Abstract
We prove how to validly quantify into hyperpropositional contexts de dicto in Transparent Intensional Logic. Hyperpropositions are sentential meanings and attitude complements individuated more finely than up to logical equivalence. A hyperpropositional context de dicto is a context in which only co-hyperintensional propositions can be validly substituted. A de dicto attitude ascription is one that preserves the attributee’s perspective when one complement is substituted for another. Being an extensional logic of hyperintensions, Transparent Intensional Logic validates all the rules of extensional logic, including existential quantification. Yet the rules become more exacting when applied to hyperintensional contexts. The rules apply to only some types of entities, because the existence of only some types of entities is entailed by a hyperpropositional attitude de dicto. The insight that the paper offers is how a particular logic of hyperintensions is capable of validating quantifying-in in a principled and rigorous manner. This result advances the community-wide understanding of how to logically manipulate hyperintensions.
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Notes
See Bealer (1982, 26) for discussion of externally quantifiable variables.
Here we use the term ‘individual concept’ in an intuitive sense. Below we are going to distinguish and rigorously define individual role or office, i.e., individual-in-intension, in opposition to individual hyperoffice, which is a hyperintension presenting an office.
A fully transparent semantics qualifies as ‘semantically innocent’ according to the letter (if not spirit) of Davidson’s characterisation, but we arrive at semantic innocence via the opposite route than Davidson’s. He attempts to make each context extensional; we generalise from hyperintensional contexts to all other contexts. See Duží et al. (2010, 12).
Morton (1969, 163) says, “treatments of non-truth-functional contexts have assimilated them to intensional contexts, either to shade them with the same dark incorrigibility [i.e., the fact that it is obscure how to calculate the truth-value of an intensional context, thus understood. The authors] or to honor them with all the mathematical and philosophical sophistication that the intensional requires.” Our conception of intensionality (actually, hyperintensionality) is the latter, which Bealer sums up thus: “[T]here is no genuinely intensional language; when prima facie intensional language is properly analysed, it turns out to be extensional language concerning intensional entities.” (Bealer 1982, 148) See also Copi (1968, 244) and Klement (2002, 99–100). When these authors speak of ‘intensionality’ they intend intensionality as understood in mathematics, which is hyperintensionality. The coarse-grained intensionality of possible-world semantics equates co-intensionality with necessary co-extensionality, thus yielding (in a logic of total functions) but one necessary proposition, but one impossible proposition, failure to distinguish between inverse relations, etc., etc.
Yalcin (2015, 207) asks, “what should the semantic analysis of attitudes de re look like from a Fregean perspective—a perspective according to which attitude states are generally relations to structured Fregean thoughts, themselves composed of senses?”. Yalcin (2015, 208) claims that “the Fregean position is underdeveloped” and left with a ‘lacuna’, because no Fregean position has so far specified how to compositionally derive truth-conditions for attitudes de re. We beg to disagree. Both Duží et al. (2010, §5.1.2.2) and Duži and Jespersen (2012) answer Yalcin’s question and address his complaint. The Quinian problem of ‘double vision’ (i.e., the Ralph/Ortcutt case; see Sect. 2.2 below) which Yalcin brings up in (2015, § 4) is solved in Jespersen (2015a, 2015b).
Since all true mathematical sentences denote the truth-value T, on an intensional reading the sentence would be a contradiction. On an intensional reading, any true mathematical sentence can be substituted for the complement, and we end up with the paradox of mathematical omniscience.
Since, by definition, the Pope and the Head of the Catholic Church are one and the same office, the sentence would be contradictionary on an intensional reading. Yet, since the sentence can be true, the attitude must be hyperintensional.
Again, on an intensional reading, the sentence would be contradictory; hence, the attitude must be a hyperintensional one.
The attitude must be hyperintensional, because on an intensional reading the sentence is contradictory. If Tilman believes that no bachelor is married then on an intensional reading he must believe any necessarily true proposition, like, e.g., that whales are mammals, and we end up with the paradox of analytical omniscience. The other undesirable extreme is the paradox of analytical idiocy, so to speak. If one believes a necessarily false proposition (e.g., that a forged banknote is a valid banknote) then one would have to believe any necessarily false proposition.
