Abstraction in Fitch's Basic Logic

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Abstraction in Fitch's Basic LogicEric Thomas Updike aa Residential Faculty, Department of Philosophy and ReligiousStudies, Glendale Community College, Glendale, AZ, 85302, USAVersion of record first published: 17 Feb 2012.

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HISTORY AND PHILOSOPHY OF LOGIC, 33 (August 2012), 215–243

Abstraction in Fitch’s Basic LogicEric Thomas Updike

Residential Faculty, Department of Philosophy and Religious Studies, Glendale Community College, Glendale,AZ 85302, USA

[email protected]

Received 19 July 2011 Revised 19 November 2011 Accepted 20 November 2011

Fitch’s basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting atype collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not workaxiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic.Fitch’s proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors thathave so far gone undetected. This paper corrects these errors and presents a reasonably intuitive proof that Fitch’ssystem K supports an implicit abstraction operation. Some general remarks on the philosophical significance ofbasic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic modelsa highly intensional form of logicism.

1. IntroductionF. B. Fitch (1908–1987) is widely recognized for developing the proof system that bears

his name and for presenting the paradox of knowability in his 1963 paper ‘A LogicalAnalysisof Some Value Concepts’. His most systematic body of work, and arguably the most impor-tant, was devoted to a close family of illative combinatory logics collectively entitled ‘basiclogic’, a contribution that, unlike the paradox of knowability, has elicited spare commen-tary from contemporary philosophers.1 Basic logic has much of the tractability of first-orderlogic while containing a non-trivial fragment of the rich expressive power of second-orderlogic, in a system which is simpler than set theory. A survey of the properties of basiclogic reveals its peculiar double nature which combines features of first- and second-orderlogics: basic logic has unlimited abstraction and comprehension, enjoys nontrivial inferen-tial power in spite of the Curry paradox, is able to define a range of concepts customarilyassociated with higher-order logics such as the ancestral of an arbitrary relation, containsa considerable quantity of mathematics, is (partially) semantically closed as it can define anotion of ‘reflective’ truth specific to a kind of abstract syntax,2 is demonstrably consistentand the base system for basic logic is complete. These facts, except completeness, wereessentially established by Fitch though not always in ways that would meet contemporarystandards of rigor.3 The fact that these properties can be combined into a consistent logicalframework which has genuine mathematical content is something of a revelation.

This evaluation of the significance of basic logic demands an explanation of the relativeneglect of basic logic by historians of logic. In this regard, I can only speculate as to therelevant intellectual history. Basic logic is a highly intensional and type-free system and

1 Fitch’s first paper on basic logic was published in 1942 and the last was published in 1984, with many intermediary contributions.

See Anderson et al. 1975 for a partial bibliography of Fitch’s work.2 The extended form of the base system of basic logic has this property. Truth is reflective in the sense that the truth operator (and

its dual) can (truly) apply to itself.3 Myhill 1950 established a completeness theorem for a system fairly similar to basic logic. Future work will show how to modify

Myhill’s completeness proof with the notion of a ‘structured variable’ to establish a completeness theorem for Fitch’s system

K , which was the first formulation of basic logic. The phrase ‘abstract syntax’ is from Cantini 1996.

History and Philosophy of Logic ISSN 0144-5340 print/ISSN 1464-5149 online © 2012 Taylor & Francishttp://www.tandfonline.com http://dx.doi.org/10.1080/01445340.2011.648312

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216 Eric Thomas Updike

Fitch was strongly committed to some species of logical realism involving genuine onto-logical commitments to propositions, attributes and essences, where the latter are assignedto objects by propositions. The intellectual environment in which basic logic developed wasnot friendly to a system with these features, facing, on the one side, the extreme extension-alism of Quine and the anti-metaphysical polemics of the logical positivists, and, on theother, standing in direct conflict with partisans of type theory. The success of set theoryin philosophical problem solving and the alleged reduction of the intensional to the exten-sional seems to have left basic logic as something of a curiosity disconnected from broaderintellectual movements in philosophical logic. Though Fitch does not appear to have haddifficulty finding outlets for publishing articles on basic logic his work was simply too farremoved from orthodoxy to invite inclusion in rational reconstructions of the developmentof 20th century logic until very recently.4

A significant barrier to appreciating Fitch’s formal work today is that he worked within thetradition of combinatory logic and his work was in various ways influenced by Whitehead’sprocess philosophy. These movements have largely fallen silent (as grand philosophicalprojects) in the contemporary philosophical scene. Fitch was also adverse to model theoryprogrammatically (and likewise to any semantic theory that relies on a denotation relation toimbue expressions with content) and philosophers accustomed to approaching philosophicallogic under the tripartite division of syntax, semantics and pragmatics may find his workincoherent.5 One reason Fitch seems to have avoided model theory was that he wanted hissolutions to philosophical problems to be free from the charge that they covertly rely onhomophonic truth conditions provided by a background model, the effect of which (so theworry goes) merely defers the problem of interest to the metatheory. The cost of this aversionto model theory was high as it made Fitch’s work inaccessible to most logically mindedphilosophers who demand an explicit interface linking syntax to the intended semantics.6

Basic logic is supposed to convey something meaning – theoretic purely in virtue of syntaxwithout relying on any kind of intermediate, either senses or a relation of denotation, toguarantee the correctness of the system.7 It is hard to see how to convince someone of thisclaim who is not already a disciple of basic logic.

Fitch’s aversion to modern semantic theory was also part of a methodological programfor philosophy whereby basic logic would serve as a neutral framework in which differentphilosophical theories could be compared and evaluated, moreover, that before discussionbetween philosophical disputants can even begin some such metalanguage for philosophy

4 So far as I know Cantini 2009 is the first systematic attempt to include Fitch’s work on type-free foundations for logic and

mathematics within the broader history of mathematical logic. Cantini also wonders why Fitch’s work on logic is forgotten

today. Cantini speculates that Fitch’s peers took his work as primarily relevant to a formalist foundational project and to illative

combinatory logic in particular (p. 937). However, Fitch seems to have understood propositions in the sense of Russell and if the

constituency relation (of an object in a proposition) finds an exact analogue in the part/whole relation of syntax then formalist

philosophical scruples may instead betray a thoroughgoing commitment to logical realism rather than to a crude formalism.

Here as in other areas Fitch does not give a great deal of guidance but some hints may be found in his ‘Propositions as the Only

Realities’ published in 1971.5 Körner 1976 records some such attitude from Geach and McDowell in their respective responses to a talk by Fitch on a

combinatory logic similar to basic logic. Haack, in her review 1978 of the proceedings recorded in the Körner volume, calls

their comments ‘somewhat chilly’.6 There are a few exceptions to the general lack of interest in Fitch’s program. The Frege structures of Aczel 1980, the systems

of Apostoli 2000, the minimal framework of Cantini 1996, and the work on type-free logics in Feferman 1984, all bear fruitful

comparisons to basic logic. Scott and Myhill (especially the latter) devoted important papers to basic logic and Gilmore 1968

(and Gilmore’s later work on partial set theory) is somewhat in the tradition of basic logic. The algorithmic logic of Aitken and

Barrett 2004, 2007a,b is closely allied with the philosophical program of basic logic.7 The base system for basic logic, Fitch’s system K , is complete with respect to what is essentially the theory of pairing (see

Myhill 1950 for a proof of this type) though Fitch did not develop basic logic in order to exemplify this kind of completeness.

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History and Philosophy of Logic 217

must be used (this is so even if the object of concern is basic logic itself). A realist semanticsfor the metalanguage in which philosophical theories are rationally surveyed is not neutral,at least if the dispute is about the background semantics itself. Unfortunately, Fitch nevermanaged to show in detail just how basic logic would serve this particular ideologicalcause and the metaphilosophical role of basic logic is by no means apparent and to manyphilosophers not even desirable.8

These issues were not the only problems basic logic had in finding a sympathetic audience.Fitch (1942) outlined a proof that basic logic has an implicit abstraction operator in the firstpaper devoted to basic logic but the methodology is obscure as it proceeds through a seriesof sometimes baffling abbreviatory devices.9 Moreover, the combinatorial basis for basiclogic is not combinatorial complete in isolation (certain logical operators are also necessaryfor establishing this metalogical fact) contra the situation with pure combinatory logic.Fitch’s choice of combinators is given very little motivation and his mixing of combinatorswith standard logical operators is highly unconventional. Fitch’s philosophical programthus begins with a certain lack of clarity which was inherited by Fitch’s various extensionsof the original system of basic logic.

This paper is intended to satisfy two formal desiderata. The first is to present a clear‘canonical’ proof of monadic and relational abstraction in basic logic which highlightsvery clearly how the clauses of basic logic work together to effect what amounts to asubstitution operation. The second desideratum is to correct some subtle flaws in Fitch’sown presentation which have, so far as I know, gone undetected for nearly 70 years. Someremarks of a clarificatory and motivational nature are included, especially regarding Fitch’schoice of combinators and the ambitious metaphilosophical role Fitch thought basic logicwould ultimately serve. Basic logic is worth a second look in the light of recent trends in thephilosophy of mathematics occasioned by the revival of Frege studies. Fitch’s abstractionis different in kind from the prevailing methodology found in the Hale–Wright approach toabstraction and some brief comments on the significance of this difference is offered at theend of the paper.

2. The system KFitch’s illative combinatory logic K is the base system for basic logic in the sense that

it (or a theorem-wise equivalent system) is extended to various other systems as Fitchexpanded his investigations to account for new mathematical and logical phenomena. K is auniversal calculus in the sense that the class of theorems of any finitary system (understoodas an ordered pair consisting of a recursive language L and an L-theory equipped with aneffective consequence relation) can be represented within it.10 Theories with this propertyare characterized as Turing complete in the Chomsky hierarchy. As K is itself finitary it canrepresent itself, hence K surveys every possible way of specifying a class of propositionsrecursively without exceeding the finitary standpoint.

