Abstract
The essay is dedicated to the memory of Jaakko Hintikka and Hilary Putnam, two logically inventive philosophers who, nonetheless, showed deep judgment in bringing to the fore the limits of reducing natural languages to formal languages, via the use of logical forms and model theory. Writing in parallel ecologies, the two proposed rather similar “limitative” theses about the popular logical-form-cum-model theory methodology.
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Notes
- 1.
Neither the Russell of the 1902 Principles of Mathematics nor the yet more influential Russell of the 1905 “On Denoting” uses the famous first order decompositions that are standardly called “Russellian analyses”. Russell’s treatment of \(\forall x\) and \(\exists x\) is high order (“f(x) is always (sometimes) instantiated”). But the lore used later first order reductions (essentially due to the focus on first order formulas by Lowenheim, Skolem and Tarski) to relate to what we might call the first-orderized Russell (e.g. in casting his theory of (definite) descriptions). The 1902 Russell—much investigated by Hintikka—avoids what he calls “the method of the variable” and the “denoting phrases” decompositions—first or high order. We return to Russell’s semantics and logic for ordinary English below. I owe the separation of the first-orderized versus high order 1905 Russell to three decades of meticulous and penetrating teaching of the 1902–1905 period by David Kaplan.
- 2.
The naturalist tradition—as it surfaces in the philosophy of mathematics—runs from Skolem to Henkin to Hintikka and Putnam, and will be dissected in the sequel paper. I will only mention here the key Skolem piece diagnosing the limits of model theory as a semantics of mathematical English, see [26]. Specifically, we refer to “Einige Bemerkungen zur axiomatischen Begriindung der Mengenlehre”, in the just mentioned 1970 collection, pp. 137–152. This is the key Skolem 1922 paper.
As for Hintikka. The reader may wish to have before him as an encapsulation of Hintikka’s standpoint the trio “Quantifiers versus quantification theory” [13] and The Principles of mathematics reconsidered [15], as well as the joint paper with Gabriel Sandu on standard versus general model theory “The skeleton in Frege’s cupboard: the standard versus nonstandard distinction” [16]. I do relate locally to pertinent papers as we advance. In the late stretches on pluralities and functions, the overview by Barbero and Sandu of IF-logic would be helpful to the reader in [8].
Second and related, I am aware that many followers of Hintikka’s own internal developments within the structural-semantical landscape—what is often called game-theoretic semantics—will point out to a much weaker and “conservative” thesis on his behalf: Hintikka was out to deny strong reductions of meaning e.g. to first order classical model theoretic ideas. But all the same, his enhanced class of logical forms (e.g. his “independence friendly” notion of logic) and enlarged class of primitive meanings were meant merely as a variation, his own proposal being cast within the model theoretic program, one pointing to generalized first order logic and its model theory as providing the reductive “oracle” as to truth and meaning. Thus, the Montague we discuss below sought the full power of the lambdas (over property variables) and full second order (“standard”) model theory, making set and property theoretic notions the key. In contrast, it may be said, Hintikka seeks Skolem functions (and in turn a restricted \(\varSigma ^{1}_{1}\) fragment of second order logic) as the key, a fragment he can in turn, represent in his IF logic without speaking of sets. So we may read Hintikka in Montague’s vein as urging a liberalization-cum-generalization within model theoretic logicism. But I am inspired, sometimes by reading between the lines, to conceptualize Hintikka in the vein of Skolem—it is not more (than elementary) model theory that is needed; rather what is needed is a separation of the algebraic intra mathematical work done by model theory and the claim it provides something of quite a different order—a semantics/meaning theory. I return to this dual reading of Hintikka below. As for Putnam, see the key paper, “Is semantics possible?” [22], especially pp. 149–150 and the final remarks on p. 152. See also Putnam’s discussion of Skolem’s morals in the philosophy of ordinary language all the way to Putnam’s Skolemite conclusion, that model theory—first or higher order—is not a theory of meaning, in the final pages of [23].
- 3.
Note that the formulation—so far—stayed silent on the matter of “unique decomposition” often thought to be built into the “fundamental theorem”. We surely all agree there is meaning-ambiguity in English words (“bank”) and in composite units such as phrases and sentences. But whether the (two) understandings of “every boy danced with some girl” or “flying planes can be dangerous” is due to alternative “structural” (“logical form”) methods of combination or alternative primal meanings (as in “I am at the bank”) is a key question addressed below. Either way, the many meanings—if such there be—of e.g. “flying planes can be dangerous” will be a result of the primal meanings and the product operation(s). For the duality (multiplicity) of meanings to be accounted for systematically we need not presuppose dual forms. We return to this matter below.