So far, so good. But beyond that, exactly what problem, or cluster of problems, is being discussed in Quine (1956), or his previous work on quantifying into modal contexts, is still not entirely clear. See, for instance, Crawford (2008). Bear in mind that we are not engaged in Quine scholarship as such, but rather in charting the systematic roots of the problem of quantifying-in in the light of how we find it most fruitful to frame it.
One argument against quantified modal logic is his example of the ‘mathematical cyclist’, which is intended to show that it is both necessary and also not necessary that an individual who is a biking mathematician is rational and bipedal. See Duží et al. (2010, §4.2.1) on how to debunk this argument along the same lines as in Stalnaker and Barcan Marcus.
See Kaplan (1986, App. B) on ‘the syntactically de re’, which is supposed to capture Quine’s relational readings. The technique consists in forming a predicate in the passive voice in the vein of ‘is believed by a to be an F’. This yields “The tall handsome stranger spotted on the beach is believed by Ralph to be a spy”.
Quine states that “no variable inside an opaque construction is bound by an operator outside. You cannot quantify into an opaque construction.” (Quine, 1960, 166) It is the right move, of course, for Quine to resist quantifying into opaque contexts. In Quine (1960) and elsewhere, he likens trying to quantify into opaque contexts to trying to quantify into quotation contexts. In Quine (1956) he gives an additional reason. In the famous Ralph/Ortcutt case, it is true that a is believed by Ralph to be a spy, that b is not believed by Ralph to be a spy, and that a = b. What happens is that quantification ‘quantifies away’ the two different guises under which Ralph has encounted a/b. There can be no individual such that it is believed, and also not believed, by someone to be a spy. So, quantifying-in would yield a paradox. See also Kaplan (1986, 269–70). But, or so we think, the fact that opacity appears to be the root cause should have given Quine pause. He ought to have reconsidered the assumptions and tenets that landed him in an (ostensibly) opaque context that suspends quantifying-in on pain of paradox. The conclusion should not have been that opacity is a fact of linguistic life, or ‘intensional’, i.e., anti-extensional, logic a fact of logical life, thus turning some contexts into no-go areas. In fact, the strategy pursued by TIL is to design a formal semantics that cannot generate opacity, again with provisos for quotational contexts. (We note that the line of reasoning found in the Ralph/Ortcutt example resembles that of the reasoning behind the ‘mathematical cyclist’; see fn. 12; see fn. 11.)
The main difference between Pickel and Cumming is that Pickel assigns a more elaborate semantics to his variables. Cumming has, as it were, got only the first half right. Pickel provides an argument to the effect that Cumming is unable to distinguish between true and false beliefs. Assume that a believes, (Ba), that the value of xb is an F. This is formalised thus: “BaFxb”. This is a closed formula, because operator B binds the variable. Assume that σ(xb) = Dublin. The formula being closed, it retains its truth-value independently of any assignment functions other than the original σ. Now let an arbitrary assignment function, τ, assign a different value: τ(xb) = Lublin. It is true, therefore, that a believes that Lublin is an F. Except, of course, it is not. Pickel’s remedy is to assign a dual semantics to variables. Whether BaFxb “is true on assignment σ depends not just on the value of xb relative to σ, but also on whether every world-assignment pair [\(\langle {w, \tau} \rangle\)], in the agent’s belief set makes true [the ‘quasi-open proposition’ Fxb]. The assignments in [believer a’s] belief set may assign different values to x and y, even though x and y co-refer on the input assignment [σ].” (Pickel, 2015, 347). TIL goes in the opposite direction. We do not want the option to change horses in midstream, so to speak, by bringing in an alternative to the ‘input assignment’ in a static context. TIL does not capture an agent’s idiosyncratic perspective by means of ‘shiftable’ assignment functions, but by means of fine-grained, structure-sensitive hyperpropositions as attitude complements.
See Duží et al. (2010, §3.3.1).
See Duží et al. (2010, §3.5) on anaphoric reference.
Whether notional/relational must map onto de dicto/de re is far from a foregone conclusion, though, as different theories will have different conceptions of the dicto/re distinction. For instance, should some form or other of acquaintance play a role in attitudes de re? [For the record: no, not in TIL. See Duží et al. (2010, 435)].
