Abstract
We demonstrate how to validly quantify into hyperintensional contexts involving non-propositional attitudes like seeking, solving, calculating, worshipping, and wanting to become. We describe and apply a typed extensional logic of hyperintensions that preserves compositionality of meaning, referential transparency and substitutivity of identicals also in hyperintensional attitude contexts. We specify and prove rules for quantifying into hyperintensional contexts. These rules presuppose a rigorous method for substituting variables into hyperintensional contexts, and the method will be described. We prove the following. First, it is always valid to quantify into hyperintensional attitude contexts and over hyperintensional entities. Second, factive empirical attitudes (e.g. finding the site of Troy) validate, furthermore, quantifying over intensions and extensions, and so do non-factive attitudes, both empirical and non-empirical (e.g. calculating the last decimal of the expansion of \(\pi \)), provided the entity to be quantified over exists. We focus mainly on mathematical attitudes, because they are uncontroversially hyperintensional.
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Notes
The first well-known modern example of quantifying into modal contexts may be the Barcan Formula: \(\lozenge \exists x\varphi \rightarrow \exists x\lozenge \varphi \). Where \(\lozenge \) represents logical possibility, BF states that if it is logically possible that something be a \(\varphi \) then something has the logical potential to be a \(\varphi \). The \(\exists \)-bound occurrence of \(x\) in the consequent falls within the scope of \(\lozenge \) (hence exemplifying Quine-style ‘third-degree modal involvement’), and the question arises how \(\exists \) succeeds in binding this occurrence.
See Quine (1960, in particular pp. 147–148).
Co-intensionality for possible-world intensions is necessary co-extensionality: where \(p\), \(q\) are mappings, if \(p(w)=q(w)\) for all worlds \(w\) then \(p=q\), i.e. logical equivalence entails identity, which amounts to an extensional principle of individuation for intensions. For a survey of the ascent from intensions to hyperintensions as meanings and attitude complements, see Jespersen (2010, 2012).
We follow Tichý in holding that it is an analytic truth that deities, including the God of medieval scholasticism, exist contingently. See his (1979) for the argument that necessary existence would detract from the greatness of a deity.
See Church (1993).
Crawford (2014) employs a different notion of propositionalism. Crawford’s notion of propositionalism is the thesis that the complement of an attitude phrased as a ‘that’ clause is a proposition and not anything else, like a sentence or, as Crawford would prefer, a plurality of entities like properties and individuals in the vein of Russell’s multiple-relation theory of judgement.
Likewise it would not do to characterize, let alone define propositional attitudes as all and only those attitudes whose complement is denoted by a ‘that’-clause. Just think of how Latin phrases propositional complements, as in Cato’s famous “Praeterea censeo Carthaginem delendam esse”. Some languages allow both a ‘that’-clause variant and a Latin-style accusative-with-infinitive variant, but Latin does not (*“Praeterea censeo ut Carthago delenda sit” is neither here nor there). For instance, in Dutch we have the choice between “Tilman vindt dit niet kunnen” and “Tilman vindt dat dit niet kan” (“Tilman doesn’t think that this is appropriate”) and in Italian between “Tilman la trova contenta” and “Tilman trova che lei sia contenta” (“Tilman thinks that she is happy”). There are no semantic or logical differences, but competent speakers may well detect an extra-semantic difference. For instance, the subjunctive ‘sia’ signals a slightly more guarded attribution, leaving more wiggle room to qualify or even retract the attribution.
In Duží et al. (2010) we use ‘notional’ after inspiration by the Czech phrase ‘pojmové postoje’ (‘conceptual attitudes’). ‘Notional’ is the English term we would have preferred, for the attitudes under scrutiny confront an agent with a concept or notion of an object, rather than with the conceptualized object directly. But ‘notional’ does not fit the bill entirely, either: also propositions are notional (because conceptual) in character, and Quine (1956) already reserves ‘notional’ for a particular sort of attitudes, to be contrasted with those he calls ‘relational’ (see Sect. 1.3 below).
In fact, there are cases, in English and other languages, where a logical analysis will translate a non-propositional locution into a propositional one. For instance, “I believe you“ is short for “I believe what you are claiming”. “You stand refuted” is another example of the agent putting forward a proposition being identified, at least linguistically (‘you’), with the proposition itself. We are not concerned with such cases in this paper.
