Vagueness as an epiphenomenon, and non-transitivity

ABSTRACT

This paper deals with the linguistic phenomenon of vagueness. Based on certain observations regarding the intuitions and linguistic practices of the philosophically informed speaker, we make a series of assumptions concerning the nature and characteristics of the phenomenon. Vagueness is treated as an emerging phenomenon, caused, in essence, by the messy way in which linguistic communities reach classificatory equilibria. Any talk of ‘meaning’, ‘truth’, and such is treated as an indirect way of attempting to describe such equilibria, and it is in regard to these that ascriptions of truth are here relativised. Given such assumptions, we then proceed to formulate and study a semantic theory for languages that contain vague predicates. Finally, we describe two proof systems, one based on trees, and a natural deduction system in the style of Fitch, and outline the appropriate proofs of soundness and completeness.

KEYWORDS:

• Vagueness
• sorites paradox
• non-transitive
• non-classical logic
• natural deduction

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 To paraphrase Friedrich Waismann, see page 21 of Waismann (1945).

2 Historically, the semantic characteristics of conditional sentences have been, and still often are, a matter of intense contention and debate. Indeed, this has been the case since ancient times, as testified by the famous epigram of Callimachus: ‘Even the crows on the roofs caw about the nature of conditionals’. Characteristically, Sextus Empiricus, in his Outlines of Scepticism, describes four semantic accounts, held by philosophical schools contemporary to him: ‘For Philo says that a sound conditional is one which does not begin from a truth and end in a falsity (e.g. – when it is day and I am conversing – ‘If it is day, I am conversing’). Diodorus says that it is one which neither could nor can begin from a truth and end in a falsity. According to him, the conditional just stated seems to be false, since if it is day but I shall be silent, it will begin from a truth but end in a falsity. But ‘If it is not the case that there are indivisible elements of existing things, there are indivisible elements of existing things’ is true – for it will always begin from something false, viz. ‘It is not the case that there are indivisible elements of existing things’, and – according to him – end in something true, viz. ‘There are indivisible elements of existing things.’ Those who introduce connectedness say that a conditional is sound when the opposite of its consequent conflicts with its antecedent. According to them, the conditionals just stated will be unsound, but ‘If it is day, it is day’ will be true. And those who judge by meaning say that a conditional is true when its consequent is contained implicitly in its antecedent. According to them, ‘If it is day, it is day’ – and every duplicated conditional statement – will no doubt be false; for it is impossible for anything to be contained in itself’ (Outlines of scepticism, Book 2, 110–112). The semantic part of the theory outlined in this paper bears, in regards to the conditions that correspond to the conditional, some similarities with Akama (1988), Almukdad and Nelson (1984), Gurevich (1977), Nelson (1949), Thomason (1969).

3 In essence, the quadruple $〈W,⊑,Va{l}^{+},Va{l}^{-}〉$ may be treated as a collection of appropriate assignments of extensions and anti-extensions to the various relational symbols, given D. Each atomic constant is assigned an object in D, with no differences between assignments emerging here. The relation $x⊑y$ conveys the fact that all extensions and anti-extensions assigned relative to y extend the corresponding ones assigned relative to x.

4 See Arrow and Debreu (1954), p. 273

5 This last characteristic could be considered as being given as an initial endowment to each agent when entering a context, in relation to which the corresponding function is to a certain extent determined

6 Where ‘${\sqcap }_{i\in \{1,\dots ,n\}}{u}_i$’ denotes a point such that for every relational b, $Va{l}^{+}({\sqcap }_{i\in \{1,\dots ,n\}}{u}_i,b)={\cap }_{i\in \{1,2,\dots ,n\}}Va{l}^{+}(u_i,b)$, and $Va{l}^{-}({\sqcap }_{i\in \{1,\dots ,n\}}{u}_i,b)={\cap }_{i\in \{1,2,\dots ,n\}}Va{l}^{-}(u_i,b)$

7 We employ the symbols ‘⇒’, ‘iff’, ‘∧’, ‘∨’ as the metalinguistic propositional symbols for the conditional, biconditional, conjunction, and disjunction, correspondingly. The metalinguistic symbols for the conjunction and disjunction of sentences coincide with the ones used in the object language, but, context is in any case sufficient for the determination of the appropriate reading.