Evidently K isolates some important class of truths specific to formal representation. Inhis 1944 paper ‘A Minimum Calculus for Logic’ Fitch described the way in which basiclogic involved some notion of truth vital to formalization in the exact sciences:

Some logical truths are in some respects more important than others. The most fun-damental, it would seem, are those which are required for formulating or expressing

8 Basic logic does have resources sufficient to establish interesting results in metalogic, in particular, a kind of �02-incompleteness

result, which generalizes Gödel’s first incompleteness theory, will be established in forthcoming work.9 In subsequent papers on basic logic Fitch refers to the first paper for the proof of abstraction so he evidently thought he had

established its proof beyond doubt.10 A precise definition of representation-in-K will be given later in this section.

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all the others. A really ‘basic logic’ is one in which every other system of logic canbe defined, without necessarily settling the question of the validity or non-validityof such other systems of logic. The theorems of a basic logic are logically valid inthe sense that they are required or presupposed in all rational discussion of othersystems of logic, since such discussion cannot be effectively carried on without alogic in which the various other systems are specified and compared. (Fitch 1944a,p. 89)

Fitch’s basic logic may be interpreted as an abstract syntax for the presentation of theoriesas formal objects and it functions as an ur-logic which captures precisely what is necessary(and sufficient) for formalization in the exact sciences.11 The theory has semantic contentof a sort as Fitch seems to understand basic logic as involving the construction of languagesin a universal syntax which yields abstract syntactic structures having the same structure aspropositions (with respect to constituency), properties (with respect to exemplification) andconcepts (with respect to the falling under relation). If propositions are combinations of con-stituents (the Russellian conception of propositions) and supposing properties are assignedto objects by propositions, then the idea finds support from the fact that K is combinatoriallycomplete, which is an immediate corollary of the proof of monadic abstraction in the nextsection. Fitch (1942) claims that K is a formulation of a system of propositions, apparentlyall that can be expressed by a sentence in a recursive language, and basic logic evidentlyexhausts possibilities for the representation of concepts by finitary means (assuming thatconcepts depend on some sort of finite specification). As briefly indicated in Section 1 itwas Fitch’s intention that basic logic would serve as a metalanguage for philosophy and theuniversal characteristics of basic logic just indicated give insight into how basic logic wasto achieve this desideratum though the associated metaphysics will be very distasteful tothose with a preference for desert landscapes.12

Basic logic is a mixed logic containing logical operators and combinators.13 The combina-tors are general function types which are intended to represent the combinatorial character ofmathematics (Russellian propositions, as briefly discussed before, are particularly amend-able to representation in combinatory logics). Fitch does not use the traditional pair of

11 Basic logic is richer than apparently similar projects involving universal grammar or universal notational systems such as

Backus-Naur notation in theoretical computer science. K contains a fragment of set theory (the extent of this is discussed

later) and extensions of K contain considerably more (perhaps all that is needed for ordinary mathematics). Indeed, from the

proof-theoretic point of view K is able to match the deductive power of any finitary theory, including ZFC. This is so since K

can represent every finitary inductive definition, indeed, every recursively enumerable set of natural numbers (the former was

essentially established in Fitch 1942, Theorem 5.16, the latter in Fitch 1944b, as Theorem III).12 Prospects for such a program obviously depends on the extent to which the axiomatic method is applicable in philosophy,

including ethics and aesthetics, of which there is substantial doubt after the demise of logical positivism. Fitch was entirely

serious about the role of logic in philosophy and in 1968 he anticipated the objection that value theory, of all things, can find

no use for symbolic logic:

Naturally [symbolic logic] meets with strong opposition from those who do not have the technical training to understandit, and from those who feel that, though science has made great advances by use of mathematics, no analogous advance,by similar use of exact methods, is to be expected in philosophy. There is also the mistaken supposition that valueconcepts (e.g., those of ethics and [a]esthetics) cannot be handled by symbolic logic and that all use of symbolic logicindicates a return to a hopeless materialism and skepticism. Quite the reverse is actually the case. The only way thatmankind can develop an ethics and a philosophy commensurate with its achievement in building the atomic bombis to make full use of symbolic logic in criticizing and correcting the past systems of ethics and philosophy and inconstructing new and better ones. (Fitch 1968, p. 545)

Similar sentiments are also found among advocates of the Unity of Science Movement, as detailed in Reisch 2005.13 The combinators were extensively investigated by Curry and were first introduced in Schönfinkel 1924. Quine’s preface to

Schonfinkel’s article in the van Heijenoort collection is a useful primer on the combinators. More advanced treatments may be

found in Curry and Feys 1958 and Curry et al. 1972.

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combinators S and K (the latter of which is distinguished from Fitch’s system with thesame name) from which the proof of abstraction in pure combinatory logic is a simpleaffair.14 Fitch uses the compositionality combinator B, the duplication/contraction combi-nator W and the distinguished combinators ε and its relational counterpart ε which modelmembership/exemplification. These combinators do not form a combinatorially completeset in isolation – the logical operators of the system are also involved in the proof that basiclogic has both monadic and relational abstraction, a feature that is unique in the literatureon combinatory logic.

The presentation of K below will follow that of Fitch 1942. Hermes 1965 and Myhill 1984present proof-theoretic alternatives which are in some respects simpler but they suppressthe role of the combinators as they treat abstraction axiomatically. As discussed below thereare philosophical disadvantages to these alternatives which arguably outweigh their gainsin formal simplicity.

Like combinatory logic, basic logic is a simple applicative language which means itsexpressions are built from a single binary operation called ‘application’ from some smallstock of primitive expressions or alphabet (which are expressions which are not combina-tions of others). Other modes of combination could, in principle, be incorporated but thebinary mode is the most basic kind. Application should not be confused with concatena-tion since the latter involves the manipulation and combination of finite sequences. It isnot assumed in advance that basic logic quantifies over finite sequences.15 Calculi withapplication as the only mode of combination for expressions are combinatory calculi; Fitch(1942) observes that all calculi can be formulated as a combinatory calculus. Fixing a finitestock of primitive expressions and considering all possible combinatory calculi constructedfrom them by application also surveys every possible kind of formal calculus in any count-able language (Fitch 1942, p. 106). Hence, every formal system becomes comparable sinceeach can be formulated from the same basic stock of expressions via the same mode ofcombination.

It is possible to choose a single primitive expression ‘σ ’ from which all calculi can, inprinciple, be constructed. ‘σ ’ is understood to be syntactically simple or atomic in the sensethat from the point of view of basic logic it has no proper parts no matter that its tokens arenecessarily complex (for example, ‘σ ’ contains microdots, among other possible physicalconstituents).16 The operation of application is represented as APP(x, y), for any expressionsx, y, and the result of this operation is represented by the notation ‘(xy)’ for convenience.In applicative languages ‘(xy)’ is conventionally read as ‘the application of y to x’, and inthe Curry tradition it is generally understood as expressing the proposition ‘y is a memberof x’ or as expressing the converse of the relation ‘y falls under x’. Expressions of the form((xy)z) are understood as the application of z to the result of applying y to x, in which casex may be interpreted as a functor. Grouping to the left suggests that an expression of theform (x(yz)) is the application of the pair yz to x, in which case x may be interpreted asa binary relation though as seen later there are other ways to represent binary relations inbasic logic.

14 In combinatory logic these combinators satisfy the equations Kxy = x and Sxyz = xz(yz). K (the constant function) allows the

system to introduce novel terms and S is carefully engineered to effect a substitution operation while simultaneously eliminating

duplications of terms.15 Fitch’s extension of basic logic to his system K ′, which adds universal quantification to K , can quantify over all finite sequences

by using Quine’s method of framed ingredients given in Quine 1981.16 It might be thought that this last point is simply too obvious to bear mention. However, basic logic is arguably a system for

the description of syntax, in particular, a description of general sign-shapes. That ‘σ ’ is to be free of any further syntactical

analysis in terms of simpler components must be explicitly assumed in advance.

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The universe U of all expressions (of basic logic) is inductively generated from the singleatomic expression σ via the binary operation of application. Elements of U (called U-expressions) will be denoted in the metatheory by lower-case Roman letters. Examples ofU-expressions are the following:

σ , (σσ), (σ (σσ)), ((σσ)σ ), (σ (σ (σσ))) . . . . (1)

For notational convenience, we associate pairs of parentheses to the left, so that abc is anabbreviation for ((ab)c) while a(bc) abbreviates (a(bc)) and a(bc)d abbreviates ((a(bc))d).By this convention, the expression (τa1 . . . an) is an abbreviation for the formally correct(. . . ((τa1)a2) . . . an). Note that expressions of the form �(aa)� are members of U and itwill be seen later that expressions of that form are provable in basic logic. This satisfiesFeferman’s criterion in Feferman 1984 for a system to be untyped, namely, that the systemmust actually have non-trivial instances of self-application as provable theses in order tobe genuinely type-free. On the Feferman criterion for type-freeness, the language of ZFCis not type-free though the expression x ∈ x is well-formed but all its instances are false ina well-founded universe.

It need not be assumed that U is a completed, infinite totality. However, the followingmetatheoretical facts imply an axiom of infinity in the metatheory since the sign for identityis interpreted as strict literal identity:

Proposition 2.1 For all a, b ∈ U , a �= (ab) and in particular σ �= (ab).

From the standpoint of basic logic, it is sufficient to assume that U is potentially infinite.We also have the useful fact that for all a, b, c, d ∈ U , if (ab) = (cd) then a = c and b = d.This is the basis for defining ordered pairs, if they are wanted.

The elements of U have been characterized as ‘expressions’ which suggests that theyare candidates for some kind of meaning. This suggestion should be resisted since it isnot clear in advance that, say, σ has any meaning at all in isolation. It is more usefulat this stage to think of the elements of U as items of an abstract syntax which will beused to represent other combinatorial calculi. It may be helpful to view the definition of K(forthcoming) as a truth-definition where truth in the sense of K does not require a relationof reference to domains as in model theory. U-expressions can play a double-role in thetheory, as expressing propositions and as constituents of the same (and by type-freenessmultiple instances of an expression in a context can play either role). The use/mentiondistinction is not articulated in terms of an explicit assignment of type (as in Gilmore 2005)but is resolved by the functional role the expression plays in the given context. This rolemust be discovered by repeatedly applying the defining clauses of basic logic, given below.

