In linguistics and philosophy, a vaguepredicate is one which gives rise to borderline cases. For example, the English adjective "tall" is vague since it is not clearly true or false for someone of middling height. By contrast, the word "prime" is not vague since every number is definitively either prime or not. Vagueness is commonly diagnosed by a predicate's ability to give rise to the Sorites paradox. Vagueness is separate from ambiguity, in which an expression has multiple denotations. For instance the word "bank" is ambiguous since it can refer either to a river bank or to a financial institution, but there are no borderline cases between both interpretations.
Vagueness is a major topic of research in philosophical logic, where it serves as a potential challenge to classical logic. Work in formal semantics has sought to provide a compositional semantics for vague expressions in natural language. Work in philosophy of language has addressed implications of vagueness for the theory of meaning, while metaphysicists have considered whether reality itself is vague.
Importance
The concept of vagueness has philosophical importance. Suppose one wants to come up with a definition of "right" in the moral sense. One wants a definition to cover actions that are clearly right and exclude actions that are clearly wrong, but what does one do with the borderline cases? Surely, there are such cases. Some philosophers say that one should try to come up with a definition that is itself unclear on just those cases. Others say that one has an interest in making his or her definitions more precise than ordinary language, or his or her ordinary concepts, themselves allow; they recommend one advances precising definitions.[1]
In law
Vagueness is also a problem which arises in law, and in some cases, judges have to arbitrate regarding whether a borderline case does, or does not, satisfy a given vague concept. Examples include disability (how much loss of vision is required before one is legally blind?), human life (at what point from conception to birth is one a legal human being, protected for instance by laws against murder?), adulthood (most familiarly reflected in legal ages for driving, drinking, voting, consensual sex, etc.), race (how to classify someone of mixed racial heritage), etc. Even such apparently unambiguous concepts such as biological sex can be subject to vagueness problems, not just from transsexuals' gender transitions but also from certain genetic conditions which can give an individual mixed male and female biological traits (see intersex).
In science
Many scientific concepts are of necessity vague, for instance species in biology cannot be precisely defined, owing to unclear cases such as ring species. Nonetheless, the concept of species can be clearly applied in the vast majority of cases. As this example illustrates, to say that a definition is "vague" is not necessarily a criticism. Consider those animals in Alaska that are the result of breeding huskies and wolves: are they dogs? It is not clear: they are borderline cases of dogs. This means one's ordinary concept of doghood is not clear enough to let us rule conclusively in this case.
Approaches
The philosophical question of what the best theoretical treatment of vagueness is—which is closely related to the problem of the paradox of the heap, a.k.a. sorites paradox—has been the subject of much philosophical debate.
Fuzzy logic
In fuzzy logic, e.g. the predicates cold, warm, and hot apply gradually (vertical axis, 0 and 1 meaning certainly not and certainly, respectively) to a given temperature (horizontal axis).
One theoretical approach is that of fuzzy logic, developed by American mathematician Lotfi Zadeh. Fuzzy logic proposes a gradual transition between "perfect falsity", for example, the statement "Bill Clinton is bald", to "perfect truth", for, say, "Patrick Stewart is bald". In ordinary logics, there are only two truth-values: "true" and "false". The fuzzy perspective differs by introducing an infinite number of truth-values along a spectrum between perfect truth and perfect falsity. Perfect truth may be represented by "1", and perfect falsity by "0". Borderline cases are thought of as having a "truth-value" anywhere between 0 and 1 (for example, 0.6). Advocates of the fuzzy logic approach have included K. F. Machina (1976)[2] and Dorothy Edgington (1993).[3]
Another theoretical approach is known as "supervaluationism". This approach has been defended by Kit Fine and Rosanna Keefe. Fine argues that borderline applications of vague predicates are neither true nor false, but rather are instances of "truth value gaps". He defends an interesting and sophisticated system of vague semantics, based on the notion that a vague predicate might be "made precise" in many alternative ways. This system has the consequence that borderline cases of vague terms yield statements that are neither true, nor false.[4]
Given a supervaluationist semantics, one can define the predicate "supertrue" as meaning "true on all precisifications". This predicate will not change the semantics of atomic statements (e.g. "Frank is bald", where Frank is a borderline case of baldness), but does have consequences for logically complex statements. In particular, the tautologies of sentential logic, such as "Frank is bald or Frank is not bald", will turn out to be supertrue, since on any precisification of baldness, either "Frank is bald" or "Frank is not bald" will be true. Since the presence of borderline cases seems to threaten principles like this one (excluded middle), the fact that supervaluationism can "rescue" them is seen as a virtue.