Quine’s original objection to quantified modal logic is that (what appears to be) the same variable will have both used and mentioned occurrences within the same context. See also Kaplan (1986, 262–63). On a similar note, Pickel (2015, 340) objects to Cumming (2008), “There is no coordination between the occurrences of x outside of the belief ascription and the x occurring within the belief ascription”. Our distinction between displayed and executed modes of occurrence of procedures, including variables, is sort of parallel to the distinction between words occurring mentioned or used, and quantifying into a displayed procedure is sort of parallel to quantifying into a quotation context. But we do not wish to push the parallel too far. Attempting to quantify into a quotation context is a no-starter, whereas the main technical point we are making here is that it is both feasible and sensible to quantify into a displayed context.
For further critical comments on contextualism, see Duží et al. (2010, 110–112).
See Turner (1992, 165) for a ‘flat version of Montague’s intensional logic’ developed within the untyped λ-calculus.
See Berto and Nolan (2021, §2.2) for examples and discussion.
‘Function’ is historically ambiguous between Frege’s Funktion (generation of mapping) and Wertverlauf (mapping). Church (1956, 16) is clear on this: “If the way in which a function-in-extension yields or produces its values for its arguments is altered without causing any change either in the range of the function or in the value of the function for any argument, then the function remains the same; but the associated function concept, or concept determining the function …, is thereby changed.” In modern-day parlance, function concepts qualify as hyperintensions. Our notion of hyperintensions is rooted in Church’s function-in-intension or functional concept, Frege’s Funktion and Sinn, as well as Turing machine. Application and abstraction are theoretical primitives, which are central to the definition of two of our hyperintensions. See Definition 1 (iii), (iv), in Sect. 3.4.
Caie et al. (2019) is an exceptionally rich paper, which we would have liked to engage with at length. For now, we are just scratching the surface and confining ourselves to the core question of the validity of Leibniz’s Law, hence of Substitution. Thus, this comparison of ‘classical opacity/transparency’ is about substitution specifically rather than quantification. We want to stress that we find it commendable that someone should try to develop a logic of opacity, which will distinguish between true and false instances of Substitution. Opacity has typically been understood purely negatively as the failure to preserve transparency, but Caie et al. (2019) helps clarify what the logical implications are of adopting opacity. Still, we disagree with treating opacity as a datum (even in the explorative spirit of Caie et al. (2019)), rather than as a symptom of a wrongheaded formal semantics.
“Water = H2O” has never sat well with us. How can a liquid be identified with a molecular structure? This smacks of category mistake, or type-theoretic incongruity. “Water = H2O” feels like a throwback to the long-gone days of materialist reductionism. We would rather say that (pure) water has H2O (namely, as its molecular structure).
See Sect. 4.1.1 on procedural isomorphism, which defines co-hyperintensionality.
See also Duži et al. (2010, 3): Propositional Hesperus/Phosphorus.
Moschovakis (2006) characterises meanings as generalised algorithms. Our procedures likewise qualify as generalised algorithms, because they are procedures that need not be effectively computable (thereby perhaps straining the idea of an algorithm a bit).
This qualifies TIL as a reductionist theory of hyperpropositions, because hyperpropositions are ‘reduced’ to instances of a more general sort of entity instead of being sui generis. TIL is also a reductionist theory of propositions, i.e., the truth-conditions of empirical sentences, because TIL identifies them with functions from possible worlds to functions from times to truth-values.
Indexicals being the only exception: while the sense of an indexical remains constant (i.e., as a free variable with a type assignment), its denotation varies in keeping with its contextual embedding. See Duží et al. (2010, §3.4).
This program of anti-actualist semantics is described in Duží et al. (2010, §2.4.1).
For further details, see Duží et al. (2010, 301–11).
First-order, higher-degree intensions are defined as functions from intensions to functions that contain an intension in their domain or range.
See Duží et al. (2010, §2.5).
For critical comments on Montague’s IL and a comparison with TIL, see Duží et al. (2010, §2.4.3).
When speaking of ‘world/time pairs’, we are allowing ourselves to pretend that a function from worlds to a function from times to entities is equivalent to a binary function from world/time pairs to entities. This pretence is innocuous in this essay, because here we are not considering the modal and the temporal dimension separately. See Duží et al. (2010, §2.5). Moreover, in a logic of partial functions, such as TIL, schönfinkelisation fails to always preserve equivalence: see Duží et al. (2010, 204–05).