Fitch’s development of basic logic is highly unconventional. He starts by choosingU-expressions denoted by =, ε, ε, B, B1, W , &, ∨, E in such a way that no one is part ofanother (this is always possible). Indeed, as shown in Hermes 1965 by letting σn bethe n-fold application of σ to itself it is easily seen that in the sequence of expres-sions σ2σ2, σ3σ3, σ4σ4, . . . no two distinct elements contain the other as a part, andconversely (p. 220). Choose then σnσn, for 2 ≤ n ≤ 10, as the U-expressions abbreviatedby =, ε, ε, B, B1, W , &, ∨, E, respectively. The remaining stock of expressions can be usedas free variables. Finally, in the definition of K below the expressions in square bracketssuch as [a = b] are more properly written as ((= a)b) but the infix notation will be usedwherever it promotes clarity (even more liberal conventions will be used frequently, forexample, often we write ‘a = b’ without any grouping symbols).

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K is defined as the smallest class of U-expressions satisfying the following inductiveclauses:17

Definition 2.2 Fitch’s Basic Logic K :

[a = b] ∈ K ⇐⇒ a,b are syntactically identical U-expressions Rule[=][aεf ] ∈ K ⇐⇒ fa ∈ K Rule[ε]Bfga ∈ K ⇐⇒ f (ga) ∈ K Rule[B]εafb ∈ K ⇐⇒ fab ∈ K Rule[ε]

B1fgab ∈ K ⇐⇒ f (ga)b ∈ K Rule[B1]Wfa ∈ K ⇐⇒ faa ∈ K Rule[W ]

[a&b] ∈ K ⇐⇒ a ∈ K and b ∈ K Rule[&][a ∨ b] ∈ K ⇐⇒ a ∈ K or b ∈ K Rule[∨]

Ef ∈ K ⇐⇒ (∃a ∈ U)fa ∈ K Rule[E]K ′ is K plus the following additional ω-style rule for the universal operator ‘A’:

Af ∈ K ′ ⇐⇒ (∀a ∈ U)fa ∈ K ′ Rule[A].Fitch presented K ′ as an extension of K with the above rule and 10 additional rules fornegation (symmetric to the rules for K and the rule for A above). This is not necessary sinceit is possible to define a materially adequate negation operator for K ′, which Fitch (1984)sketched in outline, though this is not the place for a detailed proof of this fact.

A few explanatory comments are in order. A form of membership (the sign ε is more thanincidentally related to ∈) may be expressed as the application of one term to another, ascaptured by Rule [ε] above. The rule for E is intended to express that a U-expression is non-empty or inhabited. The distinguished propositional connectives work exactly as expected.K contains combinators (like W , B, B1), which are general function types whose functionalbehavior is independent of any underlying domain in the sense that their functional behav-ior does not depend on the specification of a domain and a co-domain in advance. Thisstandpoint treats functions as rules for computation rather than the Dirichlet conception offunctions as single-valued binary relations between antecedently given sets.As shown in thenext section the effect of this is that K provides a uniform method for abstraction over anyformal theory of interest. Indeed, it may be surprising that Fitch’s combinators are sufficientfor establishing combinatorial completeness since in the standard presentation of the purecombinatory calculus W and B are defined in terms of the combinatorially complete set ofcombinators {S, K}.

K is an illative combinatory logic since it mixes combinators with logical notations,though it may be remarked that K has no explicit sign for negation nor for implication. Thelack of negation disables K from having means for representing universal quantification.Note that the equations for the standard combinators, such as Wfa = faa, are not theoremsof K . However, one may define a relation = which allows K to define term models ofcombinatory logic in the sense of Church–Rosser.

In terms of its functional behavior W contracts multiple instances of terms (it may alsobe characterized as duplicating the same), while B is the general form of compositionality

17 Fitch presents basic logic using a number of notations original with Peano. I’ve modified Fitch’s notation to promote clarity,

where I thought such was needed.

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(B1 behaves similarly).18 Since the main task of this paper is to show that K has unlimitedmonadic and relational abstraction it seems reasonable to present K with the standardcombinators S and K , as Fitch 1963 himself does for a system related to basic logic, ratherthan via Fitch’s combinators in the definition above. However, there is a disadvantage indoing this. Fitch 1936 concisely shows the role of W in the derivation of Russell’s paradox,indeed, W is the means for defining the diagonal of a function. In this paper, Fitch callsa formula ψ(x) the contraction of a formula ϕ(x, x) if for all x, ϕ(x, x) is equivalent toψ(x). If ψ is the contraction of ∼ x(x) it follows that for all x, ∼ x(x) is equivalent to ψ(x).Assuming that formulae themselves are within the range of quantification it follows that∼ ψ(ψ) is equivalent to ψ(ψ) (Fitch 1936, pp. 92–93). If the W combinator is available inthe substructural part of the logic, then for any ϕ(x, x) one may always find some ψ whichis its contraction, namely, choose ψ to be Wϕ.19 Fitch (1936) presents a system withoutan analogue to the W combinator, has negation and material implication, has a notation forclasses while avoiding the class antinomies, and has an unrestricted form of the principle ofexcluded middle. The strategy is to keep the logic classical (and the scope of quantificationunrestricted) at the expense of disabling certain kinds of definitions, even definitions whichare combinatorially very trivial (like in Russell’s paradox). Unfortunately, Fitch notes thatdeveloping mathematics in this system appears to be impossible so its role in foundationaliststudies is minimal. In particular, the lack of W apparently prevents the system from defininga fixed-point combinator so the recursion theorem will not be provable.

In nearly every presentation of K over four decades Fitch highlights the role of W in itsdefining clauses rather than taking the more expedient route of simply defining abstractionin terms of the combinators {S, K}. The cost is that extending K to a system with negationrequires some kind of restriction on the principle of excluded middle. Hence, to accommo-date the W combinator certain modifications to the background logic must be made on painof paradox. Prospects for developing a fragment of analysis in K ′ are very good, as Fitch1949, 1950, 1951 demonstrated in detail. By boldly highlighting the role of W in basic logicFitch confronts the paradoxes as natural linguistic and logical phenomena rather than aspathologies which must be excised from the system. This attitude, in a much more radicalform, is also found among proponents of dialetheism. Another advantage to keeping withFitch’s original presentation is that some results only depend on the functional behaviorof W , B and B1, and not on the membership/exemplification combinators ε and ε, that is,these results do not depend on the full power of abstraction but only a fragment.20 If thecombinators S and K were used to effect abstraction then proofs of this type would appearto require much stronger combinatorial resources than they actually do.

K is a finitary theory in the sense that it is always possible to determine in a finite numberof applications of its defining clauses whether a particular U-expression is a thesis if it isindeed so. We use the familiar notation �K ϕ to indicate that ϕ ∈ K (ϕ is a provable thesisof K). It is possible to present K proof-theoretically where the rule for ‘=’ serves as thesole axiom scheme and the remaining rules are the rules of procedure.

Implicit in the schemas defining K is an intensional equivalence relation (here denotedby ↔) factoring U-expressions according to whether they have identical truth-conditionsrelative to K . The intended relation is easily defined as the co-provability (in K) of U-expressions, for example, W&σ ↔ σ obviously obtains in the metatheory. A moment’sreflection is enough to see that ‘↔’ is an equivalence relation, indeed, from the set-theoretic

18 For example, one finds these combinators characterized in this way in substructural logics.19 It is assumed that the object theory is presented as a combinatorial calculus, so that ϕ(x, x) is written as ((ϕx)x), that is, ϕ is a

Curried relation.20 A proof of this type will appear in a forthcoming work.

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point of view its equivalence classes are elements of the associated Lindenbaum algebragenerated by K’s reduction relation. However, it is not possible to extend ‘↔’to a congruencewhich demonstrates the irreducibly intensional character of the logic.21 Fitch adopts asimilar notation in ‘A Simplification of Basic Logic’ and true instances of ϕ ↔ ψ expresswhat amounts to a ‘truth-by-definition’, a use also exploited by Fitch in his paper ‘Self-Referential Relations’. In the following section principles of abstraction will be expressedrelative to the just introduced notation.

Certain subsets of U may be intuitively represented in K , in the sense of the followingnatural definition:

Definition 2.3 (Representation) A subset S of U-expressions is represented by a term s inK if for all u ∈ U

u ∈ S if and only if �K uεs.

Fitch establishes the following theorems in ‘Representations of Calculi’ (published in1944), where the second is essentially a corollary to the first:

Theorem 2.4 (The Turing Completeness of K) The graphs of all partial recursivefunctions are K-representable and hence also all recursively enumerable sets of U-expressions are K-representable (under an arithmetical encoding of U-expressions intocombinatory numerals).

Theorem 2.5 A system is a calculus (finitary system) if and only if it is K-representable(as a combinatory system), that is, all finitary inductive relations are K-representable andconversely.

By Theorem 2.5, K is a universal calculus which can represent all finitary systems, includingitself. Hence, K can represent Church’s lambda-calculus and first-order logic, the unde-cidability of which immediately implies the undecidability of K . Incidentally K cannotrepresent its own set-complement (with respect to U) so its complement cannot form acalculus and from the finitary standpoint it is inaccessible. The set-complement of K canbe represented in K ′ and for this reason K ′ is not finitary. This may seem obvious since therule for A is an ω-style rule but this disguises a subtle fact. Introducing bound variablesto give a finitary version of K ′ runs into the problem that its variables have a syntacticallycomplex inner structure as U-expressions and the usual rule for universal introduction maynot be truth-preserving since some properties involving the syntactic structure of a variableof basic logic are not general. A finitary metalanguage for K ′ which avoids this problemwas investigated by Myhill 1952 but not all of Fitch’s results for K ′ are provable within it(a fact which Myhill acknowledges).

3. A notation for abstraction in KA general description of the logical function of principles of abstraction is that they

associate a term with some kind of higher-order entity (properties, logical classes, concepts),which brings reasoning about the latter under general reasoning involving the term structure

21 Cantini proves this fact for a system quite similar to basic logic by applying Gordeev’s paradox (Cantini 1996, p. 74). A

modification of this proof can be used to derive a similar result for basic logic, indeed, from the point of view of basic logic

there are many non-identical empty classes.