This construal is quite reasonable on independent grounds, for ‘Stalin’ (or ‘Cta
h’) was Ioseb Jugashvili’s nom de guerre denoting a particular political persona that was marked by particular properties likely to stir enthusiasm in a fellow dictator. It is this persona Hitler admired, rather than Jugashvili without any qualification of any of his capacities.Church (1951, n. 15) offers various examples of non-propositional attitudes, including the famous example of Ponce de León searching the fountain of youth.
The type of the functional values can be refined from individuals to the more complex one of locations. For details, see (Duží et al. (2010), § 5.2.2).
(Quine (1956), p. 177) points out, correctly, that “[a]ppreciation of the difference is evinced in Latin and Romance languages by a distinction of mood in subordinate clauses; thus “Procuro un perro que habl\(a\)” has the relational sense [...] as against the notional “Procuro un perro que habl\(e\)...”. ” (Italics inserted.). Italian would have “Cerco un cane che parl\(a\)” and “Cerco un cane che parl\(i\)”, resp. In the first case (using indicative) I am looking for a particular dog, which by the way also speaks; in the second case (using subjunctive) I am looking for a talking dog, and any talking dog, or canine talker, will do, although there may be none. A rough English approximation might be ‘a dog, who talks’ and ‘a dog that talks’. However, the problem as we see it is how to convert this grammatical distinction found in Romance languages into a logical distinction.
See (Duží et al. (2010), p. 435) for the exclusive and the inclusive conception of de re attitudes.
Formally speaking, extensional entities like individuals, numbers and truth-values are extreme forms of 0-ary functions, whereas sets are identified with their characteristic functions.
See also (Duží et al. (2010), pp. 124–125).
Of course, the rule ‘breaks down’ when applied to opaque/oblique contexts. But then, every rule presumably does, for there is no knowing what the logic is of such contexts, if indeed they have one. The notion of opacity/obliqueness is misplaced in a logical semantics, because its task is to enable us to draw inferences we know to be valid.
Cf. (Soames (2010), p. 114).
Triple (Quadruple, ...) Execution is a theoretical possibility, though one we have so far never had any use for. The informal explications of constructions above draw on material from Jespersen (2014).
For sure, the empirical execution of a procedure can be more effective or easier using one notational system rather than another. Brown (ibid.) calls this aspect the computational role of a particular notation. We agree on this point. Yet this is a pragmatic aspect; the semantic role of particular notations remains the same.
See Duží (2014a).
Here we use the terms ‘intensionally’ and ‘extensionally’ in the sense of occurring in an intensional or extensional context, respectively, rather than in the sense of possible-world semantics. See (Duží et al. (2010), § 2.6).
For details and proofs, see (Duží et al. (2010), § 2.6, § 2.7).
See also Duží (2014b).
We are grateful to Jakub Macek for the proposal to include \(\upbeta \)-conversion by value.
We are grateful to Jiří Raclavský for calling our attention to this problem. See also Raclavský (2010).
See (Duží et al. (2010), § 4.1).
We are using ordinary infix notation without Trivialization to make the formalization of the construction easier to read. Hence “\([x^{n}+y^{n} = z^{n}]\)” is a shorthand notation for a couple of Compositions. If Pn is the function \(n^{\mathrm{th}}\) power then in the full TIL notation we have “\([{}^{0}\!{=}\,[{}^{0}{+} [{}^{0}{Pn}\;x]\;[{}^{0}{Pn}\;y]] [{}^{0}{Pn}\;z]]\)”.
Here we are again using a shorthand infix notation without Trivialization.
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Acknowledgments
We are grateful to three anonymous referees, two of whom engaged thoroughly with the manuscript and whose comments improved the quality of the paper. This work has been supported by Internal Grant Agency of VSB-Technical University Ostrava Project No. SP2014/157, Knowledge Modeling, Process Simulation and Design (Marie Duží and Bjørn Jespersen), and Marie Curie Fellowship No. 628170, USHP: Unity of Structured Hyperpropositions, FP7-PEOPLE-2013-IEF (Bjørn Jespersen). A version of this paper was read as an invited lecture by Bjørn Jespersen at Munich Centre for Mathematical Philosophy, October 2013, and by Marie Duží at Logica 2014, Hejnice, Czech Republic, June 2014.
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This paper is intended for the special issue on Hyperintensionality.
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Duží, M., Jespersen, B. Transparent quantification into hyperintensional objectual attitudes. Synthese 192, 635–677 (2015). https://doi.org/10.1007/s11229-014-0578-z
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DOI: https://doi.org/10.1007/s11229-014-0578-z