By ‘privileged’ or ‘canonical’ form we intend the literal analysis of a sentence, where syntactically simple terms like ‘Pluto’ and ‘planet’ are paired off with a Trivialisation of the denoted object, here 0Pluto, 0Planet. For the notion of literal analyses, see Duží et al. (2010, 105, Defs. 1.10, 1.11).
This exposition relies on Duží et al. (2010, §2.4.2).
It may be instructive to consider how TIL formalises the Barcan Formula: ◇∃xFx ⊃ ∃x ◇Fx. There are two ways to go about this. Either we stick to ◇ or we turn to existential quantification over worlds (ignoring times). S5-possibility is typed as a property of propositions, of type (ο(οω)), which is not indexed to worlds, because the S5-modalities are analytic, being valid on equivalence frames. Sub-S5-modalities are not, so properties of propositions are indexed to worlds, and for this reason such properties are of type ((ο(οω))ω): see Materna (2005).
-
(i)
[[0∃λw [0∃λx [0Fw x]]] ⊃ [0∃λx [0∃λw [0Fw x]]]]
-
(ii)
[[0◇ λw [0∃λx [0Fw x]]] ⊃ [0∃λx [0◇ λw [0Fw x]]]]
Both formulas, on their intended interpretation, express that if some world has some individual with property F then some individual at some world has property F. It is obvious why the Barcan Formula (and its Converse) requires S5-possibility and that the domain function be constant: for all \(w, w^{\prime} \in W,\) D(w) = D(w′). It is also obvious (as proved by running a type check) why neither of (i), (ii) engenders the problem that a λ-bound variable has an ‘opaque’ occurrence.
-
(i)
Restricted β-reduction consists merely in the substitution of variables for variables.
The reason is that TIL is a typed λ-calculus with a Church-style semantics, in which every λ-term comes with a unique type and types are organised into disjoint layers. On the other hand, Curry systems are essentially treated as untyped λ-calculi, in which a term is associated with a set (which may be empty) of potential types. See Gordon and Melham (1993) on HOL.
Actually, the sentence “Cotg of π equals 0” comes with the presupposition that the value of the function Cotg exists at π, because this is entailed both by the sentence and its narrow-scope negation, “Cotg of π is not equal to zero”. See Duží (2017, 2018a, 2018b) for more on presupposition and negation.
We are glossing over a slight complication here. Our hyperpropositions are not truth-bearers, so they cannot, strictly and literally, be believed to be true (or false). Rather it is truth-conditions, i.e., the propositions of possible-world semantics, that are truth-bearers: a truth-condition is true when it is satisfied. So to believe an empirical hyperproposition is to believe that the truth-condition it produces is satisfied at the given world and time of evaluation; and to believe a mathematical hyperproposition amounts to believing that the procedure produces the truth-value T. This technical detail is solved in Duží et al. (2010, §5.1.6).
For details on how TIL analyses anaphoric resolution, see Duží (2018a).
For literal analysis, see fn. 39.
See Jespersen (2021) for arguments in favour of procedural isomorphism, together with a concrete application.
See Anderson (1998).
Such a constraint is known as a requisite in TIL. See Duží et al. (2010, §4.1).
See Duží et al. (2021) for our logic of analytically impossible hyperoffices.
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Acknowledgements
We are much indebted to two anonymous referees for Linguistics and Philosophy whose careful remarks helped improve the quality of this paper. The paper rounds out the quantifying-in trilogy whose other two instalments are our (2012) and (2015). This research has been supported by the University of Oxford project New Horizons for Science and Religion in Central and Eastern Europe funded by the John Templeton Foundation, as well as by Grant No. SP2021/87, VSB-Technical University of Ostrava, Czech Republic, Application of Formal Methods in Knowledge Modelling and Software Engineering IV.
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Jespersen, B., Duží, M. Transparent quantification into hyperpropositional attitudes de dicto. Linguist and Philos 45, 1119–1164 (2022). https://doi.org/10.1007/s10988-021-09344-9
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DOI: https://doi.org/10.1007/s10988-021-09344-9