There is an obvious difference between what a term designates and whatit means. At least it is obvious that there is a difference. In someway, meaning determines designation, but is not synonymous with it.After all, “the morning star” and “the eveningstar” both designate the planet Venus, but don’t have the samemeaning. Intensional logic attempts to study both designation andmeaning and investigate the relationships between them.

• 1. What is this about?
• 2. A Brief History
  • 2.1 Frege
  • 2.2 Church
  • 2.3 Carnap
  • 2.4 Marcus
  • 2.5 Montague, Tichý, Bressan, and Gallin
• 3. A Particular Intensional Logic
  • 3.1 Propositional Modal Logic
  • 3.2 The Move To First Order
  • 3.3 Adding Intensions
  • 3.4 The Semantics Formally
  • 3.5 Partial Intensional Objects
  • 3.6 Problems With Rigidity
• 4. Sense as Algorithm
  • 4.1 Motivating Examples
  • 4.2 Syntax
  • 4.3 Denotation
  • 4.4 Sense
  • 4.5 Algorithms Need Not Be Effective
• Bibliography
  • Additional Reading
• Academic Tools
• Other Internet Resources
• Related Entries

1. What is this about?

If you are not skilled in colloquial astronomy, and I tell you thatthe morning star is the evening star, I have given youinformation—your knowledge has changed. If I tell you themorning star is the morning star, you might feel I was wasting yourtime. Yet in both cases I have told you the planet Venus wasself-identical. There must be more to it than this. Naively, we mightsay the morning star and the evening star are the same in one way, andnot the same in another. The two phrases, “morning star”and “evening star” may designate the same object, but theydo not have the same meaning. Meanings, in this sense, are oftencalledintensions, and things designated,extensions. Contexts in which extension is all that mattersare, naturally, calledextensional, while contexts in whichextension is not enough areintensional. Mathematics istypically extensional throughout—we happily write“\(1+4=2+3\)” even though the two terms involved maydiffer in meaning (more about this later). “It is knownthat…” is a typical intensional context—“itis known that \(1+4 = 2 + 3\)” may not be correct when theknowledge of small children is involved. Thus mathematical pedagogydiffers from mathematics proper. Other examples of intensionalcontexts are “it is believed that…”, “it isnecessary that…”, “it is informativethat…”, “it is said that…”, “itis astonishing that…”, and so on. Typically a contextthat is intensional can be recognized by a failure of thesubstitutivity of equality when naively applied. Thus, the morningstar equals the evening star; you know the morning star equals themorning star; then on substituting equals for equals, you know themorning star equals the evening star. Note that this knowledge arisesfrom purely logical reasoning, and does not involve any investigationof the sky, which should arouse some suspicion. Substitution ofco-referring terms in a knowledge context is the problematicmove—such a context is intensional, after all. Admittedly thisis somewhat circular. We should not make use of equality of extensionsin an intensional context, and an intensional context is one in whichsuch substitutivity does not work.

The examples used above involve complex terms, disguised definitedescriptions. But the same issues come up elsewhere as well, often inways that are harder to deal with formally. Proper names constituteone well-known area of difficulties. The name “Cicero”and the name “Tully” denote the same person, so“Cicero is Tully” is true. Proper names are generallyconsidered to be rigid, once a designation has been specified it doesnot change. This, in effect, makes “Cicero is Tully” intoa necessary truth. How, then, could someone not know it?“Superman is Clark Kent” is even more difficult to dealwith, since there is no actual person the names refer to. Thus whilethe sentence is true, not only might one not know it, but one mightperfectly well believe Clark Kent exists, that is “ClarkKent” designates something, while not believing Superman exists.Existence issues are intertwined, in complex ways, with intensionalmatters. Further, the problems just sketched at the ground levelcontinue up the type hierarchy. The property of being an equilateraltriangle is coextensive with the property of being an equiangulartriangle, though clearly meanings differ. Then one might say,“it is trivial that an equilateral triangle is an equilateraltriangle,” yet one might deny that “it is trivial that anequilateral triangle is an equiangular triangle”.

In classical first-order logic intension plays no role. It isextensional by design since primarily it evolved to model thereasoning needed in mathematics. Formalizing aspects of naturallanguage or everyday reasoning needs something richer. Formal systemsin which intensional features can be represented are generallyreferred to asintensional logics. This article discussessomething of the history and evolution of intensional logics. The aimis to find logics that can formally represent the issues sketchedabove. This is not simple and probably no proposed logic has beenentirely successful. A relatively simple intensional logic that canbe used to illustrate several major points will be discussed in somedetail, difficulties will be pointed out, and pointers to other, morecomplex, approaches will be given.

2. A Brief History

Recognition that designating terms have a dual nature is far fromrecent. The Port-Royal Logic used terminology that translates as“comprehension” and “denotation” forthis. John Stuart Mill used “connotation” and“denotation.” Frege famously used “Sinn” and“Bedeutung,” often left untranslated, but when translated,these usually become “sense” and “reference.”Carnap settled on “intension” and “extension.”However expressed, and with variation from author to author, theessential dichotomy is that between what a termmeans, andwhat itdenotes. “The number of the planets”denotes the number 9 (ignoring recent disputes about the status ofbodies at the outer edges of the solar system), but it does not havethe number 9 as its meaning, or else in earlier times scientists mighthave determined that the number of planets was 9 through a process oflinguistic analysis, and not through astronomical observation. Of themany people who have contributed to the analysis of intensionalproblems several stand out. At the head of the list is Gottlob Frege.

2.1 Frege

The modern understanding of intensional issues and problems beginswith a fundamental paper of Gottlob Frege, (Frege 1892). This paperopens with a recital of the difficulties posed by the notion ofequality. In his earlier work, Frege notes, he had taken equality torelate names, or signs, of objects, and not objects themselves. Forotherwise, if \(a\) and \(b\) designate the same object,there would be no cognitive difference between \(a = a\)and \(a = b\), yet the first is analytic while thesecond generally is not. Thus, he once supposed, equality relatessigns that designate the same thing. But, he now realizes, this cannotbe quite correct either. The use of signs is entirely arbitrary,anything can be a sign for anything, so in considering \(a =b\) we would also need to take into account themode ofpresentation of the two signs—what it is that associatesthem with the things they designate. Following this line of thought,equality becomes a relation between signs, relative to their modes ofpresentation. Of course the notion of a mode of presentation issomewhat obscure, and Frege quickly shifts attention elsewhere.

A sign has both a reference, and what Frege calls asense—we can think of the sense as being some kind of embodiment ofthe mode of presentation. From here on in his paper, sense is underdiscussion, and modes of presentation fade into the background. A nameexpresses its sense, anddesignates itsreference. Thus, “morning star” and “eveningstar” have the same designation, but express different senses,representing different modes of presentation—one is a celestialbody last seen in the morning before the sun obscures it, the other isa celestial body first seen in the evening after the sun no longerobscures it. Frege goes on to complicate matters by introducing theidea associated with a sign, which is distinct from its senseand its reference. But the idea is subjective, varying from person toperson, while both sense and denotation are said by Frege to be notdependent in this way. Consequently the idea also fades into thebackground, while sense and reference remain central.

Generally when a sign appears as part of a declarative sentence, it isthe reference of the sign that is important. Both “Venus”and “the morning star” designate the same object. Thesentence “The morning star is seen in the sky nearsunrise” is true, and remains true when “the morningstar” is replaced with “Venus”. Substitution ofequi-designating signs preserves truth. But not always; there arecontexts in which this does not happen,indirect referencecontexts. As a typical example, “George knows that the morningstar is seen in the sky near sunrise” may be true while“George knows that Venus is seen in the sky near sunrise”may be false. Besides knowledge contexts, indirect reference ariseswhen a sentence involves “I believe that…”,“I think that…”, “It seems to methat…”, “It is surprising that…”,“It is trivial that…”, and so on. In such contexts,Frege concludes, not designation but sense is central. Then, since“George knows that…” is an indirect referencecontext, senses are significant. The signs “the morningstar” and “Venus” have different senses, we are notreplacing a sense by a sense equal to it, and so should not expecttruth to be preserved.

Frege notes that an expression might have a sense, but not areference. An example he gives is, “the least rapidly convergentseries.” Of course an object might have several signs thatdesignate it, but with different senses. Frege extends thesense/reference dichotomy rather far. In particular, declarativesentences are said to have both a sense and a reference. The sense isthe proposition it expresses, while the reference is its truth value.Then logically equivalent sentences have the same designation, but mayhave different senses. In indirect contexts sense, and notdesignation, matters and so we may know the well-ordering principlefor natural numbers, but not know the principle of mathematicalinduction because, while they are equivalent in truth value, they havedifferent senses.

No formal machinery for dealing with sense, as opposed to reference,is proposed in Frege 1892. But Frege defined the terms under whichfurther discussion took place. There are two distinct but relatednotions, sense and reference. Equality plays a fundamental role, and acentral issue is the substitutivity of equals for equals. Names,signs, expressions, can be equal in designation, but not equal insense. There are both direct or extensional, and indirect orintensional contexts, and reference matters for the first while senseis fundamental for the second.

2.2 Church

Frege gave the outline of a theory of intensionality, but nointensional logic in any formal sense. There have been attempts tofill in his outline. Alonzo Church (1951) went at it quite directly.In this paper there is a formal logic in which terms have both sensesand denotations. These are simply taken to be different sorts, andminimal requirements are placed on them. Nonetheless the logic isquite complex. The formal logic that Frege had created for his work onthe foundations of mathematics was type free. Russell showed hisfamous paradox applied to Frege’s system, so it was inconsistent. As away out of this problem, Russell developed the type theory that wasembodied inPrincipia Mathematica. Church had given anelegant and precise formulation of the simple theory of types (Church1940), and that was incorporated into his work on intensionality,which is one of the reasons for its formal complexity.

Church uses a notion he calls aconcept, where anything thatis the sense of a name for something can serve as a concept of thatsomething. There is no attempt to make this more precise—indeedit is not really clear how that might be done. It is explicit that conceptsare language independent, and might even be uncountable. There is atype \(\omicron_{0}\) of the two truth values. Then, there is atype \(\omicron_{1}\) of concepts of members of\(\omicron_{0}\), which are calledpropositionalconcepts. There is a type \(\omicron_{2}\) of concepts ofmembers of \(\omicron_{1}\), and so on. There a type\(\iota_{0}\) of individuals, a type \(\iota_{1}\) ofconcepts of members of \(\iota_{0}\), a type \(\iota_{2}\)of concepts of members of \(\iota_{1}\), and so on. And finally,for any two types \(\alpha\) and \(\beta\) there is a type\((\alpha\,\beta )\) of functions from items of type \(\beta\) toitems of type \(\alpha\). Church makes a simplifying assumptionconcerning functional types. In order to state it easily he introducessome special notation: if \(\alpha\) is a type symbol, for example\(((\iota_{3}\,\omicron_{2})(\omicron_{5}\,\iota_{4}))\),then \(\alpha_{1}\) is the result of increasing each subscript by1, in our example we get\(((\iota_{4}\,\omicron_{3})(\omicron_{6}\,\iota_{5}))\). (Thereis a similar definition for \(\alpha_{n}\) for eachpositive integer \(n\), but we will not need it here.) Church’sassumption is that the concepts of members of the functional type\((\alpha\,\beta )\) are the members of the type\((\alpha_{1}\,\beta_{1})\). With this assumption,uniformly the concepts of members of any type \(\alpha\) are the membersof type \(\alpha_{1}\).

