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The Use-Mention Distinction

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The Use-Mention Distinction The Use-Mention Distinction The use-mention distinction is traditionally understood as the distinction between using an expression (or phrase, etc.) and mentioning it. Quoting from Wikipedia encyclopedia: In written language, mentioned words or phrases often appear between quotation marks or in italics. Used words or phrases, being more common than mentioned ones, do not have any typographic distinction. Making a statement mention itself is an interesting way of producing logical paradoxes. Violation of the use-mention distinction can produce sentences that sound and appear similar to the original, but have an entirely different meaning. For example, "The use-mention distinction" is not "strictly enforced here". is literally true because the two phrases in it are not the same. The general property of terms changing their reference depending on the context was called suppositio (substitution) by classical logicians. It describes how one has to substitute a term in a sentence based on its meaning—that is, what referent the term has. In general, a term can be used in several ways. For nouns, they are: • Properly with a real referent: “That is my cow” (assuming it exists). • Properly with a generic referent: “Any cow gives milk.” • Properly but with a non-real referent: “Ulysses’s cow was big.” • Improperly by way of metaphor: “Your sister is a cow”. • As a pure term: “Cow has only three letters”. The last use is what gives rise to the use-mention distinction. However, a lot of misunderstanding and many paradoxes arise also from confusing different ways in which a meaningful expression can be used. Though there is a lot of dispute on using-mentioning expressions, surprisingly, little attention has been paid to the distinction between using and mentioning entities which are referred to when the expression is used. There are many examples of such paradoxes, see, e.g., Gamut (1991, pp. 203,204). To adduce one, consider the following (obviously invalid) argument: The temperature in Amsterdam equals the temperature in Prague. The temperature in Amsterdam is increasing. ⇒ The temperature in Prague is increasing. To explain the problem informally, we have to realize that there is an essential difference between the way of using ‘temperature’ in the first and second premise. Whereas in the first premise the (empirical) function T denoted by ‘temperature’ is used as a pointer to its current actual value, the second premise talks about the whole function T ascribing to it the property of being increasing. Hence in the second premise the function T itself is not used but mentioned. Here is another example: Charles calculates 2 + 5. 2 + 5 = 7 ⇒ Charles calculates 7. Again, there is a substantial difference between using the term ‘2+5’ in the first and second premise. But now, unlike the first example, the distinction does not consist in talking about a function by ascribing a property to it and using a function as a pointer to its value. The first assumption expresses Charles’ relation(-in-intension) to the very procedure of calculating 2+5; Charles is simply trying to execute the procedure, and the procedure (meaning of the expression ‘2+5’) is mentioned. On the other hand, in the second premise this procedure is used to point at the number 7. In the paper we exploit Transparent Intensional Logic (TIL, see Tichý 1988, 2004) to formulate these intuitions more precisely. We conceive meanings of natural language expressions (including mathematical expressions) as abstract procedures, known as TIL constructions. Constructions are structured from the algorithmic point of view. They are procedures, or instructions, specifying how to arrive at less-structured entities. Since constructions are abstract, extra-linguistic entities, they are reachable only via a verbal definition. The ‘language of constructions’, in which constructions are encoded, is a modified version of the typed λ-calculus, where Montague-like λ-terms denote, not the functions constructed, but the constructions themselves. Constructions qua procedures operate on input objects (of any type, even higher-order constructions) and yield as output objects of any type. It is of critical importance to distinguish between using constructions as constituents of compound constructions and mentioning constructions that enter as input into compound constructions. The latter is, in principle, achieved by using atomic constructions. A construction is atomic if it is a procedure that does not contain as a (used) constituent any other construction but itself. There are two atomic constructions that supply objects (of any type) on which complex constructions operate: variables and trivialisations. Variables are constructions that construct an object dependently on a valuation v; they v- construct. Variables can range over any type. If c is a variable ranging over constructions of order 1 (type *1), then c belongs to *2, the type of order 3, and constructs a construction of order 1 belonging to *1: the type of order 2. When X is an object of any type, the trivialisation of X, denoted ‘0X’, constructs X without the mediation of any other construction. 