8 This is a variant of a similar concept used by Raymond M. Smullyan, see Smullyan (1995). Hybrid sentences are of a philosophical interest here. Atomic hybrid sentences could be treated as acts of characterisation that may be performed by idealised agents. The objects in W could be treated as specific sets of such sentences, and in this way, linguistic equilibria could be treated as grounded on maximisation that occurs on the level of such acts. We define as an atomic hybrid formula any ordered set of the form $〈F_n,〈o_1,o_2,\dots ,o_n〉〉$, where $F_n$ is an n-ary relational symbol of the language and $〈o_1,o_2,\dots ,o_n〉$ an ordered n-tuple such that each of $o_1,o_2,\dots ,o_n$ is either a variable symbol of the language or an object belonging to D. We consider as a complex hybrid formula any ordered triad of the form $〈\to ,A,B〉$, or $〈\wedge ,A,B〉$, or $〈\vee ,A,B〉$, where ‘$\to$’ is the implication symbol of the language, ‘∧’ the conjunction symbol, ‘∨’ the disjunction symbol, and A, B are hybrid formulas, each ordered pair of the form $〈\mathrm{\neg },A〉$, where ‘¬’ the negation symbol and A a hybrid formula, and, lastly, any ordered pair of the form $〈\mathrm{\forall }x,A〉$ or $〈\mathrm{\exists }x,A〉$ where ‘$\mathrm{\forall }x$’ and ‘$\mathrm{\exists }x$’ are the corresponding strings of symbols of the language and A a hybrid formula. Any object not fitting in any of these cases will be considered not to be a hybrid formula. We will consider any occurrence of the variable x in the post-comma position, within a hybrid formula of the form $〈\mathrm{\forall }x,A〉$ or $〈\mathrm{\exists }x,A〉$, where A a hybrid formula, as bound, otherwise we will consider it as free. We denote as ‘$[A{]}_x^o$’ the hybrid formula that results from the hybrid formula A if we replace every free occurrence of the variable x within it with the object o. We will say that a hybrid formula is a hybrid sentence if and only if it does not contain free occurrences of any variable.Given an interpretation, we can assign to each formula of the language a specific hybrid formula in the following way: To each atomic formula of the form $F(a_1,a_2,\dots ,a_n)$, where F is an n-ary relational symbol and $a_1,a_2,\dots ,a_n$ are either atomic constants of the language or variables, we assign the hybrid formula $〈F,〈{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n〉〉$, where $〈{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n〉$ is an ordered n-tuple such that, for any i where $1\le i\le n$, it holds that if $a_i$ is a variable then ${\alpha }_i$ is the same variable, while if $a_i$ is a constant then ${\alpha }_i$ is the object within D that gets assigned to the pair $〈w,a_i〉$, where w is an object that belongs to W, by the function $Va{l}^{+}$.To each formula of the form $A\to B$ we assign the hybrid formula $〈\to ,〈A〉,〈B〉〉$, where $〈A〉$ is the hybrid