Quantification and implication are introduced, or rather versionsappropriate for the various types are introduced.\(\lambda\) abstraction notation is present. And finally, for eachtype \(\alpha\) it is assumed there is a relation that holdsbetween a concept of something of type \(\alpha\) and the thingitself; this is a relation between members of type\(\alpha_{1}\) and members of type \(\alpha\). Thisis denoted \(\Delta\), with appropriate type-identifying subscripts.

A fundamental issue for Church is when two names, lambda terms, havethe same sense. Three alternatives are considered. Common to all threealternatives are the assumptions that sense is unchanged under therenaming of bound variables (with the usual conditions of freeness),and under \(\beta\) reduction. Beyond these, Alternative 0 issomewhat technical and is only briefly mentioned, Alternative 1 is finegrained, making senses distinct as far as possible, while Alternative 2makes two terms have the same sense whenever equality between them is alogical validity. The proper definition of the alternatives isaxiomatic, and altogether various combinations of some 53 axiom schemesare introduced, with none examined in detail. Clearly Church wasproposing an investigation, rather than presenting full results.

As noted, the primary reference for this work is Church 1951, butthere are several other significant papers including Church 1973,Church 1974, and the Introduction to Church 1944, which contains aninformal discussion of some of the ideas. In addition, the expositorypapers of Anderson are enlightening (Anderson 1984, 1998). It shouldbe noted that there are relationships between Church’s work and thatof Carnap, discussed below. Church’s ideas first appeared in anabstract (Church 1946), then Carnap’s book appeared (Carnap 1947). Afew years later Church’s paper expanded his abstract in Church 1951.The second edition of Carnap’s book appeared in 1956. Each man had aninfluence on the other, and the references between the two authors arethoroughly intertwined.

2.3 Carnap

Church simply(!) formalized something of how intensions behaved,without saying what they were. Rudolf Carnap took things further withhis method of intension and extension, and provided a semantics inwhich quite specific model-theoretic entities are identified withintensions (Carnap 1947). Indeed, the goal was to supply intensionsand extensions for every meaningful expression, and this was done in away that has heavily influenced much subsequent work.

Although Carnap attended courses of Frege, his main ideas are based onWittgenstein 1921. In theTractatus, Wittgensteinintroduced a precursor of possible world semantics. There arestates of affairs, which can be identified with the set ofall their truths, “(1.13) The facts in logical space are theworld.” Presumably these facts are atomic, and can be variedindependently, “(1.21) Each item can be the case or not the casewhile everything else remains the same.” Thus there are manypossible states of affairs, among them the actual one, the realworld. Objects, in some way, involve not only the actual state ofaffairs, but all possible ones, “(2.0123) If I know an object Ialso know all its possible occurrences in states of affairs. (Everyone of these possibilities must be part of the nature of the object.)A new possibility cannot be discovered later.” It is from theseideas that Carnap developed his treatment.

Carnap begins with a fixed formal language whose details need notconcern us now. A class of atomic sentences in this language,containing exactly one of \(A\) or \(\neg A\) for eachatomic sentence, is astate-description. In eachstate-description the truth or falsity of every sentence of thelanguage is determined following the usual truth-functionalrules—quantifiers are treated substitutionally, and the languageis assumed to have ‘enough’ constants. Thus truth isrelative to a state-description. Now Carnap introduces a strongernotion than truth,L-truth, intended to be “anexplicatum for what philosophers call logical or necessary or analytictruth.” Initially he presents this somewhat informally, “asentence is \(L\)-true in a state description \(S\) if it istrue in \(S\) in such a way that its truth can be established onthe basis of the semantical rules of the system \(S\) alone,without any reference to (extra-linguistic) facts.” But this isquickly replaced by a more precise semantic version, “A sentence is\(L\)-true if it holds in every state-description.”

One can recognize in \(L\)-truth a version of necessary truthusing possible world semantics. There is no accessibility relation, sowhat is being captured is more like S5 than like other modallogics. But it is not S5 semantics either, since there is a fixed setof state-descriptions determined by the language itself. (If\(P\) is any propositional atom, some state-descriptionwill contain \(P\), and so \(\Diamond P\) willbe validated.) Nonetheless, it is a clear anticipation of possibleworld semantics. But what concerns us here is how Carnap treatsdesignating terms in such a setting. Consider predicates\(P\) and \(Q\). For Carnap these are intensionallyequivalent if \(\forall x(Px \equiv Qx)\) is an \(L\)-truth, that is, in eachstate-description \(P\) and \(Q\) have the sameextension. Without being quite explicit about it, Carnap is proposingthat the intension of a predicate is an assignment of an extension forit to each state-description—intensional identity means identityof extension across all state-descriptions and not just at the actualone. Thus the predicate ‘\(H\)’, human, and thepredicate ‘\(FB\)’, featherless biped, have the sameextension—in the actual state-description they apply to the samebeings—but they do not have the same intension since there areother state-descriptions in which their extensions can differ. In asimilar way one can model individual expressions, “The extensionof an individual expression is the individual to which itrefers.” Thus, ‘Scott’ and ‘the author ofWaverly’ have the same extension (in the actualstate-description). Carnap proposes calling the intension of anindividual expression anindividualconcept, andsuch a thing picks out, in each state-description, the individual towhich it refers in that state description. Then ‘Scott’and ‘the author of Waverly’ have different intensionsbecause, as most of us would happily say, they could have beendifferent, that is, there are state-descriptions in which they aredifferent. (I am ignoring the problems of non-designation in thisexample.)

Carnap’s fundamental idea is that intensions, for whatever entitiesare being considered, can be given a precise mathematical embodimentas functions on states, while extensions are relative to a singlestate. This has been further developed by subsequent researchers, ofcourse with modern possible world semantics added to the mix. TheCarnap approach is not the only one around, but it does take us quitea bit of the way into the intensional thicket. Even though it doesnot get us all the way through, it will be the primary versionconsidered here, since it is concrete, intuitive, and natural when itworks.

2.4 Marcus

Carnap’s work was primarily semantic, and resulted in a logic that did notcorrespond to any of the formal systems that had been studied up tothis point. Axiomatically presented propositional modal logics werewell-established, so it was important to see how (or if) they could beextended to include quantifiers and equality. At issue were decisionsabout what sorts of things quantifiers range over, and substitutivityof equals for equals. Quine’s modal objections needed to beaddressed. Ruth Barcan Marcus began a line of development in (Marcus1946) by formally extending the propositional system S2 of C. I. Lewisto include quantification, and developing it axiomatically in thestyle ofPrincipia Mathematica. It was clear that otherstandard modal logics besides S2 could have been used, and S4 wasexplicitly discussed. The Barcan formula, in the form \(\Diamond (\exists \alpha )\)A \(\supset (\exists \alpha )\Diamond\)A, made its firstappearance in (Marcus 1946),^[1] though a full understanding of its significance would have to waitfor the development of a possible-world semantics. Especiallysignificant for the present article, her system was further extendedin (Marcus 1947) to allow for abstraction and identity. Two versionsof identity were considered, depending on whether things had the sameproperties (abstracts) or necessarily had them. In the S2 system thetwo versions were shown to be equivalent, and in the S4 system,necessarily equivalent. In a later paper (Marcus 1953) thefundamental role of the deduction theorem was fully explored aswell.

Marcus proved that in her system identity was necessary if true, andthe same for distinctness. She argued forcefully in subsequent works,primarily (Marcus 1961), that morning star/evening star problems werenonetheless avoided. Names were understood astags. Theymight have their designation specified through an initial use of adefinite description, or by some other means, but otherwise names hadno meaning, only a designation. Thus they did not behave likedefinite descriptions, which were more than mere tags. Well, theobject tagged by “morning star ” and that tagged by“evening star ” are the same, and identity between objectsis never contingent.

The essential point had been made. One could develop formal modalsystems with quantifiers and equality. The ideas had coherence.Still missing was a semantics which would help with the understandingof the formalism, but this was around the corner.

2.5 Montague, Tichý, Bressan, and Gallin

Carnap’s ideas were extended and formalized by Richard Montague, PavelTichý, and Aldo Bressan, independently. All made use of some versionof Kripke/Hintikka possible world semantics, instead of the more specializedstructure of Carnap. All treated intensions functionally.

In part, Bressan wanted to provide a logical foundation for physics. The connectionwith physics is this. When we say something has such-and-such a mass,for instance, we mean that if we had conducted certain experiments, wewould have gotten certain results. This does not assume we did conductthose experiments, and thus alternate states (or cases, as Bressancalls them) arise. Hence there is a need for a rich modal language,with an ontology that includes numbers as well as physical objects. InBressan 1972, an elaborate modal system was developed, with a fulltype hierarchy including numbers as inPrincipia Mathematica.

Montague’s work is primarily in Montague 1960 and 1970, and has naturallanguage as its primary motivation. The treatment issemantic, but in Gallin 1975 an axiom system is presented. Thelogic Gallin axiomatized is a full type-theoretic system,with intensional objects of each type. Completeness is proved relativeto an analog of Henkin models, familiar for higher type classicallogics.

Tichý created a system of intensional logic very similar tothat of Montague, beginning, in English, in Tichý 1971, with adetailed presentation in Tichý 1988. Unfortunately his work didnot become widely known. Like Montague’s semantics,Tichý’s formal work is based on a type hierarchy withintensions mapping worlds to extensions at each type level, but itgoes beyond Montague in certain respects. For one thing, intensionsdepend not only on worlds, but also on times. For another, in additionto intensions and extensions Tichý alsoconsidersconstructions. The idea is that expressionsdetermine intensions and extensions, and this itself is a formalprocess in which compound expressions act using the simplerexpressions that go into their making; compositionality at the levelof constructions, in other words. Using this formal machinery,“\(1+4\)” and “\(2+3\)” prescribe differentconstructions; their meaning is not simply captured by theirintensional representation as discussed here.

3. A Particular Intensional Logic

As has been noted several times earlier, formal intensional logicshave been developed with a full hierarchy of higher types, Church, Montague, Bressan, Tichý for instance. Such logics can be ratherformidable, but Carnap’s ideas are often (certainly not always) at theheart of such logics, these ideas are simple, and are sufficient toallow discussion of several common intensional problems. Somehow,based on its sense (intension, meaning) a designating phrase maydesignate different things under different conditions—indifferent states. For instance, “the number of theplanets” was believed to designate 6 in ancient times (countingEarth). Immediately after the discovery of Uranus in 1781 “thenumber of the planets” was believed to designate 7. If we takeas epistemic states of affairs the universe as conceived by theancients, and the universe as conceived just after 1781, in one state“the number of the planets” designates 6 and in the otherit designates 7. In neither state were people wrong about the conceptof planet, but about the state of affairs constituting theuniverse. If we suppress all issues of how meanings are determined,how meanings in turn pick out references, and all issues of whatcounts as a possible state of affairs, that is, if we abstract allthis away, the common feature of every designating term is thatdesignation may change from state to state—thus it can beformalized by a function from states to objects. This bare-bonesapproach is quite enough to deal with many otherwise intractableproblems.