Subvaluationism is the logical dual of supervaluationism, and has been defended by Dominic Hyde (2008) and Pablo Cobreros (2011). Whereas the supervaluationist characterises truth as 'supertruth', the subvaluationist characterises truth as 'subtruth', or "true on at least some precisifications".[5]
Subvaluationism proposes that borderline applications of vague terms are both true and false. It thus has "truth-value gluts". According to this theory, a vague statement is true if it is true on at least one precisification and false if it is false under at least one precisification. If a vague statement comes out true under one precisification and false under another, it is both true and false. Subvaluationism ultimately amounts to the claim that vagueness is a truly contradictory phenomenon.[6] Of a borderline case of "bald man" it would be both true and false to say that he is bald, and both true and false to say that he is not bald.
A fourth approach, known as "the epistemicist view", has been defended by Timothy Williamson (1994),[7]R. A. Sorensen (1988)[8] and (2001),[9] and Nicholas Rescher (2009).[10] They maintain that vague predicates do, in fact, draw sharp boundaries, but that one cannot know where these boundaries lie. One's confusion about whether some vague word does or does not apply in a borderline case is due to one's ignorance. For example, in the epistemicist view, there is a fact of the matter, for every person, about whether that person is old or not old; some people are ignorant of this fact.
As a property of objects
One possibility is that one's words and concepts are perfectly precise, but that objects themselves are vague. Consider Peter Unger's example of a cloud (from his famous 1980 paper, "The Problem of the Many"): it is not clear where the boundary of a cloud lies; for any given bit of water vapor, one can ask whether it is part of the cloud or not, and for many such bits, one will not know how to answer. So perhaps one's term 'cloud' denotes a vague object precisely. This strategy has been poorly received, in part due to Gareth Evans's short paper "Can There Be Vague Objects?" (1978).[11] Evans's argument appears to show that there can be no vague identities (e.g. "Princeton = Princeton Borough"), but as Lewis (1988) makes clear, Evans takes for granted that there are in fact vague identities, and that any proof to the contrary cannot be right. Since the proof Evans produces relies on the assumption that terms precisely denote vague objects, the implication is that the assumption is false, and so the vague-objects view is wrong.
Still by, for instance, proposing alternative deduction rules involving Leibniz's law or other rules for validity some philosophers are willing to defend ontological vagueness as some kind of metaphysical phenomenon. One has, for example, Peter van Inwagen (1990),[12]Trenton Merricks and Terence Parsons (2000).[13]
Legal principle
In the common law system, vagueness is a possible legal defence against by-laws and other regulations. The legal principle is that delegated power cannot be used more broadly than the delegator intended. Therefore, a regulation may not be so vague as to regulate areas beyond what the law allows. Any such regulation would be "void for vagueness" and unenforceable. This principle is sometimes used to strike down municipal by-laws that forbid "explicit" or "objectionable" contents from being sold in a certain city; courts often find such expressions to be too vague, giving municipal inspectors discretion beyond what the law allows. In the US this is known as the vagueness doctrine and in Europe as the principle of legal certainty.