- 4.
A word about Montague’s method. In his double reformation drive—higher order and intensional—Montague was focused on countering the influential tradition of philosophical logic dear to Quine and Davidson (and how they were reading the first-orderized 1905-Russell). The first order tradition uses the logical forms of first order quantification theory—forms that were decomposing the unified determiner phrases of ordinary English (such as our above “every man”) and, in turn, were decomposing into ordinary predicates F(x), R(x, y), etc. the adverbial and adjectival modifying forms “John is a clever Belgian mathematician” and “John kissed Mary in the kitchen (in a dream)”. To go with, Montague was critical of the “merely” first order model theory that was given to these logical forms as too “weak” to (1) express the meanings of the sentences involved, (2) express (at all) related grammatical mates: “most critics are wise”, “infinitely many primes are of the \(4n+1\) form”, “John and Mary coordinated a plan” etc. See Montague’s “English as a formal language”, “Universal grammar” and “The proper treatment of quantification in ordinary English”, respectively EFL, UG, PTQ, Chap. 6–8 in [20]. A very elegant and less technically heavy critique of “first orderism” in Montague’s vein is offered by [27]. Aside of our own detailed critical examination of Montague’s treatment of proper nouns and determiner phrases below, the reader should compare with Hintikka’s critique in “The proper treatment of quantifiers in Montague semantics” in [14].
- 5.
It is not often noted but it is of course possible to treat “bank” as one word but evaluate it relative to a new parameter (on top of the world and time and context parameters), a “dictionary”. I followed this track way back in my Oxford dissertation (published in Synthese [3] with comments made in correspondence by Hintikka to myself and the late Pat Suppes). The seeds of the idea are in Montague’s EFL. See also the disambiguating method used in my “Semantical anthropology” [2] due originally to Kaplan, and exploiting the historical chain leading to the specific utterance of “bank”. I now think I was wrong, not “technically” but “conceptually”, in thinking that a single word—taken by its surface spelling only—carrying two meanings creates a “crisis” vis a vis truth evaluation. The “ambiguity” of “Aristotle was clever” (say, between two Greeks so called) may seem a “crisis” from the epistemologically ignorant point of view of an arbitrary “receiver”, who has only the spelling to go by (what I call below a “Google receiver”). But the intra-natural-history uttered word(s) proper are as determined as could be and evaluable for truth with no “crisis”. See our development below.
- 6.
In his “Linguistics and natural logic” [18], Lakoff with his remarkably “musical ear” spots examples of a similar phenomenon but goes on to compress them into his theory of underlying logical form, as he does to the semantic entailment he hears between “Mary killed John” and “Mary caused John to become not alive”—if entailment there is, reasons Lakoff, it must emanate from an underlying compound logical form of “kill”. Schematism, after all.
- 7.
Pressed on me for some years now by Jessica Pepp.
- 8.
I will not broach now the classical (reputedly) scope cases involving plural subjects as “John and Mary or Sam are in town (coauthored the paper)”. We discuss them below while discussing plural subject phrases.
- 9.
Thus sentences with the definite noun phrase (as those that famously interested Frege and Russell, “The Queen of England is happy”, belong in class (2) and not in class (1) that sports only simple proper nouns and free standing pronouns (“he is happy”).
- 10.
The classification follows key cases in the history of semantics from (Frege-)Russell to our day. I here treat as belonging to the same type of case the intransitive verb case “Barack runs” and the adjectival “Barack is happy” together (class (1)), in the wake of the common simple monadic predicative symbolization “F(a)”. Again, for myself, I do not think that the two cases have the same semantics—a verb and an adjective introduce substantially different types of meanings—and neither involves the logicized idea of a predicate (and its “extension”) or the idea of open sentence F(x). But I abstract from my own ultimate views to guide us to the key central cases using familiar terms.
- 11.
As noted, I believe “is happy” and “runs” have different types of meaning, neither of which is given by the format “F(x)”. It will become clear that the standard third type of “F(a)” sentence we-symbolizers deploy, with the re-presentation/reduction of “Michelle is a lawyer” to “Lawyer (m)” is at the heart of both the first order and higher order deformations of English DP’s. See immediately below in class (2).