In order to keep things simple, we do not consider a full typehierarchy—first-order is enough to get the basics across. Thefirst-order fragment of the logic of Gallin 1975 would be sufficient,for instance. The particular formulation presented here comes fromFitting 2004, extending Fitting and Mendelsohn 1998. Predicateletters are intensional, as they are in every version of Kripke-stylesemantics, with interpretations that depend on possible worlds. Theonly other intensional item considered here is that of individualconcept—formally represented by constants and variables that candesignate different objects in different possible worlds. The sameideas can be extended to higher types, but what the ideas contributecan already be seen at this relatively simple level. Intensionallogics often have nothing but intensions—extensions are inferredbut are not explicit. However, an approach that is too minimal canmake life hard, so consequently here we explicitly allow both objectsand individual concepts which range over objects. There are two kindsof quantification, over each of these sorts. Both extensional andintensional objects are first-class citizens.

Basic ideas are presented semantically rather thanproof-theoretically, though both axiom systemsand tableau systems exist. Even so, technical details can become baroque,so as far as possible, we will separate informal presentation, whichis enough to get the general idea, from its formal counterpart, whichis of more specialized interest. A general acquaintance with modallogic is assumed (though there is a very brief discussion to establishnotation, which varies some from author to author). It should be notedthat modal semantics is used here, and generally, in two differentways. Often one has a particular Kripke model in mind, though it maybe specified informally. For instance, we might consider a Kripkemodel in which the states are the present instant and all past ones,with later states accessible from earlier ones. Such a model isenlightening when discussing “the King of France” forinstance, even though the notion of instant is somewhat vaguelydetermined. But besides this use of informally specified concretemodels, there is formal Kripke semantics which is a mathematicallyprecise thing. If it is established that something, say\(\Box (X \supset Y) \supset (\Box X \supset \Box Y)\), is valid in all formal Kripke models, we canassume it will be so in our vaguely specified, intuitive models, nomatter how we attempt to make them more precise. Informal modelspervade our discussions—their fundamental properties come fromthe formal semantics.

3.1 Propositional Modal Logic

A propositional language is built up frompropositional letters, \(P, Q,\ldots\), using \(\wedge , \vee , \supset , \neg\) and otherpropositional connectives, and \(\Box\) (necessary) and \(\Diamond\)(possible) as modal operators. These operators can be thought of asalethic, deontic, temporal, epistemic—it will matter whicheventually, but it does not at the moment. Likewise there could bemore than one version of \(\Box\), as in a logic of knowledge withmultiple knowers—this too doesn’t make for any essentialdifferences.

Kripke semantics for propositional modal logic is, by now, a veryfamiliar thing. Here is a quick presentation to establish notation,and to point out how one of Frege’s proposals fits in. A moredetailed presentation can be found in the article on modal logic inthis encyclopedia.

3.1.1 The Informal Version

A model consists of a collection of states, some determination ofwhich states are relevant to which, and also some specification ofwhich propositional letters hold at which of these states. States couldbe states of the real world at different times, or states of knowledge,or of belief, or of the real world as it might have been hadcircumstances been different. We have a mathematical abstraction here.We are not trying to define what all these states might‘mean,’ we simply assume we have them. Then more complexformulas are evaluated as true or false, relative to a state. At eachstate the propositional connectives have their customary classicalbehavior. For the modal operators. \(\Box X\), that is,necessarily \(X\), is true at a state if \(X\) itselfis true at every state that is relevant to that state (at allaccessible states). Likewise \(\Diamond X\),possibly \(X\), is true at a state if \(X\) is true atsome accessible state. If we think of things epistemically,accessibility represents compatibility, and so \(X\) is known in astate if \(X\) is the case in all states that are compatible withthat state. If we think of things alethically, an accessible state canbe considered an alternate reality, and so \(X\) is necessary in astate if \(X\) is the case in all possible alternative states.These are, by now, very familiar ideas.

3.1.2 The Formal Version

Aframe is a structure \(\langle \bG,\bR\rangle\), where \(\bG\) is a non-empty setand \(\bR\) is a binary relation on\(\bG\). Members of \(\bG\) arestates(orpossible worlds). \(\bR\) is anaccessibility relation. For \(\Gamma , \Delta \in \bG, \Gamma \bR \Delta\) is read“\(\Delta\) is accessible from \(\Gamma\).” A (propositional)valuation on a frame is a mapping, \(\bV\), thatassigns to each propositional letter a mapping from states of theframe to truth values,true orfalse. Forsimplicity, we will abbreviate \(\bV(P)(\Gamma )\)by \(\bV(P, \Gamma )\). Apropositionalmodel is a structure \(\cM = \langle \bG,\bR, \bV\rangle\), where\(\langle \bG, \bR\rangle\) is a frame and\(\bV\) is a propositional valuation on that frame.

Given a propositional model \(\cM = \langle \bG,\bR, \bV\rangle\), the notion offormula \(X\) being true at state \(\Gamma\) will be denoted \(\cM, \Gamma \vDash X\), and is characterized by the following standard rules,where \(P\) is atomic.

Suppose we think about formulas using an intensional/extensionaldistinction. Given a model \(\cM\), to eachformula \(X\) we can associate a function, call it\(f_{X}\), mapping states to truth values, wherewe set \(f_{X}(\Gamma )\) =true justin case\(\cM, \Gamma \vDash X\).

Think of the function \(f_{X}\) as theintensional meaning ofthe formula \(X\)—indeed, think of it as theproposition expressed by the formula (relative to aparticular model, of course). At a state \(\Gamma\), \(f_{X}(\Gamma )\) is a truth value—thinkof this as theextensional meaning of \(X\) at thatstate. This is a way of thinking that goes back to Frege, whoconcluded that the denotation of a sentence should be a truth value,but the sense should be a proposition. He was a little vague aboutwhat constituted a proposition—the formalization just presentedprovides a natural mathematical entity to serve the purpose, and wasexplicitly proposed for this purpose by Carnap. It should be clearthat the mathematical structure does, in a general way, capture somepart of Frege’s idea. Incidentally we could, with no loss,replace the function \(f_{X}\) on states withthe set \(\{ \Gamma \in \bG \mid f_{X}(\Gamma) = \textit{true}\}\).The function \(f_{X}\) is simply thecharacteristic function of this set. Sets like these are commonlyreferred to as propositions in the modal logic community. In atechnical sense, then, Frege’s ideas on this particular topichave become common currency.

3.2 The Move To First Order

First we discuss some background intuitions, then introduce aformal semantics. Intensions will be introduced formally in Section3.3. The material discussed here can be found more fully developed in(Fitting and Mendelsohn 1998, Hughes and Cresswell 1996), among otherplaces.

3.2.1 Actualist and Possibilist

If we are to think of an intension as designating different thingsunder different circumstances, we need things. At the propositionallevel truth values play the role of things, but at the first orderlevel something more is needed. In classical logic each model has adomain, the things of that model, and quantifiers are understood asranging over the members of that domain. It is, of course, left openwhat constitutes a thing—any collection of any sort can serve asa domain. That way, if someone has special restrictions in mind becauseof philosophical or mathematical considerations, they can beaccommodated. It follows that the validities of classical logic are, bydesign, as general as possible—they are true no matter what wemight choose as our domain, no matter what our things are.

A similar approach was introduced for modal logics in Kripke 1963.Domains are present, but it is left open what they might consistof. But there is a complication that has no classical counterpart: ina Kripke model there are multiple states. Should there be a singledomain for the entire model, or separate domains for each state? Bothhave natural intuitions.

Consider a version of Kripke models in which a separate domain isassociated with each state of the model. At each state, quantifiers arethought of as ranging over the domain associated with that state. Thishas come to be known as anactualist semantics. Think of thedomain associated with a state as the things that actually exist atthat state. Thus, for example, in the so-called real world the GreatPyramid of Khufu is in the domain, but the Lighthouse of Alexandria isnot. If we were considering the world of, say, 1300, both would be inthe domain. In an actualist approach, we need to come to some decisionon what to do with formulas containing references to things that existin other states but not in the state we are considering. Severalapproaches are plausible; we could take such formulas to be false, orwe could take them to be meaningless, for instance, but this seems tobe unnecessarily restrictive. After all, we do say things like“the Lighthouse of Alexandria no longer exists,” and wethink of it as true. So, the formal version that seems most usefultakes quantifiers as ranging over domains state by state, but otherwiseallows terms to reference members of any domain. The resultingsemantics is often calledvarying domain as well asactualist.

Suppose we use the actualist semantics, so each state has anassociated domain of actually existing things, but suppose we allowquantifiers to range over the members of any domain, withoutdistinction, which means quantifiers are ranging over the same set, atevery state. What are the members of that set? They are the thingsthat exist at some state, and so at every state they are the possibleexistents—things that might exist. Lumping these separatedomains into a single domain of quantification, in effect, means weare quantifying over possibilia. Thus, a semantics in which there is asingle domain over which quantifiers range, the same for every state,is often calledpossibilist semantics or, of course,constant domain semantics.

Possibilist semantics is simpler to deal with than the actualistversion—we have one domain instead of many for quantifiers torange over. And it turns out that if we adopt a possibilist approach,the actualist semantics can be simulated. Suppose we have a singledomain of quantification, possibilia, and a special predicate,\(E\), which we think of as true, at each state, of the thingsthat actually exist at that state. If \(\forall\) is a quantifierover the domain of possibilia, we can think of the relativizedquantifier, \(\forall\)x\((E(x)\supset \ldots )\) as corresponding to actualistquantification. (We need to assume that, at each state, \(E\) istrue of something—this corresponds to assuming domains arenon-empty.) This gives an embedding of the actualist semantics into thepossibilist one, a result that can be formally stated and proved.Here possibilist semantics will be used, and we assume we have anexistence predicate \(E\) available.

3.2.2 Possibilist Semantics, Formally

The language to be used is a straightforward first order extension ofthe propositional modal language. There is an infinite list ofobject variables, \(x, y,x_{1}, x_{2},\ldots\), and a listofrelationsymbols, \(R, P,P_{1}, P_{2},\ldots\), of allarities. Among these is the one-place symbol \(E\) and thetwo-place symbol =. Constant and function symbols could be added, butlet’s keep things relatively simple, along with simply relative. If\(x_{1}, \ldots ,x_{n}\) areobject variables and \(P\) is an \(n\)-place relationsymbol, \(P(x_{1}, \ldots ,x_{n})\) is an atomic formula. We’ll write\(x = y\) in place of = (x,y). More complexformulas are built up using propositional connectives, modaloperators, and quantifiers, \(\forall\) and \(\exists\), in the usualway. Free and bound occurrences of variables have the standardcharacterization.