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of the language.22 Apparently the latter kind of reasoning (relying as it does on facts ofgeneral syntax) is perfectly safe: we know, of course, that under mild assumptions thatthis is not the case. The reasoning associated with the paradoxes is not applicable to basiclogic because abstracts (the products of abstraction) are treated as processes for classifyingU-expressions and not all classifications yield 2-partitions of the domain. In this respectFitch is a so-called gap theorist when it comes to solving the traditional paradoxes.

When unrestricted principles of abstraction are admitted without ad hoc restrictionsthen abstracts may fall under themselves, indeed, there are examples of non-trivialself-application in basic logic which shows that application very naturally models the self-exemplification relation of property theory. This is a form of genuine self-reference in thesense that self-application does not depend on an arithmetical coding scheme to simulate atype collapse, indeed, on this model U-expressions are best understood as universals whosemultiplicity does not affect their unity as a single thing. The outstanding feature of K is thatit has unlimited principles of abstraction and correspondingly unlimited comprehensionprinciples for logical classes. On the present point of view logical classes bear a one-manyrelation to the things that fall under them rather than the one-one relation postulated byFrege’s Law V . Logical classes, unlike mathematical classes, are best understood as plural-ities which bear a distributed form of reference to the items that fall under them. If logicalclasses are ontologically dependent on concepts, which in turn presumably depend on somekind of finite specification, then one pleasing philosophical consequence of K’s unlimitedabstraction principles is that basic logic correctly represents these dependencies as formalrelationships. K and its various extensions are revealed to be ideal frameworks for inves-tigating the complex interplay between abstraction, truth, propositions and representationvia concepts.

A principle of abstraction takes the following form in basic logic (recall the ‘↔’ con-vention of the last section). Let ϕ be any context (possibly containing parameters �p) whichcontains at least one occurrence of x and let ε be the primitive binary relation of exemplifi-cation or predication.23 A principle of abstraction yields a term, denoted in the metatheoryby x.[ϕ(x, �p)], which satisfies the condition, for all a ∈ U ,

aεx.[ϕ(x, �p)] ←→ ϕ(a, �p). (2)

The metalinguistic notation ‘ϕ(a, �p)’ denotes the result of simultaneously substituting a forx in ϕ (usually parameters will be left implicit). Abstraction in the above form is intended togive the system means for expressing higher-order theses while still maintaining a certaindegree of tractability resulting in a pleasing middle ground between first- and second-orderlogic. Myhill in his 1984 paper ‘Paradoxes’ claims that any instance of (2) determines anextra-linguistic intensional entity, namely a property (and not an extensional entity like amathematical class or set) denoted by terms of the form x.[ϕ]. Somewhat polemically he callsFitch-style abstraction ‘Frege’s principle’ (p. 130). This is a rhetorical excess on Myhill’spart since Frege’s theory of abstraction is connected to extensionality, that is, concepts forFrege have an ontologically intimate relationship with extensions while Fitch’s abstractsare highly intensional in nature.

22 For Fitch the term structure of abstracts is a reliable guide to the metaphysical structure of attributes (as assigned to objects

by propositions) so the structure of terms mirrors the ontological structure of higher-order entities. Fitch did not mark this

distinction explicitly – he seemed to have thought abstracts (which are linguistic items) just are attributes, rather than denoting

them.23 ‘ϕ’ will be a U -expression which, in a particular application, may be a combinatory analog to a sentence in some formal

language. Parameters may be drawn from U but in principle they may be assumed to be members of some set (possibly with

ur-elemente). The clauses of basic logic will treat parameters as syntactic simples when they are not themselves U -expressions.

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The particular way in which Fitch treats abstraction is very far removed from the contem-porary approach where abstraction operators are primitive terms which are to satisfy someantecedently given set of axioms (like comprehension, extensionality, or Hume’s principle).This methodology artificially separates abstraction principles from the rest of the logic, andconsequently the contemporary approach must make a special plea for the logical statusof abstraction principles, or some privileged subset of them.24 In contrast, if K is alreadyaccepted as capturing some species of logical truth then Fitch’s derived abstraction prin-ciples (as given in Theorem 3.1) will also, presumably, be counted as logical truths of thesame species (it was suggested in the last section that these truths were analytic in the senseof being true-by-definition).

An important desideratum of workers in the combinatory tradition is to show that boundvariables are merely apparent and may be eliminated with no variation in the expressivepower of the language.25 The abstracts introduced in this section take the form x.[ϕ] whichappears to introduce bound variables in violation of this desideratum. However, this ismerely apparent. Abstracts may be eliminated in favor of certain sequences of combinators(and ∨ and &) which effects a substitution operation (a somewhat similar procedure isfound in Russell’s first ‘no class’ theory, his substitutional theory as discussed extensivelyin Landini 1998).

The following theorems are often quite tedious and the proof of abstraction would berelatively mild if Fitch presented basic logic with the standard combinators S and K . Asdiscussed in the last section, the combinator W of basic logic is involved with the tradi-tional antinomies of class theory and its inclusion in basic logic highlights the fact thatFitch is willing to modify the background logic in order to accommodate unlimited abstrac-tion as a logical principle. Another advantage to Fitch’s original definition of basic logicover one using an apparently simpler combinatorial basis is that the combinators ε andε make the expression of membership (or perhaps more properly called exemplification)a natural part of basic logic. Membership amounts to true substitution and the expres-sion of this fact is nearly immediate in basic logic thanks to ε. ε plays a similar role forbinary relations. The alternative of using S and K for defining abstraction would need aseparate axiom to guarantee the intended meaning of membership and this is certainlyundesirable, at least if membership is taken to be a logical notion which should requireno special pleading as to this status (this agrees with Bealer 1982 where he claimed thatexemplification is a basic part of logic but without the need to challenge the role of settheory in the study of intensional logic). The interpretation of abstracts as notations forrepresenting properties and attributes (which Fitch repeatedly offers) becomes far moreevident than it would otherwise be if abstraction is more than an artifact of the underly-ing combinatory algebra. A certain amount of tedium and inconvenience is therefore wellworth the philosophical advantages if fidelity to Fitch’s original presentation of basic logicis maintained.

Fitch 1942 gives the following metatheoretic conventions, which are adopted in thefollowing. The notation (. . . a . . .) will be used for an arbitrary U-expression in which aoccurs at least once. The notation (. . . b . . .) when used in the same context as the previousis to be understood as the result of simultaneously substituting b for each occurrence of ain (. . . a . . .). Fitch also employs a notation for an infinite stock of (free) variables w, x, y

24 See for example the papers in Hale and Wright’s co-edited volume The Reason’s Proper Study: Essays towards a Neo-Fregean

Philosophy of Mathematics which outline the neo-logicist position in detail and attempts to make the case for the logicality of

Hume’s principle, and the non-logicality of others apparently in the same company.25 See Schönfinkel 1924, republished in the van Heijenoort volume, for the original presentation of the combinators and its

motivation in developing the minimum basis necessary for presenting a logic; Quine’s commentary is also very useful.

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and z, possibly with numerical subscripts or accented.26 These variables are U-expressionswhich are not parts of each other (this may always be done) and they are distinct from theU-expressions used in defining K (W , B, B1 and so on). Acceptable substitution-instancesof these variables are elements of U . The intended meaning of the notation (. . . x . . .) shouldbe evident.

Fitch’s conventions are liable to produce some misunderstandings. The problems mostlyhave to do with some subtleties involving substitution, which Fitch’s notation sometimesobscures.A very perspicuous notation for substitution which avoids these problems is easilydiscovered. If ϕ ∈ U , then the notation ϕ{x �→ a} is to refer to the result of simultaneouslysubstituting a for x inϕ (x need not occur inϕ for the notation to be meaningful), and similarlyfor ϕ{a �→ b}.27 Finally, ϕ{x �→ a, y �→ b} is the result of simultaneously substituting a forx and b for y in ϕ. Note that ϕ{x �→ a, y �→ b} and ϕ{x �→ a}{y �→ b} need not result in thesame expression.28

Fitch 1942 proves the following theorem, which shows that Fitch’s basic operations forma combinatorially complete set:29

Theorem 3.1 For every U-expression ϕ there are U-expressions denoted in the metatheoryby x.[ϕ] and xy.[ϕ] satisfying, for every a, b ∈ U ,

x.[ϕ]a ←→ ϕ{x �→ a} (Monadic abstraction)

xy.[ϕ]ab ←→ ϕ{x �→ a, y �→ b} (Relational abstraction)

which by the rules for ε and ε are equivalent to

aεx.[ϕ] ←→ ϕ{x �→ a} (3)

εaxy.[ϕ]b ←→ ϕ{x �→ a, y �→ b}. (4)

Both x.[ϕ] and xy.[ϕ] will be called by the common name ‘abstracts’. ‘ε’ facilitates acanonical way to present binary relations ‘R’, namely, in the form aRb (which is representedin basic logic by ε aRb). It is important to note that xy.[ϕ]ab is not necessarily the samein meaning as xy.[ϕ](ab). The former, but not the latter, has the truth condition displayedin Theorem 3.1, while the latter may have no determinate truth condition at all. Finally,↔ has the meaning conveyed in the last section, namely, as signifying equivalence oftruth-conditions relative to K .

The notation for abstracts which Fitch ultimately derives do not require the use of boundvariables in spite of the apparent similarity between the Russell–Fitch notation for abstractsand the more familiar lambda-abstracts of Church. Despite appearances any occurrences ofx in ϕ does not occur bound in either x.[ϕ] or x.[ϕ] simply because K lacks bound variables.This approach to variables (familiar from combinatory logic) liberates Fitch’s theory fromthe tiresome need of distinguishing between free and bound occurrences of a variable in agiven formula.