- 12.
Quine’s much used first order scheme \(\forall xy(x=y \rightarrow (OF(x) \leftrightarrow OF(y)))\) is not a correct articulation of Leibniz’s second order logical law of identity. Open sentence complex forms OF(x) can not be presupposed to express genuine properties (demanded by the law). Indeed even if substitution holds for some (all) of these sentential embeddings, they are due to the semantic-meaning of the new embedding operator O (negation, necessity, always-true, believes that) and not due to issues in the semantics of the referring proper noun, which is where both Quine and Montague (in their orthogonal escape routes) concentrate their their efforts. In like manner, while the Thomason-Stalnaker use of lambda abstraction to get at adverbial cases is better than the form OF(x) it is still not to be confused with subject predicate grammar. Thus [\(\lambda x\) it might have been the case that: lose (x)](Nixon) is not at all adverbial; we use one sentential operator—the Lambda—to abstract over the modal de dicto open sentence with sentential modifier “it might have been the case that”. There is no sentential modification or variable binding in “Nixon might have lost”. See more below, when we discuss anaphora, on variable binding in logical form, when there is none in English.
- 13.
Michael Bennett who was a student of both Montague and Kaplan, took notice of these Kripke/Kaplan observations about empty names (common nouns). He noticed that “unicorn” is not a predicate whose extension is actually empty but such as to have nonempty extension in other worlds—the very intension (!) of “unicorn” is deficient, in no world is there an extension for it. Bennett tried to amend the notion of logical consequence (and underlying idea of meaning) in Montague’s grammar to accommodate such ideas as of irreversibly-empty name (common noun). See on Bennett’s elegant work below.
- 14.
I note that the first path of dropping existential generalization and allowing empty proper names, “S. Holmes”, to refer to mere possibles (from domains of “other” worlds), was followed early on by Kripke (in his 1963 Acta Philosophica Fennica paper on variable domain semantics for quantified modality; a similar policy was followed by Hintikka (at the time) in his own version of free logic). The change of heart circa 1970 led Kripke and Kaplan to see that an empty name is empty forever (necessarily)—if no semantic referent gets generated in actuality, it is too late (the model theory treating all worlds—actuality included—symmetrically is missing this “semantical meanings are generated in actuality” insight). The semantic difference between “Vulcan” and “Neptune” is developed in my own “Naming without Necessity” [4]. David Kaplan amplifies in “Afterthoughts”, where he defends existential generalization as semantico-logical consequence though modally contingent. See [21]. We return below to the prior and unique role played by the actual world in generating meaning and in gauging semantic-logical consequence.
- 15.
Kaplan raised these worries regularly in lectures on Russell’s theory of denoting phrases (DP) in between the mid sixties and late eighties in many public lectures. Of course Russell himself was skeptical about the reduction in his “Principles of Mathematics” of 1902 but by the 1905 “On Denoting” he thought the method of the variable will save us from a host of paradoxes the primitive nominals (DP) induce. I note that Jaakko Hintikka, like Kaplan, was sensitive to the difference between the reductive method of the variable coming from the languages of symbolic logic and the method of denoting (determiner) phrases coming from ordinary English. Like Kaplan, he urges a re-examination of Russell’s method of DP’s in “Principles”, when our aim is to understand, not eliminate and replace, ordinary English. I should like it noted that using the common noun itself with the indefinite in tow to form an artificial predicate “is a-philosopher”, as if this is the missing adjective, is not solving Kaplan’s challenge because we run in a circle: to satisfy “is-a-philosopher” x has to be in the reference of the common noun—the philosophers—viz. to be one of the philosophers, just as to satisfy “is Barack Obama” one needs to be Barack Obama. The Russell-Quine presupposition that to be Obama or to be among the philosophers just is to satisfy a reference-of-noun free predicate is being questioned here. They both reverse the natural order: they assume common/proper nouns are nominalizations of an underlying predicate whereas in truth, in ordinary English, the noun is prior and there is no natural derived predicate that is meaning identical with it. Reference is not predicate-satisfaction, not even predicate-satisfaction that is modally stable, viz. across worlds. More on this difference between reference to an object and the extension/intension of a predicate comes up below.
- 16.