Afirst order model is a structure \(\langle \bG, \bR,\bD_{O}, \bI\rangle\) where \(\langle \bG, \bR\rangle\) is a frame, asin Section 3.1, \(\bD_{O}\) is a non-emptyobject domain, and\(\bI\) is aninterpretation that assigns to each \(n\)-placerelation symbol \(P\) a mapping, \(\bI(P)\) from \(\bG\) to subsets of\(\bD_{O}^{n}\). We’ll write \(\bI(P, \Gamma )\) as aneasier-to-read version of \(\bI(P)(\Gamma )\). It is required that\(\bI(=, \Gamma )\) is the equality relation on \(\bD_{O}\), for everystate \(\Gamma\), and \(\bI(E, \Gamma )\) is non-empty, for every\(\Gamma\). Afirst-order valuation in a model is a mapping\(v\) that assigns a member of \(\bD_{O}\) to each variable. Note thatfirst order valuations are not state-dependent in the way thatinterpretations are. A first order valuation \(w\) is an\(x\)-variant of valuation \(v\) if \(v\) and \(w\) agree onall variables except possibly for \(x\). Truth, at a state \(\Gamma\)of a model \(\cM = \langle \bG, \bR, \bD_{O}, \bI\rangle\), withrespect to a first order valuation \(v\), is characterized as follows,where \(P(x_{1}, \ldots ,x_{n})\) is an atomic formula:

Call a formulavalid if it is true at every state of everyfirst order model with respect to every first-order valuation, asdefined above. Among the validities are the usual modal candidates,such as \(\Box (X \supset Y) \supset (\Box X \supset \Box Y)\), and the usual quantificational candidates,such as \(\forall xX \supset \exists xX\). We also have mixed cases such as theBarcan and converse Barcan formulas:\(\forall x\Box X \equiv \Box \forall xX\), which are characteristic ofconstant domain models, as was shown in Kripke 1963. Because of theway equality is treated, we have the validity of both\(\forall x\forall y(x = y \supset \Box x = y)\) and\(\forall x\forall y(x \ne y \supset \Box x \ne y)\). Much has been made about theidentity of the number of the planets and 9 (Quine 1963), or theidentity of the morning star and the evening star (Frege 1892), andhow these identities might behave in modal contexts. But that is notreally a relevant issue here. Phrases like “the morningstar” have an intensional aspect, and the semantics outlined sofar does not take intensional issues into account. As a matter offact, the morning star and the evening star are the same object and,as Gertrude Stein might have said, “an object is an object is anobject.” The necessary identity of a thing and itself should notcome as a surprise. Intensional issues will be dealt with shortly.

Quantification is possibilist—domains are constant. But, as wasdiscussed in Section 3.2.1, varying domains can be brought inindirectly by using the existence predicate, \(E\), and thisallows us to introduce actualist quantification definitionally. Let\(\forall^{E}xX\) abbreviate\(\forall x(E(x) \supset X)\), and let\(\exists^{E}xX\) abbreviate\(\exists x(E(x) \wedge X)\). Then, while \(\forall x\phi (x) \supset \phi (y)\) is valid, assuming \(y\) is free for\(x\) in \(\phi (x)\), we do not have the validityof \(\forall^{E}x\phi (x)\supset \phi (y)\). What we have instead is the validityof\([\forall^{E}x\phi (x) \wedge E(y)] \supset \phi (y)\).

As another example of possibilist/actualist difference, consider\(\exists x\Box P(x) \supset \Box \exists xP(x)\). With possibilistquantifiers, this is valid and reasonable. It asserts that if somepossibilia has the \(P\) property in all alternative states, thenin every alternative state some possibilia has the \(P\)property. But when possibilist quantification is replaced withactualist,\(\exists^{E}x\Box P(x)\supset \Box \exists^{E}xP(x)\),the result is no longer valid. As a blatant (but somewhat informal)example, say the actual state is \(\Gamma\) and \(P\) is theproperty of existing in state \(\Gamma\). Then, at \(\Gamma ,\exists^{E}x\Box P(x)\)says something that actually exists has, in all alternative states,the property of existing in the state \(\Gamma\). This is true; in factit is true ofeverything that exists in the state\(\Gamma\). But\(\Box \exists^{E}xP(x)\)says that in every alternative state there will be an actuallyexistent object that also exists in the state \(\Gamma\), which need notbe the case.

3.3 Adding Intensions

In Section 3.2 a first order modal logic was sketched, in whichquantification was over objects. Now a second kind of quantificationis added, over intensions. As has been noted several times, anintensional object, or individual concept, will be modeled by afunction from states to objects, but now we get into the question ofwhat functions should be allowed. Intensions are supposed to berelated to meanings. If we consider meaning to be a human construct,what constitutes an intension should probably be restricted. Thereshould not, for instance, be more intensional objects than there aresentences that can specify meanings, and this limits intensions to acountable set. Or we might consider intensions as ‘beingthere,’ and we pick out the ones that we want to think about, inwhich case cardinality considerations don’t apply. This is an issuethat probably cannot be settled once and for all. Instead, thesemantics about to be presented allows for different choices indifferent models—it is not required that all functions fromstates to objects be present. It should be noted that, while thissemantical gambit does have philosophical justification, it also makesan axiomatization possible. The fundamental point is the same as inthe move from first to second order logic. If we insist that secondorder quantifiers range over all sets and relations, an axiomatizationis not possible. If we use Henkin models, in which the range of secondorder quantifiers has more freedom, an axiomatization becomesavailable.

Formulas are constructed more-or-less in the obvious way, with twokinds of quantified variables instead of one: extensional andintensional. But there is one really important addition to thesyntactic machinery, and it requires some discussion. Suppose we havean intension, \(f\), that picks out an object in each state. Forexample, the states might be various ways the universe could have beenconstituted, and at each state \(f\) picks out the number of theplanets which could, of course, be 0. Suppose \(P\) is a one-placerelation symbol—what should be meant by \(P(f)\)?On the one hand, it could mean that the intension \(f\) has theproperty \(P\), on the other hand it could mean that the objectdesignated by \(f\) has the property \(P\). Both versions areuseful and correspond to things we say every day. We will allow forboth, but the second version requires some cleaning up. Suppose\(P(f)\) is intended to mean that the object designatedby \(f\) (at a state) has property \(P\). Then how do we read\(\Diamond P(f)\)? Under what circumstances shouldwe take it to be true at state \(\Gamma\)? It could be understood asasserting the thing designated by \(f\) at \(\Gamma\) (call it\(f_{\Gamma })\) has the ‘possible-\(P\)’property, and so at some alternative state \(\Delta\) we have that\(f_{\Gamma }\) has property \(P\). This is thede re reading, in which a possible property is ascribed to athing. Another way of understanding\(\Diamond P(f)\) takes the possibility operatoras primary: to say the formula is true at \(\Gamma\) means that at somealternative state, \(\Delta\), we have \(P(f)\), and so at\(\Delta\) the object designated by \(f\) (call it\(f_{\Delta })\) has property \(P\). This is thede dicto reading, possibility applies to a sentence. Of coursethere is no particular reason why \(f_{\Gamma }\) and\(f_{\Delta }\) should be identical. Thede re andde dicto readings are different, both need representation, andwe cannot manage this with the customary syntax.

An abstraction mechanism will be used to disambiguate our syntax. Thede re reading will be symbolized\([\lambda x\,\Diamond P(x)](f)\)and thede dicto will be symbolized\(\Diamond [\lambda x\,P(x)](f)\). The(incomplete) expression \([\lambda x\,X]\) isoften called apredicate abstraction; one can think of it asthe predicate abstracted from the formula \(X\). In\([\lambda x\,\Diamond P(x)](f)\) weare asserting that \(f\) has the possible-\(P\) property,while in\(\Diamond [\lambda x\,P(x)](f)\) weare asserting the possibility that \(f\) has the \(P\)property. Abstraction disambiguates. What we have said about \(\Diamond\)applies equally well to \(\Box\) of course. It should be noted that onecould simply think of abstraction as a scope-specifying device, in atradition that goes back to Russell, who made use of such a mechanismin his treatment of definite descriptions. Abstraction in modal logicgoes back to Carnap 1947, but in a way that ignores the issuesdiscussed above. The present usage comes from Stalnaker & Thomason1968 and Thomason & Stalnaker 1968.

3.4 The Semantics Formally

Now the more technical part begins. There are two kinds of variables,object variables as before, andintension variables,orindividual concept variables, \(f, g,g_{1}, g_{2},\ldots\). With twokinds of variables present, the formation of atomic formulas becomes alittle more complex. From now on, instead of just being\(n\)-place for some \(n\), a relation symbol will have atype associated with it, where a type is an \(n\)-tuplewhose entries are members of \(\{O, I\}\). An atomicformula is an expression of the form \(P(\alpha_{1},\ldots ,\alpha_{n})\) where \(P\) is arelation symbol whose type is \(\langle t_{1}, \ldots ,t_{n}\rangle\) and, for each \(i\), if\(t_{i} = O\) then\(\alpha_{i}\) is an object variable, and if\(t_{i} = I\) then\(\alpha_{i}\) is an intension variable. Among therelation symbols we still have \(E\), which now is of type\(\langle O\rangle\), and we have \(=\), of type \(\langle O,O \rangle\).

Formulas are built up from atomic formulas in the usual way, usingpropositional connectives, modal operators, and two kinds ofquantifiers: over object variables and over intension variables. Inaddition to the usual formula-creating machinery, we have thefollowing. If \(X\) is a formula, \(x\) is an object variable,and \(f\) is an intension variable, then\([\lambda x\,X](f)\) is aformula, in which the free variable occurrences are those of\(X\) except for \(x\), together with the displayed occurrenceof \(f\).

To distinguish the models described here from those in Section 3.2.2,these will be referred to asFOIL models, standing forfirst order intensional logic. They are discussed more fullyin (Fitting 2004). AFOILmodel is a structure \(\cM = \langle \bG, \bR,\bD_{O},\bD_{i}, \bI\rangle\) where\(\langle \bG, \bR,\bD_{O}, \bI\rangle\) meetsthe conditions of Section 3.2.2, and in addition,\(\bD_{i}\) is a non-empty set offunctions from \(\bG\) to\(\bD_{O}\); it is theintensiondomain.

Afirst-order valuation inFOIL model \(\cM\) is a mapping that assigns to each object variable a member of \(\bD_{O}\),as before, and to each intension variable a member of\(\bD_{i}\). If \(f\) is anintension variable, we’ll write \(v(f, \Gamma )\) for\(v(f)(\Gamma )\). Now, the definition of truth, at astate \(\Gamma\) of a model \(\cM\), with respect to a valuation \(v\), meets the conditions set forth inSection 3.2.2 and, in addition, the following:

Let us agree to abbreviate\([\lambda x\,[\lambda y\,X](g)](f)\)by \([\lambda xy\,X](f, g)\), whenconvenient. Suppose \(f\) is intended to be the intension of“the morning star,” and \(g\) is intended to be theintension of “the evening star.” Presumably \(f\) and\(g\) are distinct intensions. Even so,\([\lambda xy\,x = y]\)(f, g) iscorrect in the real world—both \(f\) and \(g\) dodesignate the same object.

Here is another example that might help make thede re /de dicto distinction clearer. Suppose \(f\) is theintension of “the tallest person,” and \(g\) is theintension of “the oldest person,” and suppose it happensthat, at the moment, these are the same people. Also, let us read\(\Box\) epistemically. It is unlikely we would say that\(\Box [\lambda xy\,x = y]\)(f, g) isthe case. We can read \(\Box [\lambda xy\,x =y]\)(f, g) as saying we knowthat\(f\) and \(g\) are the same. It asserts that under allepistemic alternatives—all the various ways the world could be thatare compatible with what we know—\(f\) and \(g\) designatethe same object, and this most decidedly does not seem to be thecase. However, we do have \([\lambda xy\,\Box (x= y)]\)(f, g), which we can read as saying we knowof \(f\) and \(g\), that is, of their denotations,that they are the same, and this could be the case. It asserts that inall epistemic alternative states, what \(f\) and \(g\)designate in this one will be the same. In the setupdescribed, \(f\) and \(g\) do designate the same object, andidentity of objects carries over across states.

It should be noted that the examples of designating terms just givenare all definite descriptions. These pick out different objects indifferent possible worlds quite naturally. The situation with propernames and with mathematics is different, and will be discussed laterin section 3.6.