Fitch begins the proof of Theorem 3.1 by establishing abbreviations for abstracts x.[ϕ]and x.[ϕ] in which x occurs exactly once in ϕ. It is evident that ϕ must be of one of thefollowing forms, where x does not occur in f and g and has precisely one occurrence in hx:x, fx, hxf , f (ghx), f (hxg). The definitions of x.[ϕ] and x.[ϕ] depend on which form ϕ takes:

26 These variables will, despite certain appearances, never occur as bound in a U -expression.27 A similar convention is adopted by Humberstone 2000.28 The notation ϕ{x �→ a}{y �→ b} is sequential substitution: first substitute a for x in ϕ, then substitute b for y in ϕ{x �→ a}. If a

contains y then ϕ{x �→ a}{y �→ b} may be distinct from ϕ{x �→ a, y �→ b}.29 Fitch’s notation for abstracts should be familiar to readers of Principia.

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Definition 3.2 Suppose x does not occur in f , g and has precisely one occurrence in hx.Then x.[ϕ] and x.[ϕ] are defined as follows:

x.[x] =df (W&) x.[x] =df B(W&)

x.[fx] =df f x.[ fx] =df f

x.[hx f ] =df x.[εfhx] x.[hx f ] =df x.[εf hx]x.[ f (ghx)] =df x.[Bfghx] x.[ f (ghx)] =df x.[B1fghx]x.[ f (hxg)] =df x.[Bf hxg] x.[ f (hxg)] =df x.[B1 fhxg]

Fitch notes that if y occurs n-times in ϕ and x occurs exactly once then y still occurs n timesin both x.[ϕ] and x.[ϕ].

It is perhaps not clear how these abbreviatory devices are intended to function, sincemost of the introduced abstracts are defined in terms of other abstracts, and so on perhapsindefinitely. How is it assured that the definitions are not circular or otherwise infinitary? Itis possible to extract an algorithm from the conventions given in Definition 3.2 which takesany U-expression ϕ which contains exactly one occurrence of x and outputs a term of theform ax, where x does not occur in a. The algorithm shuffles the lone occurrence of x in ϕ

to the far-right, encoding each step involved in this movement by a combinator (which willoccur in a). a will be the abstract associated with Fitch’s abbreviations in Definition 3.2.The algorithm which matches the left-hand column of Definition 3.2 is simple and alwayshalts, as can be verified by referring to Figure 1.

The abstract a outputted by the algorithm encodes a procedure for substituting a terminto the position previously occupied by x in ϕ. Two examples should help clarify this.

Example 1 x.[x] =df W&. Then

x.[x]u ←→ (W&)u

←→ (&u)u Rule for W

←→ u&u ∈ K Rule for &

←→ u logic

←→ x{x �→ u} substitution

Example 2 Let ϕ be (xσ)σ , where x �= σ . Then applying the algorithm will giveB(εσ )(B(εσ )), which we take as the abbreviation for x.[(xσ)σ ]. Then

x.[(xσ)σ ]u ←→ B(εσ )(B(εσ ))u

←→ (εσ )(B(εσ )u) Rule for B

←→ B(εσ )uσ Rule for ε

←→ εσ (uσ) Rule for B

←→ (uσ)σ ∈ K Rule for ε

←→ ((xσ)σ){x �→ u} substitution

These examples demonstrate that the combinators are used to encode an effective descriptionof a procedure for the substitution of terms into a fixed context.30

30 This fact is used in combinatory logic to model Church’s lambda-calculus.

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Figure 1. Algorithm for Abstraction in basic logic.

The next theorem shows that monadic abstraction over ϕ with respect to x holds when xoccurs exactly once in ϕ:

Theorem 3.3 (Fitch 1942) Let x occur exactly once in ϕ. Then for all u ∈ U ,

x.[ϕ]u ←→ ϕ{x �→ u}.Proof The proof is by induction on the number n of symbols to the right of the occurrenceof x in ϕ (counting parentheses as symbols).

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n = 0: Then ϕ is x itself, so x.[ϕ] is (W&) (see Example 1).n = 1: Then the symbol to the right of x is the right-most occurrence of ‘)’ in ϕ, so ϕ

must be of the form (ax) where x does not occur in a. Then x.[ϕ] will be a and

x.[ϕ]u ←→ au

←→ ax{x �→ u}.For the inductive step there are three cases. Since there are at least two symbols to the rightof x in ϕ, ϕ must be of the form (bc). The possibilities for ϕ are exhausted by terms of theform (hxf ), f (ghx), f (hxg):

Case 1: ϕ = (hxf ). Then x.[ϕ] is x.[εfhx]. Note that the occurrence of x in εfhx hasfewer symbols to its right than its occurrence in hxf so the result follows by the inductivehypothesis and the rule for ε:

x.[ϕ]u ←→ x.[εf hx]u←→ εf (hx{x �→ u}) by the inductive hypothesis

←→ hx{x �→ u}f rule for ε

←→ (hx f ){x �→ u} as x does not occur in f

Case 2: ϕ = f (ghx). Then x.[ϕ] is x.[Bfghx]. By inspection, it is clear that the occurrenceof x in Bfghx has fewer symbols (in fact one less, again counting parentheses as symbols)to its right than its occurrence in (f (ghx)) so the result follows by the inductive hypothesisand the rule for B:

x.[ϕ]u ←→ x.[Bfghx]u←→ Bfghx{x �→ u} by the inductive hypothesis

←→ f (ghx{x �→ u}) rule for B and fact x only occurs in hx

←→ f (ghx){x �→ u} as x does not occur in f , g

Case 3: ϕ = f (hxg). Then x.[ϕ] is x.[Bfhxg] and this in turn (by Definition 3.2) isx.[εg(Bfhx)]. By inspection, it is clear that the occurrence of x in ((εg)(Bfhx)) has fewersymbols to its right than its occurrence in (f (hxg)), so the result follows by the inductivehypothesis and the rules for ε and B:

x.[ϕ]u ←→ x.[εg(Bf hx)]u←→ εg(Bf hx{x �→ u}) by the inductive hypothesis

←→ Bf hx{x �→ u}g rule for ε

←→ f (hx{x �→ u}g) rule for B

←→ f (hxg){x �→ u} since x does not occur in f , g �

The terms of the form x.[ϕ], where x occurs exactly once in ϕ, is an intermediate notationwhich will be used to model relational abstraction and monadic abstraction over contextswhich contain multiple occurrences of x. They have the following important property.

Theorem 3.4 (Fitch 1942) Let x occur exactly once in ϕ. Then for all u, v ∈ U ,

x.[ϕ]uv ←→ ϕ{x �→ u}v.

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Proof The proof is exactly analogous to the proof of Theorem 3.3. �

So far Fitch has established monadic abstraction for those contexts ϕ which contain asingle occurrence of x. To extend this result to contexts containing multiple occurrences ofx Fitch gives an additional abbreviation for abstraction over certain contexts:

Definition 3.5 Suppose x occurs at least once in ϕ = (. . . x . . . x . . .) and also supposethat y occurs exactly once in (. . . x . . . y . . .); moreover, assume that every occurrence of xin the latter is to the left of the occurrence of y. Then

x.[(. . . x . . . x . . .)] =df Wyx.[(. . . x . . . y . . .)]where expressions of the form yx.[ψ] in turn abbreviate y.[x.[ψ]].

The role of W is to contract multiple instances of x (or duplicate them), and this useful rolewill be exploited in the following (see, for example, the proof of Theorem 3.7). Unfortunatelya key step in Fitch’s proof of unlimited abstraction in K is flawed. Fitch wants to prove thatif x occurs n + 1 times in ϕ then

x.[ϕ]u ←→ ϕ{x �→ u}.The proof is by induction on n, and a key step is to claim that, where (…x …x …) is thecontext representing ϕ,

x.[(. . . x . . . u . . .)]u ←→ (. . . u . . . u . . .)

holds by the inductive hypothesis. However, if x = u, then this fails since the inductive stepcannot be taken (that is, the above claim will become precisely what Fitch needs to show).The problem is that Fitch requires that free variables be U-expressions and not merely rangeover them in a so far unspecified metatheory for K .

Indeed, it is not at all clear at this stage how Fitch intends to provide a notation forx.[(xx)] or more generally for contexts in which x occurs exactly twice. In fact the readermay verify that Wx.[x] is a good proxy for representing x.[(xx)] in the sense that the desiredapplicative behavior of the latter is produced by the former. However, this trick does notgeneralize. One might attempt to define, as in Definition 3.2, a series of abbreviations forx.[(xx)], x.[( fx)x], x.[x( fx)] and so on for every possible way in which x may occur exactlytwice in some fixed context. This is necessary since, for example, x.[(fx)x] is not coveredby Definition 3.5.

However, an immediate problem arises. Consider (fx)x where x does not occur in f .To produce the corresponding abstraction with respect to x (where we need to produce aterm of the form ax, where x does not occur in a) we must apply ε to yield εx(fx). Fromthis the only other of Fitch’s combinators that may be applied non-trivially31 is B, whichyields B(εx)fx, and none of Fitch’s combinators may be applied to the latter. Things aremuch more promising in pure combinatory logic since one would simply apply S to εx(fx)yielding Sεfx. In this case, x.[(fx)x] would be ‘abbreviated’ by Sεf . However, since Fitchdoes not have S this move is not available.

Fortunately, there is a solution which avoids the problem of extending Definition 3.2 tocontexts containing more than one occurrence of x (which apparently requires additionalcombinators), but the solution requires the temporary use of genuine variables which are

31 We could of course apply ε again, and so on ad nauseum.

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distinct from any U-expressions. These variables may occur bound in certain expressions.The new stock of variables is only introduced in order to define lambda-abstracts and Fitchessentially shows that for any abstract definable in the new vocabulary it is possible todefine a notation for an abstract in K which has exactly the same application behavior overU . This shows that abstraction is available in the object language without a detour througha metalanguage containing bound variables. This restores the combinatory tradition’s viewthat variables are nothing more than notational conveniences: in the present exercise theyare ladders that can be kicked away.

We will use the notation (. . . x . . . x′ . . .) for contexts where substituting U-expressionsfor all the variables x, x′ . . . in (. . . x . . . x′ . . .) always yields a U-expression. The system Kλ

is defined as follows.