In PTQ fn. 8 Montague says—I do not know whether tongue in cheek or seriously—that it would be “non-ethical” for him as a logician or semanticist to take a stand on whether merely possibles—like Vulcan—are to be countenanced as referents of names and values of variables of quantification. I feel quite the contrary—since when has the truth of the (any!) matter been...unethical? The truth sets us free, even in logic and semantics. If Obama and tigers exist only contingently and non-eternally, if—now without names—there could have been more individuals than there actually (presently) are and it is also possible that less individuals than those that actually (presently) are would be around, it is very “ethical” (to use a rather big word) to note the matter when we account for the truth value of related English sentences. Validating for logical simplicity’s sake analogs of the Barcan formula and its converse viz. “everyone in the room is necessarily not older than 40” therefore “necessarily everyone in the room is not older than 40” and “necessarily everyone exists” therefore “everyone necessarily exists” (and similar temporal claims) does not make one more ethical. The claims are intuitively false and avoiding striking falsehoods has, at least in the eyes of the present naturalist semanticist, its “ethical” point. We can see in this “modal-logical” famous case the difference between doing the model theory of a formal language (quantified modal logic) and giving the meaning of English sentences. With the former, it is of course “ethical” to give whatever models one finds interesting—the fixed domain—just like the variable domain—model theories are all interesting and it is “ethical” to indulge in either. When it comes to the meaning (and truth) of English sentences one is not anymore merely constructing (interesting, elegant etc.) classes of models.
- 17.
Many class even the DP-free and proper noun-only but intensional “John seeks Mary” as ambiguous between a wide scope quantifying-in reading (“there exists something such that it is \(M^*\) and John seeks her”) and the small scope (“John seeks Mary”). Of course some like Quine (and early Montague) sought to decompose “seek” into sentential operators such as “tries that he finds that”, whereby we indeed get two sentential operators, a quantifier and an sentential attitude operator, to generate scope distinctions. But even if we keep the intensional verbs such as as “seek” un-decomposed, many hear in this case, an ambiguity between a “de re” reading—Mary is such that John seeks her versus the “intensional nonspecific” reading on which John may seek Mary even if she does not exist. The discussion often conjoins here (1) specificity and (2) existence (by way of the wide scope existential quantifier). Thus, to my ear, the claim “Mary is such that John seeks her” does not involve an existential quantifier, specific as it obviously is (Russell-Quine reduce the two issues by their symbolization). We untangle these matters below.
- 18.
See my earlier “The subject verb object class I, II” [6].
- 19.
I ignore at this point the difference between the plurality of presidents and the kind. Obviously, both are not the set of (actual, possible) presidents. There are important differences, over temporal/modal predications, between the kind and the plurality. When we say “Bengali tigers might have been less rare (than they actually are)”, let alone “tigers might not have existed”, a question arises as to what is being predicated modally (or temporally, at “other times”). The answer (my answer at any rate) is—the kind is always the primary referent and the plurality (which is given as of-the-kind) is derivative, though at times checking the predicate on the plurality suffices. We return to this matter below (see the appendix on modal-existential claims). As far as our initial comment on the dual use of the CN goes, the difference between the individual versus kind referential readings may be abstracted from this further distinction, applying to the kind and the plurality of e.g. sloops.
- 20.
The role of the determiner is fixed in both the nominal and pre-nominal uses, viz. to indicate how many from the plurality—conventional or local—are to be assessed for the predicate.
- 21.
The point about translation—e.g. “Father Xmas” is translated by “Pere Noel” and not by “Robin des Bois” originates with Keith Donnellan, who influenced Michael Bennett (his student). See “Speaking of Nothing”, in Donnellan’s collected essays edited by Leonardi and Almog [12].
- 22.
- 23.
I assume the verb introduces a relation. As indicated, the semantics of verbs and adjectives is not as the forms the predicate-logic F(a), aRb, would have it. But I abstract from this in the present essay.
- 24.
The validation of “Barack sought Michelle (a lawyer)” therefore “Barack sought Meesh (an attorney)” in both the specific and nonspecific readings of the indefinite—need not suggest such a transition is semantically-valid for the subordinative-clausal “Barack believed that: Michelle (a lawyer) is interesting”. In fact, I think the latter is a truth preserving transition. But this cannot be drawn from the semantic meaning of “seek”. It requires a separate discussion of the semantics of “believes that”. See [7].
- 25.