3.4.1 A Formal Example

Here’s an example to show how the semantics works in a technicalway. An intension isrigid if it is constant, the same inevery state. We might think of a rigid intension as a disguisedobject, identifying it with its constant value. It should not be asurprise, then, that for rigid intensions, the distinction betweende re andde dicto disappears. Indeed, something abit stronger can be shown. Instead of rigidity, consider the weakernotion calledlocal rigidity in Fitting and Mendelsohn 1998:an intension is locally rigid at a state if it has the samedesignation at that state that it has at all accessible ones. To say\(f\) is locally rigid at a state, then, amounts to asserting thetruth of\([\lambda x\,\Box [\lambda y\,x= y](f)](f)\) at that state. Local rigidityat a state implies thede re /de dicto distinctionvanishes at that state. To show how the formal semantics works, hereis a verification of the validity of

In a similar way one can establish the validity of

and from these two follows the validity of

which directly says local rigidity implies thede re /dedicto distinction vanishes.

Suppose \eqref{eq2} were not valid. Then there would be a model \(\cM = \langle \bG, \bR, \bD_{O}, \bD_{i}, \bI\rangle\), astate \(\Gamma\) of it, and a valuation \(v\) in it, such that

From \eqref{eq6} we have the following, where \(w\) is the\(x\)-variant of \(v\) such that \(w(x) = v (f, \Gamma )\).

By \eqref{eq8} there is some \(\Delta \in \bG\) with \(\Gamma\bR \Delta\) such that we have the following.

Then, as a consequence of \eqref{eq7}

and hence we have the following, where \(w'\) is the \(x\)-variant of\(v\) such that \(w'(x) = v(f, \Delta )\).

Now from \eqref{eq5}, since \(w(x) = v(f, \Gamma)\), we have

and so

and hence

where \(w''\) is the \(y\)-variant of \(w\) such that \(w''(y) = w(f,\Delta )\).

We claim that valuations \(w\) and \(w'\) are the same, which meansthat \eqref{eq9} and \eqref{eq11} are contradictory. Since both are \(x\)-variants of\(v\), it is enough to show that \(w(x) = w'(x)\), that is, \(v(f,\Gamma ) = v(f, \Delta )\), which is intuitively what local rigiditysays. Proceeding formally, \(v(f, \Gamma ) = w(x) = w''(x)\) since\(w''\) is a \(y\)-variant of \(w\) and so they agree on \(x\). Wealso have \(v(f, \Delta ) = w''(y)\). And finally, \(w''(x) = w''(y)\)by \eqref{eq14}.

Having reached a contradiction, we conclude that \eqref{eq2} must bevalid.

3.4.2 Possible Extra Requirements

In models the domain of intensions is to be some non-empty set offunctions from states to objects. We have deliberately left vague thequestion of which ones we must have. There are some conditions wemight want to require. Here are some considerations along these lines,beginning with a handy abbreviation.

(where \(x\) and \(y\) are distinct object variables).

Working through theFOIL semantics, \(\cM, \Gamma \vDash_{v} D(f,x)\) is true just in case\(v(f, \Gamma ) = v(x)\). Thus\(D(f, x)\) says the intension \(f\)designates the object \(x\).

The formula \(\forall f\exists xD(f, x)\) is valid inFOILmodels as described so far. It simply says intensions alwaysdesignate. On the other hand, there is noa priori reason tobelieve that every object is designated by some intension, but underspecial circumstances we might want to require this. We can do it byrestricting ourselves to models in which we have the validity of

If we require \eqref{eq16}, quantification over objects is reducible tointensional quantification:

More precisely, the implication \(\eqref{eq16} \supset \eqref{eq17}\) is valid inFOIL semantics.

We also might want to require the existence ofchoicefunctions. Suppose that we have somehow associated an object\(d_{\Gamma }\) with each state \(\Gamma\) of a model. Ifour way of choosing \(d_{\Gamma }\) can be specified by aformula of the language, we might want to say we have specified anintension. Requiring the validity of the following formula seems asclose as we can come to imposing such an existence condition onFOIL models. For each formula \(\Phi\):

3.5 Partial Intensional Objects

“The King of France in 1700” denotes an object, LouisXIV, who does not exist, but did. “The present King ofFrance” does not denote at all. To handle such things, therepresentation of an intension can be generalized from being a totalfunction from states to objects, to being apartial function.We routinely talk about non-existent objects—we have no problemtalking about the King of France in 1700. But there is nothing to besaid about the present King of France—there is no such thing.This will be our guide for truth conditions in our semantics.

3.5.1 Modifying the Semantics

The language stays the same, but intension variables are nowinterpreted bypartial functions on the set ofstates—functions whose domains may be proper subsets of the setof states. Thus \(\cM = \langle \bG,\bR, \bD_{O},\bD_{i}, \bI\rangle\) is apartialFOIL model if it is as in Section 3.4 exceptthat members of \(\bD_{i}\) are partialfunctions from \(\bG\) to\(\bD_{O}\). Given a partialFOILmodel \(\cM\) and a valuation \(v\) in it, an intension variable \(f\)designates at state \(\Gamma\) of this model with respect to\(v\) if \(\Gamma\) is in the domain of \(v(f)\).

Following the idea that nothing can be said about the present King ofFrance, we break condition \eqref{cond1} from Section 3.4 into two parts. Givena partialFOIL model \(\cM\) and a valuation \(v\) in it:

1. If \(f\) does not designate at \(\Gamma\) with respect to \(v\),
   it is not the case that \(\cM, \Gamma \vDash_{v}[\lambda x\,X](f)\)
2. If \(f\) designates at \(\Gamma\) with respect to \(v\),
   \(\cM, \Gamma \vDash_{v}[\lambda x\,X](f)\Leftrightarrow \cM, \Gamma \vDash_{w} X\)
   where \(w\) is like \(v\) except that\(w(x) = v(f, \Gamma )\)

Thus designating terms behave as they did before, but nothing can betruly asserted about non-designating terms.

Recall, we introduced a formula \eqref{eq15} abbreviated by\(D(f,x)\) to say \(f\) designates \(x\).Using that, we introduce a further abbreviation.

This says \(f\) designates. Incidentally, we could have used\([\lambda x\,x = x](f)\) justas well, thus avoiding quantification.

It is important to differentiate between existence and designation.As things have been set up here, existence is a property of objects,but designation really applies to terms of the formal language, in acontext. To use a temporal example from Fitting and Mendelsohn 1998,in the usual sense “George Washington” designates a personwho does not exist, though he once did, while “GeorgeWashington’s eldest son,” does not designate at all. That anintensional variable \(f\) designates an existent object isexpressed by an abstract,\([\lambda x\,E(x)](f)\). We have to bea bit careful about non-existence though. That \(f\) designates anon-existent is not simply the denial of the previous expression,\(\neg [\lambda x\,E(x)](f)\). Afterall, \([\lambda x\,E(x)](f)\)expresses that \(f\) designates an existent, so its denial sayseither \(f\) does not designate, or it does, but designates anon-existent. To express that \(f\) designates, but designates anon-existent, we need\([\lambda x\,\neg E(x)](f)\). Theformula\(\forall f([\lambda x\,E(x)](f) \vee \neg [\lambda\)x E\((x)](f))\) is valid,but \(\forall f([\lambda x\,E(x)](f) \vee [\lambda x\,\neg E(x)](f))\) isnot—one can easily construct partialFOIL models thatinvalidate it. What we do have is the following important item.

In words, intensional terms that designate must designate existents ornon-existents.

3.5.2 Definite Descriptions

In earlier parts of this article, among the examples of intensions andpartial intensions have been “the present King of France,”“the tallest person,” and “the oldest person.”One could add to these “the number of people,” and“the positive solution of \(x^{2} - 9 = 0\).” All have been specified usingdefinitedescriptions. In a temporal model, the first three determinepartial intensions (there have been instants of time with no people);the fourth determines an intension that is not partial; the fifthdetermines an intension that is rigid.

So far we have been speaking informally, but there are two equivalentways of developing definite descriptions ideas formally. The approachintroduced by Bertrand Russell (Russell 1905, Whitehead and Russell1925) is widely familiar and probably needs little explicationhere. Suffice it to say, it extends to the intensional setting withoutdifficulty. In this approach, aterm-like expression,\(\atoi y\phi (y)\), is introduced, where \(\phi (y)\) is a formula and \(y\) is anobject variable. It is read, “the \(y\) such that\(\phi (y)\).” This expression is given noindependent meaning, but there is a device to translate it away in anappropriate context. Thus, \([\lambda x\,\psi (x)] \atoi y\phi (y))\) is taken to abbreviate the formula\(\exists y[\forall z(\phi (z)\equiv z = y) \wedge \psi (y)]\). (The standard device has been used of writing \(\phi (z)\) torepresent the substitution instance of \(\phi (y)\) resultingfrom replacing free occurrences of \(y\) with occurrences of\(z\), and modifying bound variables if necessary to avoidincidental capture of \(z\).) The present abstraction notation,using \(\lambda\), is not that of Russell, but he used an equivalentscoping device. As he famously pointed out, Russell’s method allows usto distinguish between “The present King of France is notbald,” which is false because there is no present King ofFrance, and “It is not the case that the present King of Franceis bald,” which is true because “The present King ofFrance is bald” is false. It becomes the distinction between \([\lambda x\,\neg\textit{Bald}(x)](\atoi y\textit{King}(y))\) and \(\neg [\lambda x\textit{Bald}(x)](\atoi y\textit{King}(y))\).

As an attractive alternative, one could make definite descriptionsinto first-class things. Enlarge the language so that if\(\phi (y)\) is a formula where \(y\) is an objectvariable, then \(\atoi y\phi (y)\) is anintension term whose free variables are those of\(\phi (y)\) except for \(y\). Then modify thedefinition of formula, to allow these new intension terms to appear inplaces we previously allowed intension variables to appear. That leadsto a complication, since intension terms involve formulas, and formulascan contain intension terms. In fact, formula and term must be definedsimultaneously, but this is no real problem.

Semantically we can model definite descriptions by partialintensions. We say the term \(\atoi y\phi (y)\) designates at state \(\Gamma\) of a partialFOIL model \(\cM\) with respect to valuation \(v\) if \(\cM, \Gamma \vDash_{w} \phi (y)\) forexactly one \(y\)-variant\(w\) of \(v\). Then the conditions from section 3.5.1 areextended as follows.

3. If \(\atoi y\phi (y)\) does not designate at \(\Gamma\) with respect to \(v\),
   it is not the case that \(\cM, \Gamma \vDash_{v} [\lambda x\,X](\atoi y\phi (y))\)
4. If \(\atoi y\phi (y)\) designates at \(\Gamma\) with respect to \(v\),
   \(\cM, \Gamma \vDash_{v} [\lambda x\,X](\atoi y\phi (y)) \Leftrightarrow \cM, \Gamma \vDash_{w} X\)
   where \(w\) is the \(y\)-variant of \(v\) such that \(\cM, \Gamma \vDash_{w} \phi (y)\)

One can show that the Russell approach and the approach just sketchedamount to more-or-less the same thing. But with definite descriptionsavailable as formal parts of the language, instead of just asremovable abbreviations in context, one can see they determineintensions (possibly partial) that are specified by formulas.

