivistic norms just cannot match, and this is the real root that produces boundarylessness. It is constitutive of the phenomenon of vagueness to have boundarylessness in this sense.


-                                             The need for respect of norms, although they do not match.

One thought here might be that norms do not have to be respected, if they produce an incoherent situation. So, one or the other kind of norms has perhaps to be abandoned.

            The claim here though is that both of the norms have to be respected. There is nothing wrong with any of them. Given the presuppositions coming along with the individualistic assessment of the sorites sequence, it is hard to provide objections. It would be difficult to put into question the modus ponens form of reasoning. So it is quite improbable that there would be something wrong with the individualistic reasoning as it stands.

            And there is also nothing wrong with the collectivistic view of the sorites sequence. The directly observed areas do indeed contain just two kind of semantic values. Mostly so, but rightly so. Mostly so, because the common sense approach does not pride itself to be really strict.

            Individualistic and collectivistic assessments are just guided by different kinds of normativities. Both of these have to be respected. And if in their togetherness they form an incoherent whole, this just shows how the phenomenon of normative boundarylessness and thereby of vagueness is produced. Norms should be respected, although they do not match.

            Such a benevolent accepting of incoherent situation might not be something really exceptional. In language and though it rather seems to be ubiquitous. The words as they are used most of the time compete for their meaning under the pressure of several and often opposed norms. Or better expressed, they sharpen their meaning as they come to be involved into several competing normative requirements. But isn’t this just their vagueness?


-                                             Transvaluationism as the view of vagueness building on impossibility to satisfy two kinds of norms governing the sorites sequence.

The attitude to embrace the view that incoherent norms usually produce something viable, and not something that needs to be rejected, may be entitled transvaluationism.

            Because transvaluationism respects boundarylessness, it has a certain tendency to deal with the forced march, an attempt to evaluate as confirmation or rejection instance any second premise in the heap of repeatedly occurring individualistic modus ponens reasonings. Transvaluationism refuses to be engaged into activity of providing an answer to each of those consecutive steps in the sorites sequence, accomplished with the questions such as “Is the man with one hair bald?, “Is the man with two hairs bald?”. In this way, transvaluationism respects the collectivist attitude while it treats the questions of individualistic line. The refusal to answer any one of these questions may be called Zen attitude.

            Transvaluationism obtains its name from his way of transcending all the values, in this case the semantic values. “Is it true that the man with n hairs is bald?” is thus a question that is transcended for each possible substitution of n. And boundarylessness is adopted therewith, as the constitutive mark of vagueness.


-                                             Failure of epistemicism to provide an account of normativity in description of the sorites sequence.

Besides to transvaluationism, there is another general approach to the phenomenon of vagueness, with the name of epistemicism. Epistemicism’s main claims are that there does exist a sharp boundary or a cut-off point at the sorites sequence, and that this boundary is epistemically necessarily closed to cognitive beings.

            Because of the first claim, boundarylessness is rejected. But if boundarylessness is the mark of vagueness, then the question imposes itself whether vagueness is respected at all by epistemicism.

            The answer to this is that boundarylessness tries now to be assessed as a phenomenon of knowledge. There is a cut-off point on the sorites sequence, which sounds even more appealing as the classical logic with its two semantic values, true and false, may be retained. But vagueness nevertheless comes as impossibility for us to know about the cut-off point.

            Do any of these main claims of epistemicism respect normativity? The answer to this question is negative. Just take a look at the individualistic assessment of the sorites sequence according to epistemicism. If there is individualistic proceeding from one pair of cases of baldness to the next, it will have to be assessed with something like a negated knowledge operator K, figuring in the general statement covering all the cases:


            ~Ka (n) (Bn & ~Bn+1)


Thus, a person a does not know where there is a boundary between the cases ob baldness and the cases of lack of baldness. The general statement actually denies the knowledge of each of the consecutive individualistic pairs in the sorites sequence.

            One may claim that there is boundarylessness here, a case of epistemic boundarylessness that is not so far away from the usual boundarylessness involving predicates. And finally, neither our knowledge nor our predicates are here meant to be directly related to the ontology, to the world as it is independently of language and thought.