This reveals to us that the e.g. for temporal/modal adverbial predications (modality de re) “Barack might have (once in the past) lost the election” and the standard reduction to a quantifying into de dicto form, using sentential operators “there exist something such that: it is \(B^*\) and it is possible (once was the case) that: it loses the election” are not synonymous. This has been noticed by Cartwright and Wiggins and could have averted much of the Quine-Kripke dispute about quantifying into modal operators (while they were discussing in effect ordinary English modal adverbials, as in “Nixon might have lost”). But the irreducibility of the subject-predicate to de dicto operator form has not been appreciated by the many popular discussions of this dispute.
- 26.
See [13]. Kaplan’s point is made in a letter to Quine of 1970. It is hoped that this elegant letter will be published soon (Boolos’ well known reproduction of the proof in the late 80s does not not get at many subtleties in the original letter). I expound and analyze Kaplan’s “proof” of non-elementarity in [5].
- 27.
This material of Kaplan is unpublished though discussed for many years in classes at UCLA. I dissected it in a laudatio for his 80th birthday (2013) called “The symbolizer’s travels”, also unpublished.
- 28.
This way of putting things I owe to David Kaplan. It is originally present in Russell’s 1902 Principles and noticed by Hintikka.
- 29.
The reader will guess that quite apart of the relation to the non-numerical “some critics...”, this second order formula [LF-2Kaplan-numbers] is, on my view, merely model theoretically equivalent and not at all giving us the meaning of the subject predicate mathematical English “some numbers precede only one another” (which has nothing like the singularist disjunctive \(x=0 \vee x=y+1\) showing up in this surrogate).
- 30.
Hintikka introduces a richer syntax into the quantifier notation to indicate dependence patterns, what he calls the “slash notation”. In a standard set up, an existential quantifier \(\exists y\) in the syntactic scope of a universal quantifier \(\forall x\), is forcing the semantical treatment of the existential to depend on that of the universal—syntactic dependence in “scope” induces semantic dependence in value choice. Thus in our famed “for every x, there is y” sentences discussed above, we in effect have, following an idea of Skolem, the occurrence of a function f which is the controller of the existential quantifier. Once we evaluate the universal quantifier and assign some “arbitrary” individual u to its variable x, the function f selects f(u), an individual to witness the existential quantifier. These functions are called “Skolem functions”.
In the Hintikka enriched syntax, we aim to liberate the idea of semantic dependence (and the mode Skolem functions operate) from that of scopal syntactic dependence. Using the slash notation, we can see that in the formula \(\forall x \forall y \exists z/y f (x,y,z)\), the existential quantifier over z is syntactically-scopally dependent on both x and y, but is marked as independent of y. In the semantical evaluation of the truth of the quantifier formula this means that the Skolem function selecting a value for z will depend on the choice for x but not for y.
- 31.
As Gabriel Sandu points out, even on just the finite models, the encodability of existential second order logic in Hintikka’s partially ordered IF logic has important consequences. For example, the feature of a program solving a problem in non-deterministic polynomial time is known to be characterized in existential second order logic over the finite models. Indeed a form of expression of ill-foundedness of a two place relation is also available (the key idea in the Geach-Kaplan mathematical original “some numbers precede only one another”).
- 32.
Sandu points out—again correctly—that even on finite models the Hintikka notation is more expressive than standard first order notation. Indeed. But this in no way supports the Hintikka-Kaplan intimation that the relevant “new” logical notation—second order quantifiers or function quantifiers or slash notation on first order quantifiers—is involved in giving the meaning of the troubling English sentences. I am very grateful to correspondence with Sandu and his many apercus.
- 33.
Of course we can’t apply deductive consistency to the English sentence with its interpreted lexica. The formula acting as its logical form, when schematized can be shown to be interpretable in infinite models (by which time its satisfiers are not critics!).
- 34.
It may well be true—inside our metaphysics of sets that for any limited (infinite) plurality (barring absolutely infinitely pluralities), that there exists a set comprising the items as members. But this set existence axiom is not coming from the semantics of plurals.
- 35.
Note that in English, the Kaplan non-standard integers sentence “some numbers precede only one another” is true but the formal variant but lexically different “some natural numbers precede only one another” is not true. Lexical meanings of ordinary English achieve what the logical forms of purely schematic predicate letters governed by “weak” schematic first order axioms or Henkin-style second order axioms (no full induction) cannot discern.
- 36.