A property need not hold of the corresponding definite description,that is, \([\lambda x\,\phi (x)](\atoi x\phi (x))\) need not be valid. This is simply because the definite descriptionmight not designate. However, if it does designate, it must have itsdefining property. Indeed, we have the validity of the following:

One must be careful about the interaction between definitedescriptions and modal operators, just as between them and negation.For instance, \(D(\atoi x\Diamond \phi (x)) \supset \Diamond D(\atoi x\phi (x))\) is valid, but its converse is not. For a more concrete example ofmodal/description interaction, suppose \(K(x)\) is aformula expressing that \(x\) is King of France. In the presentstate, \([\lambda x\,\Diamond E(x)](\atoi xK(x))\) is false, because the definite description has no designation, but \(\Diamond [\lambda x\,E(x)](\atoi xK(x))\)is true, because there is an alternative (earlier) state in which thedefinite description designates an existent object.

3.6 Problems With Rigidity

It was noted that for rigid terms the de re/de dicto distinctioncollapses. Indeed, if \(f\) and \(g\) are rigid, \([\lambda xy\x=y](f, g)\), \({\square}[\lambda xy\ x=y](f, g)\) and \([\lambda xy\{\square}x=y](f, g)\) are all equivalent. This is a problem that setsa limit on what can be handled by the Carnap-style logic as presentedso far. Two well-known areas of difficulty are mathematics and propernames, especially in an epistemic setting.

3.6.1 Mathematical Problems

How could someone not know that \(1 + 4 = 2 + 3\)? Yet it happensfor small children, and for us bigger children similar, but morecomplex, examples of mathematical truths we don’t know can befound. Obviously the designations of “\(1 + 4\)” and“\(2 + 3\)” are the same, so their senses must bedifferent. But if we model sense by a function from states todesignations, the functions would be the same, mapping each state to5. If it is necessary truth that is at issue, there is no problem; wecertainly want that \(1 + 4 = 2 + 3\) is a necessary truth. But ifepistemic issues are under consideration, since we cannot have apossible world in which “\(1 + 4\)” and “\(2 +3\)” designate different things, “\(1 + 4 = 2 + 3\)”must be a known truth. So again, how could one not know this, or anyother mathematical truth?

One possible solution is to say that for mathematical terms,intension is a different thing than it is for definite descriptionslike “the King of France.” The expression “\(1 +4\)” is a kind of miniature computing program. Exactly whatprogram depends on how we were taught to add, but let us standardizeon: \(x + y\) instructs us to start at the number \(x\) and count offthe next \(y\) numbers. Then obviously, “\(1 + 4\)” and“\(2 + 3\)” correspond to different programs with the sameoutput. We might identify the program with the sense, and the outputwith the denotation. Then we might account for not knowing that \(1 +4 = 2 + 3\) by saying we have not executed the two programs, and socan’t conclude anything about the output.

Identifying the intension of a mathematical term with itscomputational content is a plausible thing to do. It does, however,clash with what came earlier in this article. Expressions like“the King of France” get treated one way, expressions like“\(1 + 4\)” another. For any given expression, how do wedecide which way to treat it? It is possible to unify all this. Hereis one somewhat simple-minded way. If we think of the sense of“\(1 + 4\)” as a small program, there are certainlystates, possible worlds, in which we have not executed the program,and others in which we have. We might, then, think of the intension of“\(1 + 4\)” as a partial function on states, whose domainis the set of states in which the instructions inherent in “\(1+ 4\)” have been executed, and mapping those states to 5. Then,clearly, we can have states of an epistemic possible world model inwhich we do not know that “\(1 + 4\)” and “\(2 +3\)” have the same outputs.

This can be pushed only so far. We might be convinced by somegeneral argument that addition is a total function alwaysdefined. Then it is conceivable that we might know “\(1 +4\)” designates some number, but not know what it is. But thiscannot be captured using the semantics outlined thus far, assumingarithmetic terms behave correctly. If at some state we know \(\existsx([\lambda y\ x = y](1 + 4))\), that is, we know “\(1 +4\)” designates, then at all compatible states, “\(1 +4\)” designates, and since arithmetic terms behave correctly, atall compatible states “\(1 + 4\)” must designate 5, andhence we must know \([\lambda y\ 5 = y](1 + 4)\) at the originalstate. We cannot know “\(1 + 4\)” designates withoutknowing what.

It is also possible to address the problem from quite a differentdirection. One does not question the necessity of mathematicaltruths—the issue is an epistemic one. And for this, it has longbeen noted that a Hintikka-style treatment of knowledge does not dealwith actual knowledge, but with potential knowledge—not what weknow, but what we are entitled to know. Then familiarlogicalomniscience problems arise, and we have just seen yet anotherinstance of them. A way out of this was introduced in Fagin andHalpern 1988, calledawareness logic. The idea was to enrichHintikka’s epistemic models with an awareness function, mappingeach state to the set of formulas we are aware of at that state. Theidea was that an awareness function reflects some bound on theresources we can bring to bear. With such semantical machinery, wemight know simple mathematical truths but not more complex ones,simply because they are too complex for us.

Awareness, in this technical sense, is a blunt instrument. Arefinement was suggested in van Benthem 1991: useexplicitknowledge terms. As part of a project to provide a constructivesemantics for intuitionistic logic, a formal logic of explicit proofterms was presented in Artemov 2001. Later a possible world semanticsfor it was created in Fitting 2006. In this logic truths are known forexplicit reasons, and these explicit reasons provide a measure ofcomplexity. The work was subsequently extended to a more general familyofjustification logics, which are logics of knowledge inwhich reasons are made explicit.

In justification logics, instead of the familiar \(KX\) ofepistemic logic we have \(t{:}X\), where \(t\) is an explicitjustification term. The formula \(t{:}X\) is read,“\(X\) is known for reason \(t\).” Justification termshave structure which varies depending on the particular justificationlogic being investigated. Common to all justification logics is thefollowing minimal machinery. First there are justification constants,intended to be unanalyzed justifications of accepted logicaltruths. Second, there are justification variables, standing forarbitrary justifications. And finally there are binary operations,minimally \(\cdot\) and \(+\). The intention is that if \(s\)justifies \(X \supset Y\) and \(t\) justifies \(X\), then \(s\cdott\) justifies \(Y\), and also \(s+t\) justifies anything that \(s\)justifies and also anything that \(t\) justifies. There are very closeconnections between justification logics and epistemic logics,embodied inRealization Theorems. This is not the appropriateplace to go into details; a thorough discussion of justificationlogics can be found in this encyclopedia's entry on justification logic.

If one follows the justification logic approach one might say, of\(1 + 4 = 2 + 3\) or some more complicated mathematical truth, that itis knowable but too hard for us to actually know. That is, thejustification terms embodying our reasons for this knowledge are toocomplex for us. This follows the general idea of awareness logic, butwith a specific and mathematically useful measure of the complexity ofour awareness.

3.6.2 Proper Names

Proper names are even more of a problem than mathematicalexpressions. These days proper names are generally understood to berigid designators but, unlike mathematical terms, they have nostructure that we can make use of. Here is a very standardexample. Suppose “Hesperus” is used as a name for theevening star, and “Phosphorus” for the morning star. Itshould be understood that “the evening star” isconventional shorthand for a definite description, “the firstheavenly body seen after sunset” and similarly for “themorning star”. Definite descriptions have structure, they pickout objects and in different possible worlds they may pick outdifferent objects. But proper names are not like that. Once thedesignations of “Hesperus” and “Phosphorus” arefixed—as it happens they both name the planet Venus—thatdesignation is fixed across possible worlds and so they are rigiddesignators. It follows that while the morning star is the eveningstar, that identity is not necessary because definite descriptions arenot rigid, but Hesperus is Phosphorus and that identity is a necessaryone. How, then, could the identity of Hesperus and Phosphorus not be aknown truth, known without doing any astronomical research?

There is more than one solution of the dilemma just mentioned. Oneway is very simple indeed. Possible world models can be used torepresent various kinds of modalities. They provide mathematicalmachinery, but they do not say what the machinery is for. That is upto the user. So, we might want to have such a model to representnecessary truth, or we might want to have such a model to representepistemic issues. The argument that proper names are rigid designatorsapplies to models representing necessary truth. It does not followthat this is the case for epistemic models too. Here is a quote from(Kripke 1980) that sheds some light on the issue.

But being put in a situation where we have exactly the sameevidence, qualitatively speaking, it could have turned out thatHesperus was not Phosphorus; that is, in a counterfactual world inwhich ‘Hesperus’ and ‘Phosphorus’ were notused in the way that we use them, as names of this planet, but asnames of some other objects, one could have had qualitativelyidentical evidence and concluded that ‘Hesperus’ and‘Phosphorus’ named two different objects. But we, usingthe names as we do right now, can say in advance, that if Hesperus andPhosphorus are one and the same, then in no other possible world canthey be different. We use ‘Hesperus’ as the name of acertain body and ‘Phosphorus’ as the name of a certainbody. We use them as names of these bodies in all possible worlds. If,in fact, they are thesame body, then in any other possibleworld we have to use them as a name of that object. And so in anyother possible world it will be true that Hesperus is Phosphorus. Sotwo things are true: first, that we do not knowa priori thatHesperus is Phosphorus, and are in no position to find out the answerexcept empirically. Second, this is so because we could have evidencequalitatively indistinguishable from the evidence we have anddetermine the reference of the two names by the positions of twoplanets in the sky, without the planets being the same.

In short, proper names are rigid designators in models where thepossible worlds represent logically alternative states. They need notbe rigid designators in models where the possible worlds representepistemically alternative states. Hesperus and Phosphorus are thesame, necessarily so, but we could have used the names“Hesperus” and “Phosphorus” differentlywithout being able to tell we were doing so—a state in which wedid this might be epistemically indistinguishable from the actualone. There can be necessary identities that we do not know becausenecessary truth and known truth do not follow the same rules.

3.6.3 Non-Carnapian Approaches

The formal machinery behind the discussion above traces back toideas of Carnap. In this tradition possible worlds are central, andsense or intension is a function from possible worlds todenotations. Senses determine denotations, but detailed machineryaccounting for how this happens is not made concrete (except fordefinite descriptions). One need not do things this way. If the Churchapproach is followed, one can simply say that “Hesperus”and “Phosphorus” have the same designation rigidly, hencenecessarily, but even so they do not have the same sense. This ispossible because senses are, in effect, independent and not derivedthings. Senses can determine the same extension across possible worldswithout being identical.

A logic breaking the Carnapian mold, that is thorough and fullydeveloped, can be found in Zalta 1988. In this a class of abstractobjects is postulated, some among them being ordinary. A distinctionis made between an objectexemplifying a propertyandencoding it. For instance, an abstract object mightperfectly well encode the property of being a round square, but couldnot exemplify it. A general comprehension principle is assumed, in theform that conditions determine abstract individuals that encode (notexemplify) the condition. Identity is taken to hold between objects ifthey are both abstract and encode the same properties, or they areboth ordinary and exemplify the same properties. In effect, this dealswith problems of substitutivity. The formal theory (more properly,theories) is quite general and includes both logical necessity andtemporal operators. It is assumed that encoding is not contingent,though exemplifying may be, and thus properties have both anexemplification extension that can vary across worlds, and an encodingextension that is rigid. With all this machinery available, a detailedtreatment of proper names can be developed, along with much else.