            But it would be unusual to see some normative rule implied in the lack of knowledge of the boundary. You will just not know where the boundary is. Although in reality there is a boundary there. ”You will not b able to cognitively assess the boundary for any pair of cases” would be a strange normative requirement.

            What about collectivistic normativity in the case of epistemicism? An epistemicist will acknowledge overall differences for non-sharp regions in the collectivist look at the sequence. But why would an epistemicist need normativity here? Why would he spell out the need for determining general areas where semantic values apply, without the cut-off point being important? Epistemicist just does not need normativity, at least not in this way, because he does believe in the existence of the boundary – although he does not claim to know it.


-                                             Because of its lack to give an account of normativity, epistemicism misses the phenomenon of vagueness – for vagueness is the result of mismatch between two forms of normativity.

We have determined vagueness as boundarylessness, which proceeds from mismatch of individualistic and collectivistic accounts of the sorites series. Now for an epistemicist, there does exist a boundary. So, he cannot recognize vagueness as boundarylessness. The recognition of boundary goes along with the lack of recognizing two kinds of normativity such as they are implied into the individualistic and collectivistic assessments of the sorites sequence.


-                                             All other accounts of vagueness – such as supervaluationism – do give an account of normativity, although in a disguised form. If the basis of those vagueness accounts which form a complement to epistemicism is straightened, they rather reveal themselves as forms of transvaluationism.

We have envisioned just transvaluationism and epistemicism as two alternatives for an account of vagueness. What about other approaches now, such as supervaluationism?

            One may tackle the issue by trying to answer the question whether such approaches recognize normativity. The answer is that they do recognize normativity, but not in a direct, rather in a disguised form. So supervaluationism recognizes a semantic value for each point on the sorites sequence, and this as a sharply determined value: supertruth or superfalisity: in the case where all permissible interpretations of a statement turn out to be true or false. But although there is no vagueness here at first sight, it comes back with the recognition of the boundarylessness, for assignments do and do not hold for each of the cases, including also iterated vagueness at the meta-levels. If this gets spelled out, then supervaluationism turns out to be just a species of transvaluationism. The normativity implied in transvaluationism then proceeds in the usual way.



Churchland, Paul (1981). “Eliminative Materialism and the Propositional Attitudes.” Journal of Philosophy 78, no. 2: 67-90.

Horgan, Terry (1994). “Transvaluationism: a Dionysian approach to Vagueness”. The Southern Journal of Philosophy. Vol. XXXIII. Supplement: 97-126.

Potrč, Matja˛ (1998): “Nejasnost je odporna.“  Anthropos, no. 4-6:. 91-98.

Sellars, W. (1962). “Philosophy and the Scientific Image of Man.” In R. Colodny, ed. Frontiers of Science and Philosophy, 35-78. Pittsburgh: University of Pittsburgh Press.

[1] Epistemicism acknowledges the cut-off point, and it nevertheless recognizes vagueness. But this is an epistemic kind of vagueness. It does not steam from the sorites sequence itself, rather from our epistemic inability to locate the cut-off point at the sorites sequence.

[2] Talking about the heap of modus ponens reasonings would not be really appropriate according to the reading embraced here. For each modus ponens reasoning is a grain in the individualistic proceeding along the sequence. Whereas talking about the heap has connotations that are closer to the collectivistic view of things.

[3] Individualistic proceeding is fuelled by the desire to reach the cut-off point, which it tries to accomplish by shifting the cut-off point candidates all the way down throughout all of the sequence. Rushing after the cut-off point, individualistic proceeding finishes by constantly jumping over just the possibility and not the actuality of reaching it. Collectivistic proceeding on the contrary just does not care about the cut-off point, being satisfied with a direct apprehension of the situation, without any splitting of the hairs.

[4] A very simple view may take already commitment to the explanation via modus ponens reasoning to be a sufficient mark of the scientific. Counting, accounting for and similar stuff figure as the mark of the scientific in  Heidegger’s treatment of Gestell, linking technology and science.

[5] Kathy Wilkes reasons in such a way about common sense generalizations as both being in contradiction with but nevertheless functioning fine for the purposes of common sense. This is her criticism of the approach by Churchlands according to which common sense generalizations are to be treated as exceptionless rules, from which eliminativism would follow.