Definition 3.6 (Definition of Kλ) The language of Kλ properly extends the language of Kby adding infinitely many variables x, x′, x′′ . . . which are distinct from any U-expression (anddo not contain any U-expression as a part), and by adding primitive abstraction terms of theform λx.[ϕ], where ϕ ranges over the full language of Kλ. It is assumed that lambda-abstractsare not U-expressions, and both the newly introduced variables and lambda-abstracts donot contain any U-expressions (that is, they are distinct as symbols). The new variablesrange over U (in this respect Kλ is an interpreted language). The system Kλ includes all therules defining K along with the following schema for abstraction:

λx.[ϕ]v ∈ Kλ ⇐⇒ ϕ{x �→ v} ∈ Kλ.

Kλ adds primitive notations for abstracts and the corresponding abstraction principles to K ,and the variables are genuine (what Russell called umbral symbols or apparent variables).These variables are not eliminable within sentential contexts as in ordinary combinatorylogic. Kλ has abstraction as a primitive feature of its rule set, an obvious theft over thehonest toil of actually showing that K has terms modeling the intended behavior of lambda-abstracts.

What Fitch (1942) essentially shows is that for every lambda-abstract in the expandedlanguage there is a corresponding Fitch abstract (either a notation corresponding toDefinition 3.2 or Definition 3.5) definable in K which has the same application behaviorwhen the respective terms are applied to U-expressions:

Theorem 3.7 Let (. . . x . . . x . . .) be a context, where the last displayed occurrence of x isits right-most occurrence in the context (if it occurs more than once). Then, for all u ∈ U ,where x.[(. . . x . . . x . . .)] is defined as in Definition 3.5

λx.[(. . . x . . . x . . .)]u ∈ Kλ ⇔ x.[(. . . x . . . x . . .)]u ∈ K .

Proof The proof is by induction on the number of occurrences m > 0 of x in (. . . x . . . x . . .).The proof is structurally the same as Fitch 1942 proof of his theorem 4.8. The base casem = 1 follows from Theorem 3.3. For m = n + 1 consider

λx.[(. . . x . . . x . . .)]u ∈ Kλ ⇐⇒ (. . . u . . . u . . .) ∈ Kλ by Kλ-abstraction

⇐⇒ λx.[(. . . x . . . u . . .)]u ∈ Kλ by Kλ-abstraction as

x is not part of u

⇐⇒ x.[(. . . x . . . u . . .)]u ∈ K by the IH since u �= x

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⇐⇒ yx.[(. . . x . . . y . . .)]uu ∈ K by Theorem 3.4

⇐⇒ Wyx.[(. . . x . . . y . . .)]u ∈ K W -rule

⇐⇒ x.[(. . . x . . . x . . .)]u ∈ K by Definition 3.5 �

By Kλ-abstraction and Theorem 3.7, we almost have monadic abstraction:

ϕ{x �→ u} ∈ Kλ ⇐⇒ x.[ϕ{x �→ x}]u ∈ K . (5)

But note that if ϕ{x �→ u} ∈ Kλ (where ϕ{x �→ u} ∈ U) then this is only so because of rulesalready available in K . Indeed, it may be shown by a double induction on the length of proofsand the complexity of formulae that Kλ is conservative over K .32 Hence, if ϕ{x �→ u} ∈ Uthen

ϕ{x �→ u} ∈ Kλ ⇐⇒ ϕ{x �→ u} ∈ K . (6)

Hence from Theorem 3.7, (6) and Kλ-abstraction the following has been demonstrated:

(∀u ∈ U)[x.[ϕ]u ←→ ϕ{x �→ u}] (7)

which is the first part (monadic abstraction) of Theorem 3.1.To finish proving the second part of Theorem 3.1 (relational abstraction) Fitch introduces

yet another definition:

Definition 3.8 Suppose x occurs more than once in (. . . x . . .), where the latter context isa U-expression. Then

x.[(. . . x . . .)] =df x.[y.[u.[(. . . u . . .)y]x]].

Fitch wants to show that the new definition satisfies the condition given in Theorem 3.4except that now x can occur more than once:

x.[(. . . x . . .)]ab ←→ ((. . . a . . .)b). (8)

Again there is a problem with Fitch’s proof since he assumes that in Definition 3.8 x, y, uare all U-expressions. This is catastrophic since, for example, Fitch claims in his attemptedproof of Theorem 3.9 that

y.[u.[(. . . u . . .)y]a]b ←→ u.[(. . . u . . .)b]a (9)

follows from monadic abstraction. However this may fail if y occurs in a, which it may asy ∈ U . All that follows from monadic abstraction is the weaker claim

y.[u.[(. . . u . . .)y]a]b ←→ u.[(. . . u . . .){y �→ b}b]a{y �→ b}. (10)

The solution is to work in a language like Kλ with genuine bound variables and then showthat abstraction in the new theory yields abstracts which are co-extensive with abstractsdefinable in K (via Fitch’s various definitions). To simplify the exposition we take the un-Fitchean step of assuming that x, y, u are not U-expressions (and do not contain any memberof U nor are members of each other) and any expression of the form ϕ{x �→ v} is always a

32 This is not quite right, since we have presented neither K nor Kλ proof theoretically – hence the appeal to proofs in the

unspecified metatheory. However, the point should be clear enough.

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U-expression. These variables are being treated as a temporary convenience which may beeliminated in principle. The proof of the following is then an easy affair:

Theorem 3.9 (Fitch 1942)

x.[(. . . x . . .)]ab ←→ ((. . . a . . .)b).

Proof The proof is an induction on the number of occurrences n of x in (. . . x . . .). Forn = 1, the theorem follows by Theorem 3.4. For n = m + 1 we reason as follows:

x.[(. . . x . . .)]ab ←→ x.[y.[u.[(. . . u . . .)y]x]]ab by Definition 3.8

←→ y.[u.[(. . . u . . .)y]a]b by Theorem 3.4

←→ u.[(. . . u . . .)b]a by monadic abstraction

←→ (. . . a . . .)b by monadic abstraction

Note that the second step in the proof is valid because x occurs exactly once inx.[y.[u.[(. . . u . . .)]y]x]. �

Nearly all the ingredients have now been gathered for proving relational abstraction. Weonly need the following lemma, which Fitch neglects to prove:33

Lemma 3.10 Let x, y occur at least once in ϕ. Then for all a ∈ U

y.[ϕ]{x �→ a} = y.[ϕ{x �→ a}]. (11)

Proof The proof requires a fine-grained analysis of the structure of abstracts, involvingthe definitions in 3.2, 3.5 and 3.8. Rather than attempting that here note that an abstractof the form y.[ϕ] has the same number of occurrences of x as the original ϕ (this may beproved by induction on the number of occurrences of y in ϕ). The process of abstractingover ϕ with respect to y yields a term c which is a combination of combinators, x andother subexpressions of ϕ which lack y. This process moves y to the far right and removesduplications of y with the combinator W . The resulting abstract does not contain y and, asnoted before, encodes an algorithm for substituting a term for y in ϕ. Abstraction is notsensitive to the structure of x or to any subexpression of ϕ which lacks y, in the sense thatthe abstracts y.[ϕ] and y.[ϕ{x �→ a}] are isomorphic (where only the application structureis being preserved). By the same reasoning y.[ϕ] and y.[ϕ]{x �→ a} are isomorphic, hencealso y.[ϕ{x �→ a}] and y.[ϕ]{x �→ a}. However a moment’s reflection should be enough toconvince one that in fact y.[ϕ{x �→ a}] = y.[ϕ]{x �→ a}. �

Theorem 3.11 (Fitch 1942) For all a, b, ϕ ∈ U , where x and y are distinct variables (inthe sense discussed above) :

xy.[ϕ]ab ←→ ϕ{x �→ a, y �→ b}.

33 Another reason why variables like x, y must not be U -expressions is that the lemma may fail otherwise. To see this, suppose

x, y ∈ U where x and y do not occur as parts of each other (so, in particular, x �= y). Let ϕ = yx. Then y.[ϕ] = εx. So,

y.[ϕ]{x �→ y} = εy but y.[ϕ{x �→ y}] = y.[(yy)] = Wε, and Wε �= εy as W �= ε. Fitch did not seem to be aware of this problem.

Because of this we must either work in a system like Kλ and then show by an induction that an analogue to the lemma also

holds for K or treat x, y as U -expressions which function as free variables (and these variables do not occur as parts of each

other) and where we must stipulate that in the lemma a �= x, y. This is a restriction on the substitution operation (and implicitly

a restriction on abstraction).

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Proof

xy.[ϕ]ab ←→ y.[ϕ]{x �→ a}b by Theorem 3.9

←→ y.[ϕ{x �→ a}]b by Lemma 3.10

←→ ϕ{x �→ a}{y �→ b} by monadic abstraction

←→ ϕ{x �→ a, y �→ b} as y does not occur as part of a �

Without question Fitch’s proof of abstraction is not elegant and some of his reasoning isfaulty. However it should be apparent that K is a work of a certain degree of genius – Fitchhas only postulated what is needed to prove abstraction and his notation (ε in particular)guarantees that membership/exemplification is a part of logic (where by ‘logic’one means amaximally general and consistent system for representing all possible forms of reasoning).Rather than using the artifice of separating the logic of membership (sets and classes) fromthe rest of logic Fitch’s approach is integrative, offering quite remarkable formal power.

4. Comments on Fitch’s theory of abstractionThe proof of Theorem 3.1 can be extended to show that K ′ has monadic and relational

abstraction. In a monadic or relational abstract of Fitch’s logic bound variables are beingsimulated – they are entirely dispensable as in combinatory logic. Given the availability ofmonadic abstraction it is easy to introduce a notation for quantification modeling standardfirst-order logics. Monadic abstracts intuitively represent classes (of U-expressions) and Eand A assert, of particular classes, that they are non-empty or universal, respectively (thisis their semantic description in model theoretic semantics). Fitch introduces the followingabbreviations which recover the natural notation for existential quantification in K :

(∃x)(. . . x . . .) =df Ex.[. . . x . . .](∃x, y)(. . . x . . . y . . .) =df (∃x)(∃y)(. . . x . . . y . . .)