Vagueness is primarily a filter[14][failed verification] of natural human cognition, other tasks of vagueness are derived from that, and they are secondary.[15] The ability to cognition is the basic natural equipment of human (and other creatures) allowing him to orient and survive in the real (material) world. The task of cognition is to obtain from the epistemologically incalculable (immensely vast and deep) reality to a human its cognitive (knowledge) model, containing an only finite amount of information. For this purpose, there must be a filter performing selection and thus reduction of information. It is the vagueness[14] with which man perceive and then remember information about the real (material) world. Some information gained with less vagueness, others with greater one, according to the distance from the center (focus) of attention occupied by man during his act of cognition. Human is unable to acquire information other than vague one by his natural vague cognition. It is necessary to distinguish the internal cognitive model, i.e. the intrapsychic, stored and processed in human consciousness (and probably also in the unconscious), in hypothetical intrapsychic languages: imaginary, emotional and natural and in their mixture, and then the external model, represented in a suitable external language of communication.
Vagueness internal and external
Cognition and language (Law of maintaining accuracy of information): Communication language should have the same amount of vagueness, as have information gained by cognition (source of the information). That means, language must be tuned to appropriate cognition considering vagueness. This is one of secondary tasks of vagueness.
A person is able to speak about his inherently vague knowledge (contained in the intrapsychic cognitive model represented in hypothetical intrapsychic languages) in natural (generally informal language, e.g. Esperanto), of course only vaguely.[14] The vagueness of knowledge caused by the filter of knowledge is primary, we call it internal vagueness (i.e. intrapsychic). The vagueness of a person's subsequent utterance is secondary vagueness. This utterance (transformation from intrapsychic languages to external communicative languages - it is called a formulation, see the semantic triangle) cannot reveal all the content of the personal intrapsychic cognitive model with all its inherent vagueness. The vagueness contained in the linguistic utterance (of external communication language) is called external vagueness.
Linguistically, only external vagueness can be grasped (modeled). We cannot model internal vagueness; it is part of the intrapsychic model, and this vagueness is contained in (vague, emotional, subjective and variable during time) interpretation of constructs (words, sentences) of informal language.[16] This vagueness is hidden for the other human, he can only guess the amount of it. Informal languages, such as natural language, do not make it possible to distinguish between internal and external vagueness strictly, but only with a vague boundary.[17][18]
Fortunately, however, informal languages use appropriate language constructs making meaning a little uncertain (e.g. indeterminate quantifiers POSSIBLY, SEVERAL, MAYBE, etc.). Such quantifiers allow natural language to use external vagueness more strongly and explicitly, thus allowing internal vagueness to be partially shifted up to external vagueness. It is a way to draw the addressee's attention to the vagueness of the message more explicitly and to quantify the vagueness, thus improving understanding in communication using natural language. But the main vagueness of informal languages is the internal vagueness, and the external vagueness serves only as an auxiliary tool.
Formal languages, mathematics, formal logic, programming languages (in principle, they must have zero internal vagueness of interpretation of all language constructs, i.e. they have exact interpretation) can model external vagueness by tools of vagueness and uncertainty representation: fuzzy sets and fuzzy logic, or by stochastic quantities and stochastic functions, as the exact sciences do. Principle is: If we admit more vagueness (uncertainty), we can gain more information during cognition. See e.g. possibilities of deterministic and stochastic physic. In other cases cognitive model of certain part of real world may be simplified, such a way, that certain amount of deterministic information is possible to replace by fuzzy or stochastic one.
Reducing internal vagueness
The internal vagueness of one person's message is hidden from another person, he can only guess that. We either have to accept internal vagueness, which is human, or we can try to reduce it, or completely eliminate it, which is scientific.
Demands on the accuracy of the formulation of scientific knowledge and its communication require minimizing the internal vagueness with which one connotes (vaguely, emotionally and subjectively interprets)[16] linguistic constructs of the communication language, and thus improve the accuracy of the message. Various scientific procedures aim to improve the credibility and accuracy of the scientific knowledge obtained.