Let it not be surmised in the metaphysics the plurality may exist without the function. The coexisting is symbiotic as indeed in the case Dedekind models here in abstract terms viz. 1, successor (1), successor (successor (1))...No numbers, no successor function, no successor function, no numbers. But we must separate semantics—what our discourse refers to from metaphysics—what exists. The situation is analogous to my referring to Barack Obama, with my noun “Obama”. In the metaphysics the man BO exists iff the set \(\{ BO \}\) exists (this much is of the strictest necessity). Nonetheless, when I say “Obama”, I do not refer to the singleton or to the infinite plurality of such singleton of singleton induced in the metaphysics. I elaborate on the metaphysics of pluralities, functions, sets and the absolutely infinite universe comprising them all in my Kit Fine festcschrift piece “One absolutely infinite universe to rule them all” edited by Mircea OUP, to appear.
- 37.
I do want to agree here with Hintikka (who expressed himself consistently (to no avail) on the “meaning” of the “continuum hypothesis” statement): the set-free 1874–1878 talk—the pre-menge talk (let alone embedding that menge-talk in axiomatic set-theoretic foundations as urged by Zermelo 1908) is the original form of comparing the reals and the countable ordinals. Hintikka made the point theoretically and ahistorically as part of his set theory critique, I here second it by pointing that the actual historical figures Cantor and Dedekind (and in my view Skolem 1922) did not use an intra set-theoretic casting of such correlations. The 1870s casting of the continuum hypothesis as a match of reals and countable ordinals (in this plurals involving language) is the original continuum problem. It is quite different from the set-theoretic statement let alone the problem it had to come to be, viz. the essentially proof theoretic-problem whether inside the language of set theory (for that matter: extended by large cardinals) and given a certain axiomatization using this set language (first or second order), we can settle—prove from axioms—the set-language coding of the continuum question (now cast as about set-cardinals). I believe Skolem tried to tell us this—the original issue here is not an intra-set-theoretic first order axiomatization question—in his above cited anti-foundationalist paper of 1922.
- 38.
My thesis of 30 years ago [1] looked at the 1922 Skolem unto-to-Henkin (standard vs. general model theory) theme and moved on next to the late 60s Putnam-Hintikka lineage of this pessismism-motif about the reduction of meaning theory to algebraic-model theory. In the three decades since that thesis, I have gathered many debts on this topic: to the late Jon Barwise for being so kind when in the early 90s I turned to studying the Geach-Kaplan plural subjects sentence “some critics admire only one another” and Hintikka-sentences with branching subjects. I owe many thanks over three decades to Hans Kamp and Barbara Partee, the late Keith Donnellan, the late Hilary Putnam, Tyler Burge, Tony Martin, and most of all, David Kaplan, whose ideas about the non-elementarity of marketplace English’s semantics are dissecte in some detail. In this essay and linked to Hintikka’s (in a way that (oddly) never occurred to me in the last 30 years). The material on Kaplan is drawn from the laudatio on his 80th birthday fest called “The symbolizer’s travels”. For late help, I owe thanks to Olli Koistinen, Jessica Pepp, Andrea Bianchi, Vesa Halava, Jani Sinokki, Hans van Ditmarsch and especially regarding determiner phrases and scope to Gabriel Sandu.
- 39.
We assume here the modal truth that If P, it is necessarily possibly P, a modal truth verified by the natural actuality-generated references of Kripke and Kaplan (thus sustaining the projection from P’s truth in actuality, the possible truth of P in any world). In any event, Carnap-Montague worked throughout in a modal framework presupposing S5 which verifies this truth.
- 40.
I urge that Montague’s surrogate is even more problematic than the ordinary descriptive surrogate provided for names by the classical Frege/Russell description theory of names. The classical descriptive surrogate “the x F(x)”, whether intended as rigid or not, is given for “Barack” and has a problem—in a varying domain set up—locating in the Obamaless-world w a value for the variable to satisfy “F(x)” (be “F(x)” a rigid predicate or not). I note that this could be solved by using Kaplan’s Dthat functor: “Dthat (the x F(x))” sends us back to the real world whereupon “the x F(x)” has Barack in the domain which we now can take with us as we evaluate of him predicates in w (a similar “backward (to actuality) looking functors” type of solution was made available in the late seventies in the Hintikka-Saarinen treatment). But we can’t save in this way “S-Obama”. As Dthat applies to “the set of all properties of Obama” \( \lambda P (P (\) BO)), we now carry from the actual world not Obama but the set of his properties in actuality and this local set is not the right one for evaluations of predications of him in w. Pointing out, in the Carnap-Church’s vein, that the individual concept—the function on worlds giving us in each world the local Obama-oracle—is undefined at the Obamaless world is not capturing the intuitive meaning: this man, Obama, does not exist in that world w. I note that this kind of problem multiplies when we look at Montague’s treatment of “John and Mary” by means of (intersected, unioned etc.) property sets. The treatment of “John and (or) Mary might not have existed” is not following the intuitive evaluation of the modal feature as of the actual referential plurality—John, Mary.