...

and analogously for the introduction of ‘∀’ for K ′.3-ary abstraction in K and K ′ is also available, by the following trick given by Fitch 1953,

p. 319. Define a 3-ary abstraction term as follows:

B1uz.[(∃x, y)[(. . . x . . . y . . . z . . .)&u = xy]]and then verify that the newly introduced term works as intended:

B1uz.[(∃x, y)[(. . . x . . . y . . . z . . .)&u = xy]]abc

←→ uz.[(∃x, y)[(. . . x . . . y . . . z . . .)&u = xy]](ab)c

←→ z.[(∃x, y)[(. . . x . . . y . . . z . . .)&ab = xy]]c←→ (∃x, y)[(. . . x . . . y . . . c . . .)&ab = xy]←→ (. . . a . . . b . . . c . . .)

The last step is justified since ab = xy if and only if a = x and b = y.Fitch did not believe that basic logic had resources for representing abstraction in types

higher than n = 3 (for example, he explicitly claims this in ‘A Simplification of Basic Logic’(p. 320)). However, it most evidently has this property. This may be shown explicitly by

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defining a general operator for effecting simultaneous substitution or by observing thatsimultaneous substitution may be defined recursively and K can represent all recursiverelations in the natural numbers.34 Hence, there are abstracts definable in K and K ′ of theform x1x2 . . . xn.[ϕ(x1, x2, . . . , xn)] which work in the expected way in the theory.35

Fitch’s theory of abstraction is completely type-free, for example, the abstract α =df

x.[x = x] obviously satisfies its own defining condition hence �K αεα. The algorithm givenabove for monadic abstraction may itself be encoded as a monadic abstract of basic logicand this abstract is self-exemplifying. The type-freeness of the theory makes it an adequateframework for developing ideas belonging to naïve class theory and a well-founded universeof sets may be defined by interpreting membership as the part–whole relation of general syn-tax (sets under this interpretation are well-founded since expressions cannot literally containthemselves as proper parts). Details of how this is done must be deferred to future work.

The abstracts of basic logic may be interpreted as rules for computing membership in alogical class, in particular, they are classifications of U- expressions into disjoint (but notnecessarily exhaustive) extensions and anti-extensions. Because of the type-freeness of thetheory these classifications can classify other classifications, including themselves. In gen-eral, only K ′ has resources for representing the so-called anti-extensions, and in general K ′is able to represent every inductive and co-inductive set in the natural numbers, as shown byLorenzen and Myhill (1959). Abstracts may also be interpreted as properties, concepts, andother notations for intensional objects which have a complex inner structure matching themetaphysical structure of whatever they purportedly represent. Fitch did not seem to recog-nize a metaphysical barrier between abstracts and what they represent, or between name andthing signified: an abstract literally is a property or other repeatable entity. This is the basis fora robust logical realism which has an obvious counterpart in medieval theories of universals.

K is a type-free higher order combinatory logic and relating its class notations to themathematical notion of class or set may be useful. Let ZK be the structure of all K-representable sets under inclusion. A description of this space is given as follows. Theempty set may be represented in K by x.[x = (xx)] and the universe U by x.[x = x]. Allfinite subsets of U are representable in K , for example, the subset {u0, u1, . . . , un} is K-represented by x0x1 . . . xn.[x0 = u0 ∨ x1 = u1 ∨ . . . ∨ xn = un] (all co-finite subsets of Uwill be K ′-representable). If T1, T2 ⊆ U are K-represented by τ1, τ2, respectively, then theirunion is K-represented by x.[xετ1 ∨ xετ2] and their intersection by x.[xετ1&xετ2]. Similardefinitions will handle finite union and intersection of already represented sets. Associa-tivity, commutativity, absorption, and distribution all work correctly when restricted to themembers of K-representable sets. 〈ZK , ∪, ∩〉 then forms a bounded distributive lattice (underinclusion) with top and bottom interpreted as the universal and empty sets, respectively.

With respect to the philosophy of mathematics Fitch’s investigations exploit the con-nection between abstraction as defined in illative combinatory logic and those classabstraction principles which populate the early attempts at logistic reconstructions of clas-sical mathematics.36 It may appear that this alleged connection makes a category mistakebut the relationship between them has an historical basis found in Russell’s early work onlogicism and in the work of Scott and Aczel. In a series of papers between Principles and

34 The proof of this fact is greatly facilitated by the availability of relational abstraction in K ; otherwise, the proof is nearly

unmanageable.35 The expression ‘x1 x2 . . . xn.[ϕ(x1, x2, . . . , xn)]′ is an abbreviation for x1.[x2.[. . . xn.[ϕ] . . .]].36 Neo-logicists have not, to my knowledge, seriously examined illative combinatory logic as a way to recover some of the Fregean

standpoint, indeed, Burgess (2005) does not seem to take the idea too seriously in his reconstruction and evaluation of their

program.

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Principia Russell worked on systems which were type-free in the sense that type distinc-tions were not ad hoc restrictions on expressibility but naturally occur as built-in featuresof the grammatical structure of abstracts. In these papers, as Klement 2003 showed, Rus-sell anticipated Church’s lambda-calculus (which can be modeled by combinatory logic andconversely). Indeed, Landini (1998) already took this investigation very far and he describedRussell’s type-free systems as an intensional form of logicism wherein logical classes aredefined relative to a substitution operator yielding principles of abstraction not dissimilarto Fitch’s in effect.37 Scott (1975) investigated the relationship between abstraction in basiclogic and general class theory (he lamented that the work of Fitch was not well known) anddemonstrated that Fitch’s combinatory systems were essentially term models in which aself-referential (and necessarily partial) theory of truth could be developed which containsa non-trivial fragment of the theory of logical classes.38 A few years later Aczel investigatedwhat he called Frege structures. Frege structures are built over a model of the lambda-calculus giving a system which satisfies naïve comprehension. Aczel used Frege structuresto give an innovative analysis of the sources of Russell’s paradox in class theory, a sourcewhich was found in Frege’s claim that all propositions can be factored into the true and thefalse (propositions) rather than finding fault with either comprehension or abstraction.

The historical scenes briefly surveyed above suggests that abstraction in combinatorylogic and its functional counterpart in the lambda-calculus have features congruent withthe use of abstraction principles in logicism. As such, it is possible to draw meaningfulcomparisons between basic logic and a highly intensional form of logicism. In basic logicabstraction is a basic part of logic (where ‘logic’ is understood in Fitch’s sense as a universalcalculus for the representation of all finitary theories, including itself, a theory which ispresumed to survey the limits of representation). Abstraction is a general logical operationfor the introduction of terms which intuitively represent properties and concepts in the sensethat syntactical relations definable in K are structurally analogous (have the same applicationbehavior) as property-exemplification and the falling-under relation. The unlimited powerof abstraction in K is a model of the total conceptual freedom of the mathematician inthe sense that no restriction on concepts is needed to tame the paradoxes. Since basiclogic is a universal framework in which all and only all formal systems (including itself)are definable the formal properties of the resulting theory capture precisely the essentialingredients belonging to formal representation (in any language for any subject matter ofinterest). If moreover the theory contains some quantity of mathematics then at least somemathematics is presupposed in formalization, indeed, we find that numbers are a basic partof our most general framework for the representation of thought linguistically since basiclogic contains as a subtheory a combinatorial version of arithmetic. No special plea onbehalf of numbers as logical objects is required – their necessity is a feature grounded in themost basic forms of representation. Numbers are understood as part of the formal ontologyand this level of ontological commitment does not make numbers so epistemically remotefrom human experience that reference to them becomes inscrutable.39

37 Curry’s early work on the combinators was intended to give a precise analysis of the role of substitution in Principia. According

to Landini the relevant analysis of substitution is found in Russell’s substitutional theory though some of the relevant work on

this topic was only available posthumously, as first reported in Grattan-Guinness 1974.38 Fitch’s work on systems which contain their own truth-predicate, nearly unacknowledged by contemporary philosophers,

preceded Kripke’s own work by over a decade though it must be admitted that Kripke’s work is elegant and clear while Fitch’s

formal work is unusually complex.39 The argument clearly has a transcendental flavor and basic logic under this interpretation is a novel mixing of Fregean and

Kantian themes. Note that this is not a version of the Maddy–Putnam–Quine indispensability arguments given in, for example,

Maddy 1992, where the practical use of numbers in the sciences gives them a utility which is sufficient to imply their (merely

contingent?) existence.

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True sentences of (combinatory) arithmetic would presumably be classified as syntheticrather than analytic in basic logic.40 They have this status in the sense that mathematics isa necessary feature of representation (part of the laws of thought in Frege’s nomenclature)and secures its objectivity from the fact that basic logic is a necessary precondition forour ability to work with concepts and combine them into a unified axiomatic form asa system for thinking about a particular subject domain. However, some of the Fregeanstandpoint is preserved in basic logic. Frege wanted above all else to secure arithmetic fromany connection with Kantian intuition and Fitch shows how mathematics can be objectivewithout assuming that its inter-subjective character depends on the possession of mindswith a specific cognitive structure-all that is assumed is that finite creatures must work withconcepts as they build coherent pictures of the universe.

Some additional features of Frege’s logicism can be accommodated in basic logic in away that maintains Frege’s contention that concepts and extensions are intimately related. Inbasic logic concepts have a dependency on definability in some formal language. Conceptsare denoted by abstracts (under this interpretation) and to each concept is naturally associatedan extension, namely, the set of U-expressions that it represents in K . The relationshipbetween concepts and their extensions is such that everything in the extension necessarilyhas the associated concept, assuming concepts are defined without parameters which refer tocontingent entities (note that the mapping from concepts to their associated extensions is, onthis model, many-one, which relaxes some of the intensional features of the system). UnlikeFrege’s theory basic logic handles properties in a completely type-free environment so thereare meaningful instances of self-exemplification. If membership (in a class) is understood astrue substitution then concepts in basic logic are akin to propositional functions. Rather thana metaphysical barrier obtaining between concepts and objects there is a semantic distinctionbetween the use and mention of abstracts which are distinguished by the functional role ofabstracts in the theory.41 As the number of abstracts (and hence properties) are countablethere is no worry that projecting the higher-order domain of properties into the class ofextensions will exhaust the latter. Any alleged concepts (arbitrary subsets of U) which areleft out of consideration would have to exceed definability by finitary resources and it isnot clear that these answer to Frege’s use of the term of art ‘concept’ as the denotations ofpredicate expressions.42 To the objection that concepts suffer a form of what Dummett calls‘indefinite extensibility’ note that the W -combinator effects diagonalization, which is builtright into basic logic, so the diagonalization operation is a kind of fixed-point in basic logicthat cannot be used to go outside the space of finitarily representable concepts.