To formulate them, however, it is necessary to build a more precise language, with less (internal) vagueness of message than is common in daily life. This is done by purposefully (branch) constructed terminology allowing to more accurately describe the researched reality and the acquired knowledge about it. People properly educated in the field of terminology know it with little internal vagueness, so they know accurately what the individual terms mean. Basic concepts are always formed on the basis of consensus, the other derived from them by definition, to avoid to Circular definition. To improve the accuracy of research and communication (reducing the internal vagueness of connotation), tools such as classification schemes are used, such as the taxonomy of organisms by Carl von Linné.
This is how descriptive (non-exact) sciences do it. Thus, they use natural human cognition (with the Russell’s filter of vagueness[19]) and refined natural language.
There is another continuation of the reduction of internal vagueness. The method of reducing internal vagueness to the extreme, that is zero, was realized by I. Newton.[20] It is an epochal idea, and it needs to be explained how it can be realized.
It follows from the above-mentioned Law of maintaining accuracy of information (optimization of the truthfulness of the message) saying, if we require to eliminate the internal vagueness in the knowledge completely (to zero) then of course it must first be completely eliminated in cognition (the source of information). This means that one (Newton) must avoid the intrusion of internal vagueness, that is, to choose some filter of cognition other than vagueness. Thus we pass from the natural human world to the artificial one. We call it the exact world and we will explain why.
In the case of natural language, it is not possible to completely remove (nullify) the internal vagueness, but it is possible to build artificial formal languages (mathematics, formal logics, programming languages) that have zero internal vagueness of connotation (so they have an exact interpretation) and cannot have another in principle. (Newton for this purpose has created formal language – theory of flux - theory of flowing – infinitesimal calculus). Languages with zero internal vagueness of their interpretation, i.e. the meaning of their linguistic constructions, have the property that all these constructions are understood by every appropriately educated person with absolutely precise, i.e. exact meaning. That is why they are part of the exact world. Thus, we have some language that is able to represent knowledge with zero internal vagueness. But these must first be acquired by adequate cognition, providing cognition also with zero internal vagueness, i.e. also from the exact world. And it is already evident that we are on the way to the creation of the scientific method that creates science belonging to the exact world, that is, an exact science is born.
It is still necessary to explain how to realize Newton`s exact cognition, that is, the cognition when the knowledge obtained from the real world is part of the exact world. The miraculous bridge between the real and exact worlds that makes this possible is called a quantity (e.g. electric field intensity, velocity, nitric acid concentration, etc.). It is common to both worlds, because in the exact world it is precisely delineated (every knowing person knows them with no doubts, so exactly), and in the real world it is an elemental measurable probe into that, and thus its elemental measurable representative. The quantity is the elementary building block of exact science. In exact sciences, it is always precisely defined, either consensually (basic set) or the other derived - International System of Units. And what about the artificial filter that allows Newton to avoid internal vagueness? For every problem of the real world that is to be grasped by Newton’s method of the exact science, it is necessary to choose a group of suitable quantities, find the natural laws that apply in the real world between them, and describe them in mathematical language. We get a mathematical knowledge (cognitive) model of a given part of the real world. A group of selected quantities forms a discrete Newtonian filter (sieve) through which man ‘’look’’ at a given part of the real world. Thus, in exact science, a given part of the real world is represented by a group of suitably chosen quantities and mathematically (programming language) described relations between them (more precisely between their names – symbols denoting them).
Exact scienceis a method that allows knowledge about the real world to be acquired and recorded so that it is part of the exact world. That is the method of modeling the real world by means of exact world, in other words, a method to mathematize the science.
Even exact science needs to have a tool, with which it can describe the uncertainty of the results (obtained – knowledge), whether out of necessity or the need to abandon excessive precision. Since that cannot (must not) be an internal vagueness, it can only use linguistically graspable uncertainty (external vagueness). For this purpose it has for its disposal a description of fuzzy or stochastic values of quantities, and fuzzy or stochastic relations (represented by mathematical functions) between quantities.