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Appendices
Appendix
Reference in Ordinary English Versus Sublimated Surrogates: “Obama” Versus Montague’s “S-Obama”
Consider the sentence “Obama is happy” with its natural semantic meaning: the referent, this man B. Obama, has the quality referred to by “is happy”. Consider now Montague’s sublimated surrogate “S-Obama is happy”. We won’t tinker with “is happy”—it has as its ordinary referent (by which I mean: its common semantic meaning)—referring to the property of being happy. Montague’s reformative treatment of the name sublimates it twice over—first “set-theoretically” and still extensionally to the set of properties had—in a given world w—by this man Obama, call this the Obama-extensional oracle; then, Montague further intensionalizes the sublimation one level up, to the intension that gives every possible world the Obama-oracle in that world.
Let me take the two sublimations separately, even though Montague unifies them. Montague takes the most general case—as if it involved modal or other other intensional “contexts”—and works backwards from it to simple “extensional” cases such as “Obama is happy”. This obscures a range of issues. So, I start with “Obama is happy”, without any modal/intensional “contexts” (yet). Already at this stage, the extensional sublimation fails to capture “Obama”’s ordinary English behavior. The intensionalization only aggravates the rift between the natural name and its sublimated surrogate. I will use a stipulated artificial name “S-Obama” for the treatment Montague stipulates.
“S-Obama is happy” is true in w iff being-happy is a member of the w-oracle of Obama. Now consider any possible world w. If Obama is happy in w, being-happy is in w is a member of the w-Obama oracle ; and vice versa. So in all worlds w, “Obama is happy” and “S-Obama is happy” share truth value. They are modally equivalent.
Do they have the same semantic meaning? Notice this question is asked pre any embedding in the context of modal or intensional operators. The question is in the vein of Kripke/Kaplan’s asking vis a vis the Frege/Russell “description theory” surrogate of “Aristotle”, say “the teacher of Alexander the great”, whether the semantic-content (what is said) by “Aristotle was fond of dogs” is semantically equivalent to “the teacher of Alexander was fond of dogs”. This question precedes any issue regarding behavior in the context of object language embeddings due to modal and intensional operators. The answer of Kripke/Kaplan to Frege/Russell is “no, there is no synonymy”. And our answer to Montague—who presents in effect a sophisticated variation on Frege/Russell—is “no” again.
Indeed “Obama” is rigid but “S-Obama” is not—in each local world it designates a different set of properties.
But now what if we let Montague introduce his second sublimation and shift the value of “S-Obama” to the intension that in each world provides the local property set, the w-oracle of Obama? Will not this “individual concept” mimic the rigidity simply by being a trans world intension from outset?
The answer is “no” and twice over. To calculate here things we must make a “decision” about Montague’s possible worlds semantics/model theory—does it employ a fixed domain for all possible worlds (as he seems to do by default in PTQ) or do we allow a variable domains semantics, with the domain of each local world w containing those individuals that exist locally in w?
The choice involves Montague in a hard dilemma: if he keeps to a fixed domain, he will misrepresent various existential matters arising out of Obama’s (and by analogy, the tigers’) merely contingent existence. Furthermore, beyond the contingent existence of Obama and the tigers, we have more modal facts that are pertinent; because of the nature of the actual world—there is no Vulcan in it and no unicorns in it—there may well be none in any world (this is the Donnellan-Bennett challenge mentioned above in the text).
If on the other hand, Montague allows variable domains (as any sensible metaphysical encoding must if it is not to make all existents necessary existents), he misrepresents the truth of modal existentials in yet other ways. Since the variable domain semantics is the more natural one (and richer in options) I will assume it.