4. Sense as Algorithm

Following Frege, the mathematical expressions “\(1+4\)”and “\(2+3\)” have the same denotation but differentsenses. Frege did not actually say what a sense was, though it wasclear that, somehow, sense determined denotation. Earlier we talked ofcomputations associated with “\(1+4\)” and“\(2+3\)”, but what we presented was quitesimple-minded. Tichý introduced the idea ofaconstruction with these two expressions prescribingdifferent constructions. A much more formal version of this appears ina series of papers, (Moschovakis 1994; Moschovakis 2006; Kalyvianakiand Moschovakis 2008), all of which trace back to (Moschovakis1989). In these is a very sophisticated formalism in which the senseor intension of an expression is an algorithm, and algorithm executiondetermines denotation. In what follows we sketch the ideas, skimpingon most technical details.

To keep things relatively simple we confine our discussion tosentences of a formal language for which, again following Frege,denotation is simply a truth value. Both “there are infinitelymany primes” and “there are infinitely many evennumbers” agree on denotation—both are true—butclearly have different senses. All the basic ideas of Moschovakis arealready present at the sentence level, though the ideas extendbroadly. We quote from (Moschovakis 1994), on which our presentationis based.

The mathematical results of the paper are about formal languages,but they are meant to apply also to those fragments of naturallanguage which can be formalized, much as the results of denotationalsemantics for formal languages are often applied to fragments ofnatural language. In addition to the language of predicate logic whosesense semantics are fairly simple, the theory also covers languageswith description operators, arbitrary connectives and modal operators,generalized quantifiers, indirect reference and the ability to definetheir own truth predicate.

If sense is to be identified with algorithm, perhaps the most basicquestion is: what is an algorithm. For Moschovakis, as for manyworking mathematicians, an algorithm is an abstract mathematicalobject, in the same way that a number is. Of course one uses specialnotation to work with a number or an algorithm, but notation issyntactic while mathematical objects are semantic (evenideal). Algorithmic subject matter may vary: an algorithm for baking acake does not operate in the same space as an algorithm for solvingquadratic equations. Some formalism is needed so that algorithms canbe specified, and this machinery should be suitable for all subjects,yet as simple as possible. There are several general, but equivalent,approaches to algorithmic specification across a range of subjectmatters. Moschovakis (1994) introduces a very simple, direct mechanismwhich he calls theLower Predicate Calculus with Reflection,wherereflection essentiallymeansself-reference. Of course not all algorithms terminate,and consequently the underlying truth value space needs someconsideration, but a solution along Kripke’s lines in his Theoryof Truth works well. We lead up to a general definition via some (forthe time being) informal examples.

4.1 Motivating Examples

Suppose we have a structure with a given domain and some givenrelations of various arities, say \(\langle \bD, \bR_1,\ldots, \bR_n\rangle\). And suppose we have a first-orderlanguage formed in the usual way, with relation symbols, \(R_1\),…, \(R_n\) whose arities match those of the given relations. Wewill generally use the typographical convention that \(\bR\) isa relation and \(R\) is the associated formal symbol interpreted bythat relation. In the usual way we can build up a first-order languagethat talks about the structure, where atomic formulas involve \(R_1\),…, \(R_n\) and \(=\). Constants can be simulated by the use ofunary relations that are true of single things. For example, inarithmetic we can have a relation \(\bZ\) such that\(\bZ(x)\) holds only when \(x=0\). In the interests ofreadability, in such a case we would act as if we had a constantsymbol in our language that was interpreted by 0. Such informalsimplifications make formula reading a bit easier, while nothingsignificant is lost.

What is added to the usual first-order machinery is a\(\textsf{where}\) construction. We will give a proper definitionshortly but first, here is a concrete example. Let us assume we have astructure for arithmetic, \(\langle \{0,1,2,\ldots\}, \bS,\bZ\rangle\). Here \(\bS\) is the two-place successorrelation on the domain, that is, we have \(\bS(0,1)\),\(\bS(1,2)\), …. We also assume \(\bZ\) is trueuniquely of 0 and, in accord with what we said above about relationsand individual constants, we act as if we had a constant symbol 0 inthe formal language. Consider the following formula, where \(S\) is atwo-place relation symbol interpreted by \(\bS\), and \(E\) and\(O\) are auxiliary one-place relation symbols.

For the time being think of \(\simeq\) as something like “isdefined to be”. This will be discussed further later. Think of\(E(x)\) as representing the ‘output’ relation. It isdefined in terms \(O\), where \(O\) is defined in terms of\(E\). Mutual recursion is involved. Even at this informal stage it isnot hard to see that \(\even\) defines the set of evennumbers, in the sense that \(\even(x)\) evaluates to true foreven \(x\) and to false for odd \(x\). Here is an informal calculationshowing that \(\even(2)\) evaluates to true. In it we use\(\Leftarrow\) for reverse implication. Also we write members of thedomain (numbers) directly into formulas, rather than using themachinery of valuations assigning numbers to free variables.

We used the clause three times, replacing \(E(2)\), \(O(1)\), and\(E(0)\). The final line is true because \(S(1,2)\), \(S(0,1)\), and\(0=0\) are true.

This example is a start, but it is misleadingly simple. Themachinery is rich enough to allow formulation of the liar sentence. Inthe following, \(P\) is an auxiliary relation symbol of arity 0, thatis, a propositional letter. We have written just \(P\) instead of\(P()\).

Clearly an evaluation attempt of the sort shown above will notterminate. A solution to non-termination is familiar from classicalrecursion theory, and also from work on the theory of truth: allow therelations defined by our formal machinery to bepartial. Notall instances of a relation have to receive a truth value. But theseare semantic issues and before getting to them we need to give aproper syntactic definition of the language within which our formulaswill be written.

4.2 Syntax

Above we spoke of a first-order language appropriate for astructure \(\langle\bD, \bR_1, \ldots,\bR_n\rangle\), enhanced with clauses, but these clauses wereonly shown via examples. Here is a proper definition. The LowerPredicate Calculus with Reflection (LPCR) for \(\langle\bD,\bR_1, \ldots, \bR_n\rangle\) is the language built upusing the machinery of ordinary first-order logic with equality,together with the following formation clause. If \(\phi_0\),\(\phi_1\), …, \(\phi_k\) are formulas and \(P_1\), …,\(P_k\) are (new) auxiliary relation variables, the following is aformula.

In this each \(\bx_i\) is a sequence of variables whose lengthis the arity of \(P_i\). The \(P_i\) may appear in the formulas\(\phi_0\), …, \(\phi_k\) themselves, and so we have aself-referential set of defining equations, with \(\phi_0\) as‘output’. Note that with \eqref{whereformula} added to thedefinition of formula, \(\textsf{ where }\) conditions can appear in some of the\(\phi_i\), and so means of preventing inappropriate interactionbetween nested conditions is needed. This is done through the familiarmachinery of free and bound variables. The symbols \(P_1\), …,\(P_k\) are taken to be relationvariables, and areconsidered to be bound in \eqref{whereformula}. Likewise the occurrencesof individual variables in \(\bx_i\) are understood to be boundin \(P_i(\bx_i) \simeq \phi_i\). In effect, these are localvariables.

Now the language LPCR has been defined, and we turn to notions of sense and reference.

4.3 Denotation

We have been discussing sentences and more generally formulas withfree variables. The familiar Tarskian semantics provides a basis forunderstanding here, but we need modifications and extensions to dealwith the construct.

Apartial function on a space \(S\) is a function thatassigns values to some, but not necessarily to all, members of\(S\). Said otherwise, it is a function whose domain is a subset of\(S\). For a partial function \(f\), \(f(x)\simeq y\) means \(x\) isin the domain of \(f\) and \(f(x) = y\). (Finally we have a properaccounting of our use of \(\simeq\) in the examplesearlier.)Partial relations are partial functions from\(k\)-tuples to \(\{{\textsf{t}}, {\textsf{f}}\}\). Thegivenrelations of our structures are relations in the usual sense, but itis partial relations that we may find ourselves defining.

Assume we have a structure \(\langle\bD, \bR_1,\ldots, \bR_n\rangle\), and suppose we have an LPCR languageassociated with it. Avaluation \(v\) in this structure is amapping from individual variables to members of \(\bD\) andfrom auxiliary relation symbols to partial relations on\(\bD\). We would like to associate with each valuation \(v\) amapping \(T_v\) from formulas of LPCR to truth values but since thingslike the liar sentence are formulable, \(T_v\) must be a partialfunction, and so we must be careful even about familiar things likepropositional connectives. Various three valued logics have beendeveloped; perhaps the most common is Kleene’s strongthree-valued logic, motivated by recursion theory and familiar frommuch work on the Theory of Truth. The following table says howconnectives and quantifiers behave. Cases that are not explicitlycovered are understood to be those for which a truth valuation is leftundefined. (For instance, if the truth value of \(X\) is undefined,the same is the case for \(\lnot X\).)

This still leaves formulas to deal with. Suppose we have the following.

We make two simplifying assumptions to keep our discussion frombeing too intricate. We assume no \(\phi_i\) contains a nested\(\textsf{ where }\) clause. The basic ideas are amply illustrated with this conditionimposed, but everything extends to the general case without too muchdifficulty. It is a general requirement that the variables in\(\bx_i\) are ‘local’ to \(P_i(\bx_i) \simeq\phi_i\), that is, they are considered to be bound in this formula. Tothis we add another simplifying assumption: the variables in\(\bx_i\) are theonly variables that may occur freein \(\phi_i\). Roughly this means that we have no parameters, onlylocal variables. This serves to allow us to discuss things with lessclutter. Again, everything extends to the more general case with nofundamental changes.

Continuing with \eqref{Eexample}, consider the following associated set\(E\) of equations.

The difficulty, of course, is that each \(P_i\) is allowed to occurin one or more \(\phi_j\), possibly even in \(\phi_i\), and so \(E\)is self-referential. In many computer programming languages one seesthings like \(x = x+1\). It is explained to beginning programmers thatthis takes the current value of \(x\), adds 1, and calls the result\(x\) again. Occurrences of \(x\) on the right have‘before’ values, occurrences on the left have‘after’ values. Analogously, let us think of the membersof \(E\) as (simultaneous) assignment statements. Occurrences of\(P_i\) on the right of \(\simeq\) are current values, occurrences onthe left are next values. Taking all of \(P_1\), …, \(P_k\)into account, we can think of \(E\) as defining a functional that maps\(k\)-tuples of partial relations (‘before’ values ofthese relation symbols) to \(k\)-tuples of partial relations(‘after’ values of these relation symbols). Now here arethe details a bit more formally.

Suppose we have a \(k\)-tuple \(\langle\bP_1, \ldots,\bP_k\rangle\) of partial relations, where for each \(i\) thearity of \(\bP_i\) matches that of the partial relationvariable \(P_i\). This is our input (‘before’ values). Foreach \(i\) we want to define an output partial relation which we call\(\bP'_i\), of the same arity as \(\bP_i\), so that\(\langle\bP'_1, \ldots, \bP'_k\rangle\) serves as ouroverall output (‘after’ values). To do this we must saywhen \(\bP'_i(\bd)\) maps to \({\textsf{t}}\), when itmaps to \({\textsf{f}}\), and when it is undefined, for each\(\bd\) with components from \(\bD\). Well, take \(v\)to be a valuation assigning to each auxiliary relation symbol \(P_i\)the corresponding partial relation \(\bP_i\) (this is how‘before’ values for our partial relation symbols come in),and assigning to the variables in \(\bx_i\) the correspondingmembers of \(\bd\). Now, simply let\(\bP'_i(\bd)\simeq T_v(\phi_i)\). In this way a newpartial relation \(\bP'_i\) is specified, and more generally avector of them, \(\langle\bP'_1, \ldots,\bP'_k\rangle\). The set of equations \(E\) can be thought ofas specifying afunctional transforming \(k\)-tuple\(\langle\bP_1, \ldots, \bP_k\rangle\) into\(\langle\bP'_1, \ldots, \bP'_k\rangle\). Let us callthis functional \([E]\), and write \([E](\langle\bP_1, \ldots,\bP_k\rangle) = \langle\bP'_1, \ldots,\bP'_k\rangle\).