The difference between the non-exact sciences (called descriptive) and the exact sciences is that the former use natural human cognition (with the Russell’s filter of vagueness) and refined natural language, and the exact sciences use cognition based on the use of Newtonian discrete filter and thus the use of quantities, and artificial formal language. Artificial formal language also brings a powerful tool to exact science, which is formal inference (information formal processing) known from mathematics.
The above-mentioned tools of exact and non-exact science are general principles, and different branches of science use them in combination with both. They have their parts exact and inexact. Purely exact sciences, such as theoretical physics or mathematics, use natural language as meta-language.
Exact science provides most trustworthy knowledge. The question can certainly be raised as to whether all science can be transformed into an exact science. The answer is not. The condition for the establishment of an exact science is to find suitable quantities, and this is possible only for a small part of the real world and for specific views of it. In other words, the filter of vagueness makes it possible to vaguely know many; Newton's discrete filter makes it possible to know only little but exactly.
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
In logic, the semantic principleof bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic.
Truth or verity is the property of being in accord with fact or reality. In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs, propositions, and declarative sentences.
Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.
A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are also often characterized as being the kind of thing that declarative sentences denote. For instance the sentence "The sky is blue" denotes the proposition that the sky is blue. However, crucially, propositions are not themselves linguistic expressions. For instance, the English sentence "Snow is white" denotes the same proposition as the German sentence "Schnee ist weiß" even though the two sentences are not the same. Similarly, propositions can also be characterized as the objects of belief and other propositional attitudes. For instance if one believes that the sky is blue, what one believes is the proposition that the sky is blue. A proposition can also be thought of as a kind of idea: Collins Dictionary has a definition for proposition as "a statement or an idea that people can consider or discuss whether it is true."
The sorites paradox is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a single grain does not cause a heap to become a non-heap, the paradox is to consider what happens when the process is repeated enough times that only one grain remains: is it still a heap? If not, when did it change from a heap to a non-heap?
In logic, a predicate is a symbol that represents a property or a relation. For instance, in the first-order formula , the symbol is a predicate that applies to the individual constant . Similarly, in the formula , the symbol is a predicate that applies to the individual constants and .
Bernard Bolzano was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his liberal views.
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic, which includes additional topics like how to define logic or a discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic.
In classical logic, propositions are typically unambiguously considered as being true or false. For instance, the proposition one is both equal and not equal to itself is regarded as simply false, being contrary to the Law of Noncontradiction; while the proposition one is equal to one is regarded as simply true, by the Law of Identity. However, some mathematicians, computer scientists, and philosophers have been attracted to the idea that a proposition might be more or less true, rather than wholly true or wholly false. Consider My coffee is hot.
A fuzzy concept is a kind of concept of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all. This means the concept is vague in some way, lacking a fixed, precise meaning, without however being unclear or meaningless altogether. It has a definite meaning, which can be made more precise only through further elaboration and specification - including a closer definition of the context in which the concept is used. The study of the characteristics of fuzzy concepts and fuzzy language is called fuzzy semantics. The inverse of a "fuzzy concept" is a "crisp concept".
Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components. Thus, logical truths such as "if p, then p" can be considered tautologies. Logical truths are thought to be the simplest case of statements which are analytically true. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.
In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, the constitution of sentences, concepts, learning, and thought.
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.
A probabilistic logic network (PLN) is a conceptual, mathematical and computational approach to uncertain inference; inspired by logic programming, but using probabilities in place of crisp (true/false) truth values, and fractional uncertainty in place of crisp known/unknown values. In order to carry out effective reasoning in real-world circumstances, artificial intelligence software must robustly handle uncertainty. However, previous approaches to uncertain inference do not have the breadth of scope required to provide an integrated treatment of the disparate forms of cognitively critical uncertainty as they manifest themselves within the various forms of pragmatic inference. Going beyond prior probabilistic approaches to uncertain inference, PLN is able to encompass within uncertain logic such ideas as induction, abduction, analogy, fuzziness and speculation, and reasoning about time and causality.
Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application. This involves questions about how logic is to be defined and how different logical systems are connected to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines. According to a common characterisation, philosophical logic is the part of the philosophy of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. But other theorists draw the distinction between the philosophy of logic and philosophical logic differently or not at all. Metalogic is closely related to the philosophy of logic as the discipline investigating the properties of formal logical systems, like consistency and completeness.
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics.
In logic, a finite-valued logic is a propositional calculus in which truth values are discrete. Traditionally, in Aristotle's logic, the bivalent logic, also known as binary logic was the norm, as the law of the excluded middle precluded more than two possible values for any proposition. Modern three-valued logic allows for an additional possible truth value.
In logic, an infinite-valued logic is a many-valued logic in which truth values comprise a continuous range. Traditionally, in Aristotle's logic, logic other than bivalent logic was abnormal, as the law of the excluded middle precluded more than two possible values for any proposition. Modern three-valued logic allows for an additional possible truth value and is an example of finite-valued logic in which truth values are discrete, rather than continuous. Infinite-valued logic comprises continuous fuzzy logic, though fuzzy logic in some of its forms can further encompass finite-valued logic. For example, finite-valued logic can be applied in Boolean-valued modeling, description logics, and defuzzification of fuzzy logic.
References
↑ Williamson, T. 1994. Vagueness, [pageneeded]. London: Routledge. The history of the problem of vagueness is traced, from the first Sorites Paradox to contemporary attempts to deal with higher-order vagueness such as many-valued logic, supervaluationism, and fuzzy logic. Technicalities are kept to a minimum to favour a clear account, extremely useful to both students and researchers.[citation needed]
↑ Křemen, J.: Modely a systémy (Models and Systems), ACADEMIA, Publishing house of the Czech Academy of Sciences, Prague 2007, ISBN 978-80-200-1477-1
1 2 Osgood C. E, Suci G., Tannenbaum P.: The Measurement of Meaning. Urbana, Illinois, University of Illinois Press, 1957
↑ Bek R., Křemen J.: Logic, Cognitive Model and Reality, in: Proceedings of the 8th international symposium LOGICA ‘94, T.Childers, O. Majer ed., publ. by PHILOSOPHIA, The institute of Philosophy of the Academy of Sciences of Czech Republic, Prague 1995, pp. 193-199.
↑ Křemen, J.: Notes on Vagueness of Knowledge: Fuzzy Tools. In: Acta Polytechnica, Vol. 39, No 4/1999, pp. 81- 91. ISSN 1210-2709, Czech Technical University in Prague
↑ Russell B.: Vagueness. In: The Australasian Journal of Psychology and Philosophy 1, June 1923, pp. 84--92.
↑ Newton I: Philosophiae Naturalis Principia Mathematica in 1687
Further reading
Deemter, Kees van. Not Exactly: In Praise of Vagueness (Oxford University Press; 368 pages; 2010). Considers vagueness as both a useful and unavoidable aspect of realms from everyday life to computing.
Keefe, R.; Smith, P., eds. (1997). Vagueness: A Reader. MIT Press. The editors' long introduction gives a clear and very useful overview of theories of vagueness, and they collect many classic papers on the subject.
Keefe, R. 2000. Vagueness. Cambridge: Cambridge University Press. Technicalities are kept to a minimum to favour a clear account, extremely useful to both students and researchers.
Rick Nouwen; Robert van Rooij; Uli Sauerland; Hans-Christian Schmitz, eds. (Jul 2009). International Workshop on Vagueness in Communication (ViC; held as part of ESSLLI). LNAI. Vol.6517. Springer. ISBN978-3-642-18445-1.
Ronzitti, Giuseppina (ed.), Vagueness: A Guide. Logic, Epistemology, and the Unity of Science, Springer, Dordrecht, 2011.
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