formula corresponding to the formula A, while $〈B〉$ is the hybrid formula corresponding to the formula B. To each formula of the form $A\wedge B$ we assign the hybrid formula $〈\wedge ,〈A〉,〈B〉〉$. To each formula of the form $A\vee B$ we assign the hybrid formula $〈\vee ,〈A〉,〈B〉〉$. To each formula of the form $\mathrm{\neg }A$ we assign the hybrid formula $〈\mathrm{\neg },〈A〉〉$. To each formula of the form $\mathrm{\forall }xAx$ we assign the hybrid formula $〈\mathrm{\forall }x,〈A〉〉$, where $〈A〉$ the hybrid formula that corresponds to the formula A. Finally, to each formula of the form $\mathrm{\exists }xAx$ we assign the hybrid formula $〈\mathrm{\exists }x,〈A〉〉$, where $〈A〉$ the hybrid formula that corresponds to the formula A.Moreover, we define the following: If $〈F^n,〈o^1,o^2,\dots ,o^n〉〉$ is an atomic hybrid sentence then it is the case that $〈w,a〉\nearrow 〈F^n,〈o^1,o^2,\dots ,o^n〉〉$ if and only if $〈Va{l}^{+}(w,o^1),Va{l}^{+}(w,o^2),\dots ,Va{l}^{+}(w,o^n)〉\in Va{l}^{+}(w,F^n)$, $〈w,a〉\searrow 〈F^n,〈o^1,o^2,\dots ,o^n〉〉$ if and only if $〈Va{l}^{+}(w,o^1),Va{l}^{+}(w,o^2),\dots ,Va{l}^{+}(w,o^n)〉\in Va{l}^{-}(w,F^n)$. Let B, C be hybrid sentences. We define the following: $w\in a\Rightarrow [〈w,a〉\nearrow 〈\to ,B,C〉\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{\forall }x(x\in a\wedge w⊑x\Rightarrow \mathrm{\neg }(〈x,a〉\nearrow B\wedge 〈x,a〉\searrow C)\left)\right]$, $w\in a\Rightarrow [〈w,a〉\searrow 〈\to ,B,C〉\mathrm{i}\mathrm{f}\mathrm{f}〈w,a〉\nearrow B\wedge 〈w,a〉\searrow C〉$, $,$ $w\in a\Rightarrow [〈w,a〉\searrow 〈\wedge ,B,C〉\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{\forall }x(x\in a\wedge w⊑x\Rightarrow \mathrm{\neg }〈x,a〉\nearrow B\wedge 〈x,a〉\nearrow C\left)\right)]$, $w\in a\Rightarrow [〈w,a〉\nearrow 〈\vee ,B,C〉\mathrm{i}\mathrm{f}\mathrm{f}〈w,a〉\nearrow B\vee 〈w,a〉\nearrow C]$, $,$ $w\in a\Rightarrow [〈w,a〉\nearrow 〈\mathrm{\neg },B〉\mathrm{i}\mathrm{f}\mathrm{f}〈w,a〉\searrow B]$, $w\in a\Rightarrow [〈w,a〉\searrow 〈\mathrm{\neg },B〉\mathrm{i}\mathrm{f}\mathrm{f}〈w,a〉\nearrow B]$. Let B be a hybrid formula that contains free occurrences only of the variable x, or no free occurrences of any variable, so that the hybrid formula $〈\mathrm{\forall }x,B〉$ is a hybrid sentence. We define: $w\in a\Rightarrow [〈w,a〉\nearrow 〈\mathrm{\forall }x,B〉\text{ iff }$for every object o that belongs to D it is the case that $〈w,a〉\nearrow \left[B{]}_x^o\right]$, $w\in a\Rightarrow [〈w,a〉\searrow 〈\mathrm{\forall }x,B〉\text{ iff }$for some object o that belongs to D it is the case that $,$$w\in a\Rightarrow [〈w,a〉\nearrow 〈\mathrm{\exists }x,B〉\text{ iff }$for some object o that belongs to D it is the case that $〈w,a〉\nearrow \left[B{]}_x^o\right]$, $w\in a\Rightarrow [〈w,a〉\searrow 〈\mathrm{\exists }x,B〉\text{ iff }$for every object o that belongs to D it is the case that $〈w,a〉\searrow \left[B{]}_x^o\right]$.