If we are to have equations \(E\) behave well in a logic setting,each \(P_i\) should have the same valuation no matter where we seeit—there should be no distinction between what we have beencalling left and right sides; \(\bP_i\) and \(\bP'_i\)should be the same. In other words, we would like to have partialrelations \(\bP_1\), …, \(\bP_k\) to interpret\(P_1\), …, \(P_k\) so that \([E](\langle\bP_1, \ldots,\bP_k\rangle) = \langle\bP_1, \ldots,\bP_k\rangle\)—‘before’ and‘after’ values agree. This is called afixedpoint of \([E]\). So, we need to know that \([E]\) has a fixedpoint, and if it has more than one then there is a plausible candidatewe can choose as the best one.

If \(f\) and \(g\) are two partial functions from a space \(S\) to\(R\), one writes \(f\subseteq g\) to mean that whenever \(f(x)\simeqw\) then also \(g(x)\simeq w\). Then for two partial relations\(\bP\) and \(\bQ\) of the same arity,\(\bP\subseteq\bQ\) means that whenever\(\bP(\bd)\) is defined, so is\(\bQ(\bd)\), and both have the same truth value. We canextend this to \(k\)-tuples by setting \(\langle\bP_1, \ldots,\bP_k\rangle\subseteq\langle\bQ_1, \ldots,\bQ_k\rangle\) if \(\bP_i\subseteq\bQ_i\) foreach \(i\). It is not terribly difficult to show that the functional\([E]\) defined above, and based on \eqref{Eexample}, hasthemonotonicity property: if \(\langle\bP_1, \ldots,\bP_k\rangle\subseteq\langle\bQ_1, \ldots,\bQ_k\rangle\) then \([E](\langle\bP_1, \ldots,\bP_k\rangle)\subseteq[E](\langle\bQ_1, \ldots,\bQ_k\rangle)\). There is a very general theory of monotonemappings like this, from which it follows that \([E]\) does have afixed point. Moreover, if there are more than one then there is aunique one that is least, in the sense that it is in the \(\subseteq\)relation to any other. This least fixed point is precisely the bestcandidate we mentioned above. It contains the information that anyfixed point must have.

Now we finish saying how to evaluate the formula\eqref{Eexample}. First, construct the associated set of equations,\(E\). Next, construct the functional \([E]\). There is a least fixedpoint for \([E]\), let us say it is \(\langle\bF_1, \ldots,\bF_k\rangle\). Finally, evaluate \(\phi_0\) using\(\bF_i\) to interpret \(P_i\) for each \(i\). The resultingtruth value, or undefined, is the value (denotation) associated with\eqref{Eexample}.

We have now said how to associate a truth value, or undefined, withevery formula of LPCR (under our simplifying assumptions). We have(partial) denotations.

4.4 Sense

Each formula of LPCR specifies an algorithm for its evaluation,that is, for the determination of its truth value (ifpossible). Moschovakis identifies thesense of a formula withthat algorithm. Two formulas that evaluate to the same result, thushaving the same denotation, may have different senses because theassociated algorithms are different. For example, in \eqref{evenone} wegave a formula that defines the even numbers. Here is another suchformula.

We leave it to you to verify that \eqref{eventwo} also defines the evennumbers. It is intuitively plausible that \eqref{evenone} and \eqref{eventwo}evaluate using algorithms that differ, and so have differentsenses. But of course this must be made precise. What is needed is auniform method of comparison between algorithms. Here we just brieflysketch the ideas.

There is very general machinery, from Moschovakis 1989, called theFormal Language of Recursion, FLR. Using it a thorough exploration ofrecursive definitions and fixpoints is possible. The language thatconcerns us here, LPCR, embeds into FLR, even allowing nested clausesand parameters, something we ignored in our discussion ofdenotation. In FLR there is a method for converting recursivedefinitions into a normal form, which cannot be further reduced. Thatnormal form has a very simple structure, consisting of a set ofself-referential equations with no nesting present at all. Normalforms reveal essential evaluation structure most clearly. When workingwith a single structure, \(\langle\bD, \bR_1, \ldots,\bR_n\rangle\), all normal forms will be built from a commonset of functionals. This makes it easy to compare normal forms. Theidea is that if two formulas of LPCR, when embedded into FLR, havediffering normal forms, the two formulas have different senses. Ofcourse this must be taken with some reasonable flexibility. Forinstance, two sets of equations that differ only by renaming variablesor switching order of equations do not differ in any fundamentalway. With this understood, if two LPCR formulas, when embedded intoFLR, have truly distinct normal forms, the two LPCR formulas aredefined to have different senses. This meets all the informalconditions one wants a notion of sense to have. Moschovakis evenproves the important theorem that equality of sense, as just defined,is decidable under natural conditions.

4.5 Algorithms Need Not Be Effective

The word “algorithm” suggests something effective, buthere it is being used in a more general sense, as a set ofinstructions that, for reasons of our finitistic limitations, we maynot be able to actually carry out. Consider again the paradigmformula, \eqref{whereformula}. If one of the \(\phi_i\) contains anexistential quantifier in a positive position (or a universalquantifier in a negative position) it can be thought of as invoking asystematic search through the domain \(\bD\) for a verifyingwitness. This is plausible for reasonable domains. But if \(\phi_i\)should contain a universal quantifier in a positive position or anexistential quantifier in a negative position, something must beverified for every member of the domain and unless the domain isfinite, this is not a human task. Nonetheless, we generally believe weunderstand quantification. What we are dealing with is algorithmsrelative to that understanding.

The problem with quantifiers is inescapable for much that weroutinely discuss using sense and reference. Consider Russell’streatment of definite descriptions. In this “the \(A\) hasproperty \(B\)” is replaced by “exactly one thing hasproperty \(A\) and it has property \(B\)”. To say that only onething has property \(A\) one says that something has property \(A\)and everything else does not. The first part of this involves anexistential quantifier and the second part a universal one. Then ifthe definite description occurs in a positive location we have apositive occurrence of a universal quantifier, and if it occurs in anegative location we have a negative occurrence of an existentialquantifier. Essential problems arise either way. Moschovakis is notclaiming to turn sense and reference into something computable, butsimply to provide mathematical machinery that can plausibly formalizethe ideas involved using a generalized notion of algorithm.

There is a second, related problem where lack of effectivenesscomes in. In our discussion of denotation we considered a set \(E\) ofequations \eqref{equationsE} and a functional \([E]\) associated withthem. Recall that \([E]\) mapped \(k\)-tuples of partial relations to\(k\)-tuples of partial relations. We noted that \([E]\) wouldbemonotone, and by very general results such functionalsalways have least fixed points. There is more than one way of showingthis. One well-known argument has a decidedly algorithmic flavor toit. It goes as follows. Start with the smallest \(k\)-tuple of partialrelations—this is the one where every partial relation is alwaysundefined. Call this \(T_0\). Apply the functional \([E]\) to \(T_0\),getting \(T_1\). Apply the functional \([E]\) to \(T_1\) getting\(T_2\), and so on. It is easy to show that \(T_0\subseteqT_1\subseteq T_2\subseteq\ldots\). We have that \(T_0\subseteq T_1\)because \(T_0\) is in the \(\subseteq\) relation to every\(k\)-tuple. By monotonicity we then have \([E](T_0)\subseteq[E](T_1)\), but this says \(T_1 \subseteq T_2\). And so on. Continuewith this increasing sequence and eventually the least fixed point of\([E]\) will be reached.

But this is very misleading. What does “continue” mean?We have \(T_0\), \(T_1\), \(T_2\), …. None of these may be afixed point. For instance, suppose we carry out this construction withthe functional arising from \eqref{evenone} for \(\even(x)\). Then\(T_0\) will be \(\langle E_0, O_0\rangle\), where both \(E_0\) and\(O_0\) are the everywhere undefined 1-place relation. We leave it toyou to check that we get successive \(T_i=\langle E_i, O_i\rangle\)where we have the following, with cases not displayed beingundefined.

None of \(T_0\), \(T_1\), \(T_2\), …is a fixed point, butthere is a clear notion of a limit, called \(T_\omega\), thataccumulates the results produced along the way. It is the least fixedpoint in this example.

But iterating and taking a limit may not be sufficient. Considerthe following elaboration of \eqref{evenone}.

The set of equations arising from \eqref{evenmore} has the two membersof \eqref{evenone}, and one more for \(A\). Using these equations, inorder to conclude \(A(1)\) we must already have one of \(E(y)\) or\(O(y)\) evaluating to \({\textsf{t}}\) for every number \(y\). If wecarry out the construction outlined above, we won’t have thisfor \(E\) and \(O\) until stage \(\omega\), and so we must go one morestep, to what is called \(T_{\omega+1}\), before we reach a fixedpoint.

More and more extreme examples can be given. The fixed pointconstruction may have to be continued to larger and larger transfiniteordinals. This is a well-known phenomenon, especially in areas likethe Theory of Truth. It cannot be avoided. Incidentally, it should benoted that the machinery introduced by Kripke in his treatment oftruth has a natural embedding into LPCR, but we do not discuss thishere.

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Additional Reading

• Aczel, P. (1989). “Algebraic semantics forintensional logics,” I. In G. Chierchia, B. Partee, andR. Turner (Eds.),Properties, Types and Meaning.Volume I: Foundational Issues, pp. 17–45, Dordrecht:Kluwer.
• Bealer, G. (1998). “Intensional entities,” in E. Craig(ed.),Routledge Encyclopedia of Philosophy, London:Routledge.
• Fitting, M. (2002).Types, Tableaus, and Gödel’sGod, Dordrecht: Kluwer.
• Kripke, S. (2008). “Frege’s Theory of Sense and Reference:Some Exegetical Notes,”Theoria, 74: 181–218.
• Menzel, C. (1986). “A complete, type-free second order logicand its philosophical foundations,” Technical Report CSLI-86-40,Stanford: Center for the Study of Language and InformationPublications.
• Searle, R. (1983).An Essay in the Philosophy of Mind,Cambridge: Cambridge University Press.
• Svoboda, V., Jespersen, B., Cheyne, C. (Eds.) (2004):PavelTichý’s Collected Papers in Logic and Philosophy, Prague:Filosofia, and Dunedin: Otago University Press, Dunedin.
• Thomason, R. (1980). “A model theory for propositionalattitudes,”Linguistics and Philosophy, 4:47–70.
• van Benthem, J. (1988).A Manual of IntensionalLogic, Stanford: CSLI Publications.

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Other Internet Resources

• Intensional Entities, preprint of G. Bealer, in E. Craig (Ed.),Routledge Encyclopedia ofPhilosophy, London: Routledge (1998).
• An Introduction to Pavel Tichý and Transparent Intensional Logic, Andrew Holster, 2003.

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