When we write the modal predication “Obama might have lost in 2012”, this is true intuitively because there is a world w wherein the man Obama, the one and only actual referent of “Obama”, loses in 2012 in w; we bring from actuality to w the actual referent of Obama—the man—and check “is lost” of him in w. Now, Montague cannot cite the occurrence of the name “Obama” inside a modal operator scope to activate any intensionality shift or for that matter to block the untainted occurrence in subject position from simply referring to its ordinary referent—the man Obama—and not to the fancy individual concept. Of course, Montague may add a reduction thesis that any such subject predicate “Obama might have F-ed” is reducible to a sentential operator case “it might have been the case that: Obama F’s”. I will not battle with this reduction conjecture but ask the reader to consider some simple examples below that will convince him the thesis is false for proper nouns as for compound nominals.
Consider a world w where Obama does not exist. In the Kripke-Kaplan account, the natural varying of domains across worlds is meant to capture the natural de re modal fact “Obama (I, you etc.) might not have existed” and related modal facts e.g. “Obama might not have been married to Michelle”. Consider now the Obama-less world w viz. Obama is not in the local domain of w. On the natural Kaplan/Kripke semantics, we take the referent of Obama—Barack—from the actual-world domain into w and evaluate of it “does not exist” at w. The sentence “Obama does not exist” is true at w and “Obama might not have existed” is thus true in actuality (as is intuitively suggested). In like manner “might have been married to Michelle” is true of Barack at w because in the actual world he is married to Michelle.Footnote 39
On the other hand, “S-Obama” has genuine trouble providing the natural truth values for these subject-predicate (“de re”) modal sentences—there is simply no set of properties of Barack in w.Footnote 40
And so it goes with common nouns too. The plural subject predicate “tigers might not have existed” or the singular “some tiger might not have existed” is not represented by Montague’s oracles or the intension yielding them in a given local world. Take our last sentence, on both of its readings, the individual specific (where we add: some tiger, viz. Tony, might not have existed) and the kind specific reading, where we say: of that kind—tigers—some animal or other that is actually of the kind might not have existed. If we consider now a possible world w without tigers, both of our readings are true (we need not assume the essentiality to any actual tiger of its tigerhood; the individual—specific or any old one—we check for existence in w is literally said to be a tiger only in the actual world. The modal adverb modifies only the negated verb). And so it goes all the way to familiar problems with (ordinary English) analogs of the Barcan formula and its converse. If we say “every paper of Jaakko is necessarily one of those in the complete bibliography” we assert a truth but this does not entail “necessarily every paper of Jaakko is one of those in the complete bibliography”, since he might have authored additional papers; nor should the truism “necessarily every paper published by the Synthese-editor is published by the Synthese-editor” entail the modally significant (and false) “every paper published by the Synthese-editor is necessarily published by the Synthese-editor”. The sublimated individual concept interpretation does not provide us with the natural consequence relations here.
Problems multiply when we cross over from (1) predications of actual existents who might not have existed to (2) predications of actual nonexistents. Technically, the intensional sublimation need not embrace such seemingly merely possible items as Vulcan and unicorns. But in his philosophical work in which he considered such items, e.g. “On the nature of certain philosophical entities” (ch. 5 in [20]), Montague naturally—for someone so concerned—let the intensions pick up such merely possibles, items that lodge in “merely possible” worlds, if not in the actual world. Sentences such “Vulcan (unicorns) might have existed” seem then naturally true, provided “existence in a world” (including actuality) is deformed into: the pertinent intension has a nonempty extension at the given world.
On the actuality generated relation of reference for proper and common nouns—an account recognized by Montague’s perceptive student Michael Bennett as true to ordinary English (whatever constructions we may wish for our intensional logics)—“Vulcan (unicorns) might have existed” is thus doubly mistreated by the intensionalist. First, as before, these empty nouns do not occur within a modal context and frankly there is no reason whatsoever to “bump them up” to the intensional level—the modal applies to the verb only. Secondly, because reference is what it is and not another thing—the having of a nonempty extension by a prefabricated intension—viz. because reference is a relational process in actual history loading nouns with referents and ferrying them for later uses in which refer back to those original ferried referents, the actually empty nouns are doomed—“Vulcan” and “unicorns” are essentially reference-failures.
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Almog, J. (2018). Is Natural Semantics Possible?—Ordinary English, Formal Deformations-cum-Reformations and the Limits of Model Theory. In: van Ditmarsch, H., Sandu, G. (eds) Jaakko Hintikka on Knowledge and Game-Theoretical Semantics. Outstanding Contributions to Logic, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-62864-6_3
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