0X is the primitive, non-perspectival mode of presentation of X. There are two compound constructions: composition and λ-closure. Composition is the procedure of applying a (partial) function f to an argument A; i.e., the instruction to apply f to A to obtain the value (if any) of f at A. λ-closure is the procedure of constructing a function by abstracting over variables; i.e., the instruction to execute such an abstraction in order to obtain a function. Finally, higher-order constructions can be used twice over as constituents of compound constructions. This is achieved by means of a fifth construction called double execution. TIL constructions, as well as the entities they construct, all receive a type. The definitions proceed inductively. First, we define simple types of order 1, second, constructions operating on (members of) types, and finally the whole ontology of entities organised into a ramified hierarchy of types. The formal ontology of TIL is bidimensional. One dimension is made up of constructions. The other dimension encompasses non-constructions, i.e., partial functions mapping (Cartesian product of) types to types. The framework of this ontology is a ramified hierarchy of types. It enables us to logically handle structured meanings as higher-order, hyperintensional abstract objects, thereby avoiding inconsistency problems stemming from the need to mention these objects within the theory itself. Any higher- order object can be safely, not only used, but also mentioned within the theory. On the ground level of the type-hierarchy, there are entities unstructured from the algorithmic point of view belonging to a type of order 1. Given a so-called epistemic base of atomic types (ο-truth values, ι-individuals, τ-time points, ω-possible worlds), mereological complexity is increased by an induction rule for forming partial functions: where α, β1,…,βn are types of order 1, the set of partial mappings from β1 ×…× βn to α, denoted ‘(α β1…βn)’, is a type of order 1 as well. A collection of constructions that construct entities of order 1, denoted by ‘*1’, serves as a base for the induction rule: any collection of partial functions (α β1…βn) involving *1 in their domain or range is a type of order 2. Constructions belonging to a type *2 that identify entities of order 1 or 2, and partial functions involving such constructions, belong to a type of order 3. And so on ad infinitum. The rich ontology of TIL makes it possible to distinguish between two basic ways of using an expression E in a reasonable sentence S: let CE be the meaning of E and CS the meaning of S. Then A. The meaning CE of E is mentioned by S iff CE itself is constructed within the CS (i.e., constructed by a constituent of CS). B. The meaning CE of E is used by S to identify an object O (if any) iff CE is not constructed within the CS. Another feature of the TIL language of constructions is its transparency1: all the semantically salient features are explicitly present in terms encoding constructions. In particular, explicit intensionalisation and temporalisation, which is achieved by using variables (w, w1,…, t, t1, …) ranging over possible worlds and time points respectively, enable us to further distinguish two ways in which the meaning CE of E can be used: B1. The meaning CE of E is used by S with the de dicto supposition (i.e., the respective function denoted by E is just mentioned) B2. The meaning CE of E is used by S with the de re supposition (i.e., the respective function denoted by E is used as a pointer to its value) We define a hyperintensional, intensional and extensional context of a sentence S as follows: Let a construction CE be the meaning of E. Then • If CE is mentioned by S, we say that CE and all its sub-constructions occur in S in the hyperintensional context. 1 Pavel Tichý, the author of TIL, understood under ‘transparent’ the anti-contextualism of TIL; however, he agreed with the above way of viewing transparency as well. • If CE is used by S with the de dicto supposition, we say that CE and all its sub- constructions occur in S in the intensional context. • If CE is used in S with the de re supposition, we say that CE and all its sub- constructions occur in S in the extensional context. Finally, valid substitution rules are defined so that a derivation of an invalid conclusion leading to the above-mentioned paradoxes is blocked. Roughly, a “homogeneous substitution”, i.e., substitution of an extension to an extensional context, intension to an intensional context, or hyperintension (construction) to a hyperintensional context is not a problem.
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