9 It follows from our definitions that, given a sentence B and an interpretation M, it may be the case that the argument with no premises and B as a conclusion is valid in relation to M, without it being also the case that B is logically true in relation to M.

10 Logical theories that suggest some kind of non-transitive relation of consequence have been proposed several times in the past. Historically, the first case of such a theory seems to be that of Bernard Bolzano, in his monumental work Wissenschaftslehre (Theory of Science). In this, in addition to defining a logical consequence relation in a way that one could argue is in the same spirit as the definitions given by Tarksi, he also formulates another consequence relation between the sentences of the language, which he calls ‘Abfolge’, usually translated as ‘grounding’ in English. Following Aristotle (Posterior Analytics, I.13), Bolzano will distinguish between two kinds of evidence, evidence showing that the conclusion is true and evidence showing why it is true. Bolzano will claim that in the latter case it is the relationship of grounding that is predominant. His emphasis on the latter is clear: he writes characteristically in paragraph 198: ‘Of all the relations that hold between the truths, the most worthy of my attention is that of ground and consequence, by virtue of which certain propositions are the ground of certain.’ (Taken from the English translation Bolzano, 1973) Noting, however, that the relation of grounding is not transitive, he writes in paragraph 213: ‘It seems to me that the relation of ground and consequence is of such a kind that one cannot say of a consequence of a consequence, just because it is a consequence of a consequence, that is the consequence of the ground of its ground, without altering the concept’. For reasons that may be considered to be analogous, several philosophers working in the field of Relevant Logic have also proposed systems of logic that admit of a non-transitive consequence relation. In particular, it is claimed that in order for a proposition B to follow logically from another proposition A, there must be some meaning connection between them. For propositions A, B, C, however, it might be the case that there is a meaning connection between propositions A, B, a meaning connection between B, C, but none between A and C. This approach is exemplified by the articles Geach (1970), Smiley (1958), and Epstein (1979), among others.

11 Given these observations, it follows that in some cases there may appear to exist a gap between the way that the apparent paradoxicality of a soritical argument, formulated in the object language, is resolved, and the way that any corresponding metalinguistic argument, formulated in terms of a properly relativised truth predicate, is treated in the crisp metalanguage. However, any vagueness that characterises an expression belonging in a vague object language, should, normally, also infect corresponding ascriptions of truth in a metalanguage where the intended model can be described. After all, in order to approximate such ascriptions in a non vague metalanguage we had to employ a notion, that was expressed by assigning to the corresponding metalinguistic expression appropriate subspaces of W, termed ‘conversational contexts’, and one could argue that, normally, any description of such a context should also be properly open-textured; indicating in this way that the intended model of the object language can only be described properly in a vague metalanguage. Furthermore, it should be expected that the conditional of such a metalanguage would behave in a manner similar to that of the object language, and therefore that in certain contexts truth ascriptions will display signs of tolerance. Given that what we usually interpret as the property of truth preservation is defined via the metalanguage's conditional, it seems reasonable to expect that preservation of vague truth should display signs of non-transitivity which also carry over to necessary preservation of vague truth. Moreover, if certain rules of truth substitutability are valid in the vague metalanguage, truth of a vague conditional should turn out to be equivalent to preservation of vague truth, and, provided the close ties between our definitions concerning truth of conditionals and validity, it could be argued that consequence is equivalent to preservation of vague truth, and that the non transitivity of consequence is a reflection of the tolerant characteristics of vague truth. Now, given that the philosophical portion of the theory places emphasis on specific underlying processes that are treated as leading to the partial determination of the truth predicate's extension, processes constituted by specific interactions between maximizing agents within linguistic communities, such metalinguistic speech could be rephrased in terms of phenomena that are also exhibited on a ground level. In this way, it may be hoped that similar speech in regard to the open texture of the proper truth predicate could also be rephrased in such terms, in a way that dispells the appearance of a need for a vague metalanguage. Indeed, the extension of the truth predicate in a specific context is seen as being determined by the community reaching a linguistic equilibrium. In regard to the latter, however, and since the related preference relations may in turn be based on semiorders on W, a corresponding form of elasticity may emerge, meaning that, given an equilibrium tuple, in many cases small perturbations of it will also result in an equilibrium tuple. By imposing appropriate conditions on the functions associated with the notion of affordability, and relating them to the indifference portion of agent preferences, an alignment between the tolerant characteristics of the object language conditional and this elasticity could in the end be seen as following. Given a soritical argument formulated within the object language, it could then be claimed that the corresponding metalinguistic argument properly results when matters are framed in such terms. Finally, the notion of equilibrium employed allows for discrepancies between the linguistic acts the various agents may perform, and this could also be seen as being related to what we call higher order vagueness.

12 First introduced in Fitch (1952). A Fitch style presentation was here chosen as an attempt to keep concerns regarding logic and its relation to reasoning, as it actually occurs, in the forefront.