Let me briefly compare Fitch’s approach to abstraction with that of contemporary logi-cism. An important contrast with neo-logicism is that basic logic does not treat abstractionaxiomatically. The axiomatic approach separates abstraction principles from the backgroundlogic – the status of these principles (as logical or analytical) must be given a special plea,indeed, if the background logic is varied the stock of acceptable abstraction principles maycorrespondingly wax and wane.

The neo-logicist starts with the Fregean claim that the second-order domain of concepts(now interpreted as arbitrary subsets of the domain of individuals) is disjoint from the

40 The version of logicism that basic logic instantiates is in this sense closer to Russell’s version than Frege’s.41 This could be explicitly modeled by using polymorphic typing as in Gilmore 2005.42 For Frege a concept is signified by incomplete predicate expressions and predicates are linguistic items. On the Fregean model

concepts are propositional functions but it is not clear to me that Frege intended all arbitrary maps from the domain of individuals

to the truth values to form the space of all concepts, though it is often said that the Russell paradox reveals this to be the case.

This is wrong. The Russell paradox (and the generalized form of Cantor’s paradox) have easily derived analogues in theories

with a substitution operator (and no corresponding theory of sets), as can be checked by examining their derivations.

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domain of individuals and the problem is to coordinate equivalencies about the formerwith objectual-identity statements between singular terms referring to individuals whichstand as first-order proxies to concepts.43 The neo-logicist understands by a principle ofabstraction a factoring of higher-order concepts under an equivalency between them and achoice of representatives in the domain of individuals for each equivalence class. Formally,an abstraction operator is a function FR : D2 → D1, where D1 is the domain (of individuals),D2 ⊆ P(D1) and R is an equivalence on D2, which satisfies for all X, Y ∈ D2

44

FR(X) = FR(Y) ⇐⇒ R(X, Y).

A principle of abstraction is defined as a choice for FR which satisfies AbR. It is wellknown that not all principles of abstraction are consistent, and nor are all instances of AbR

pairwise consistent with each other. On this view of things the abstracts are representableas singular terms of the form FR(X) and it is understood that the denotations of abstractsfall under the range of the first-order quantifiers. The mathematical characterization ofabstraction principles which are defined by some formula of second-order logic is of greatinterest since the strength of such principles depends on what kinds of parameters (if any)are allowed in definitional matrices. Note finally that logic is prior to abstraction on theneo-logicist view – the implication structure is not affected by which abstraction principlesare taken as mathematically significant.

Among all possible principles of abstraction only a very few will be classified as logicalby the neo-logicists. Hume’s principle, which relates equinumerosity between two conceptswith the identity of their number, is so privileged since by Frege’s Theorem the Peano axiomscan be inferred from relational second-order logic from Hume’s principle and suitabledefinitions of arithmetical notions in second-order logic. Frege’s inconsistent Law V wasused to derive Hume’s principle (which is consistent) and if the latter is a logical notionthen a substantial component of Frege’s philosophy of arithmetic is recovered as a plausibleview.

Hume’s principle is defined formally as follows. For any relation R(X, Y) betweenconcepts X, Y let ≈R be the first-order statement (where the background language hassecond-order free variables) which expresses that R is a 1–1 correspondence between X andY (this obviously yields an equivalence between concepts). Then Hume’s principle (HP) isthe following second-order sentence:

#X =c #Y ⇔ (∃R)(X ≈R Y).

According to Hale and Wright 2001, 2009 this biconditional expresses an implicit definitionof the cardinality operator (#), indeed, identities involving this operator are coordinatedwith statements already understood (the right-hand side of HP). Moreover, and this is mostimportant, ‘we can now exploit this prior ability [of understanding the right hand side ofHP] in such a way as to get to know of identities and distinctions among the referents ofthe [#]-terms – entities whose existence is assured by the truth of suitable such identitystatements’ (Hale and Wright 2009, p. 179). From the fact that every set S is equinumerouswith itself (as witnessed by the identity function restricted to S) it follows on this viewthat the truth of ‘#S =c #S’ guarantees that the singular term ‘#S’ is a genuinely referentialexpression.

To gain its status as part of logic a candidate principle of abstraction must satisfy an addi-tional mathematical constraint since there are many other candidate principles of abstraction

43 The distinction between concepts and individuals is not metaphysical but is made relative to a choice of individuals. What are

concepts in one framework may function as individuals in another.44 This definition is taken from Antonelli 2010 (p. 3).

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(like those yielding order and isomorphism types) which seem to do a great deal of math-ematical work and as they enjoy near universal application they might be viewed as beingin the same good company as Hume’s principle.45 This additional constraint on instancesof AbR to be counted as logical is the preservation of the identity/non-identity betweenabstracts under arbitrary permutations of the domain of individuals, which meets Tarski’scriterion for logicality.46 Hume’s principle does indeed satisfy Tarski’s criterion since thecardinality of a set is preserved under every permutation of the domain of individuals whileorder types do not have this property since a permutation of, say, the natural numbers intheir natural order may yield a new well-ordering with order type ω + 1. Note that theargument that order types are not logical objects under Tarski’s criterion must assume thatsome well-ordered sets are infinite since all well-orderings of a finite set share the sameunderlying order-type. It is well-known that HP is only sound with respect to infinite modelsso the previous consideration is no worry to the neo-logicist though of substantial worry tothose with more finitist sympathies.

Tarski’s criterion has disastrous consequences on Fitch’s understanding of abstraction.Fitch’s abstracts contain logically significant inner structure (as canonical names for con-cepts and properties) and arbitrary permutations of the domain will not respect this featureof the logic. This inner structure becomes accessible by applying Fitch’s principles ofabstraction and these principles will fail to yield materially true equivalencies under Tarski’scriterion. Abstracts are complex terms which encode an algorithm or other classificatoryprocess for determining membership or exemplification in a class or satisfaction by a prop-erty. For example, combinatory numerals (like Church’s numerals) treat numbers are indicesmarking stages of an unfolding iterative process, marking explicitly the role of numbersin iterated functional applications.47 In contrast to combinatory numerals the products ofHume’s principle, whatever they are, satisfy an essential ingredient of a cardinal assignmentfunction (relating it to equinumerosity) without giving any indication of their mathemati-cal structure (which could be transitive sets of transitive sets in a well-founded universe,Zermelo’s numbers, initial ordinals, alephs, or some other cardinal representation in settheory). Evidently the cardinal numbers yielded by HP have no logically or mathematicallysignificant inner structure.

The latest neo-logicist defense argues not so much for the logicality of Hume’s principleas such but rather etextitasizes its role as a sortal concept in arithmetic, in particular, itsrole in fixing the intended meaning of identity claims where both the identified relataare of the form ‘the number of C’s’, for C a concept. Hume’s principle is supposed toenjoy a privileged kind of analyticity, privileged because unlike analytic truths involvingdescriptive constants like the predicate ‘x is a bachelor’Hume’s principle fixes a crucial partof the semantics for the language of arithmetic by guaranteeing that number terms (of theform above) bear genuine reference to numbers understood as logical objects. Arithmeticaldiscourse is meaningful to the extent that its numerical terms refer to the latter, and thisreference is mediated by Hume’s principle as a kind of Fregean sense.

This is an admirable defense but there is the persistent worry that the practical utilityof a cardinal scale is to measure the sizes of some frequently encountered sets like R.However, this cardinal scale can vary with remarkable ease, consistent with ZFC, undermodest constraints. For example, the continuum function 2κ of ZFC restricted to regular

45 The problem with adding order types as logical objects is finding a principled way to avoid the Burali–Forti paradox while also

admitting transfinite order types as genuine entities.46 This criterion is supposed to ensure the independence of a logical principle from the nature of the individuals, insuring the

priority of logic over metaphysics.47 The cardinal notion of number can also be developed within basic logic by using the techniques of Myhill 1952.

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cardinals can behave in any way consistent with König’s Theorem (this result is due toEaston; an accessible proof is given in Jech 2002 (p. 232). Hume’s principle, on the otherhand, will yield a rigid cardinal scale but to actually calibrate the scale against, say, R, onemust bring in strong set-existence assumptions (or work in a particular kind of universe,like Gödel’s constructible universe L). Surely, it is no controversy to observe that these areextra-logical assumptions, but some such assumption must be made to ensure that Hume’sprinciple yields the correct stock of mathematical truths understood as necessary truths oflogic. If the concept of necessity is to be dispensed with it is not so clear why Hume’sprinciple is not classified as a synthetic principle and the revival of logicism merges intoKantianism.

The neo-logicist will likely counter that all Hume’s principle is intended to do is toguarantee that cardinals are logical objects, not fix a scale actually useful in mathematicalpractice. Such a defense depends on a radically anti-verificationist criterion for mean-ing, in particular, meaning involves nothing more than reference to a particular class ofobjects mediated, it is alleged, by Hume’s principle. The space of functions used to wit-ness equinumerosity claims is given no further description beyond their role in definingHP.48 If Hume’s principle involves the notion of analyticity and allied concepts like neces-sity it is very hard to see how to accommodate such a view with the apparent plasticityfound in the notion of ‘cardinality’, as witnessed by Easton’s theorem, without insistingthat the standard axiomatization of set theory is faulty. It is doubtful that the neo-logicistwants to encourage radical reform of mathematical practice, indeed, the desideratum ismerely to show that mathematics (as ordinarily understood) is at root logical. The attemptto show this by taking a detour through a general semantic theory involving genuine ref-erence to cardinals must also verify that there is a single intended (class) model (whichgoes way beyond the requirement of categoricity) for set theory which settles questionsof cardinality independently of strong set existence principles. If this is not done, then thefinal verdict on the neo-logicist endeavor is that their philosophical defense of HP con-fuses a necessary condition for developing a cardinal assignment function in set theorywith a sufficient condition – much more work must be done in order to get a sufficientlyfine-grained theory of cardinality as rich as its counterparts in higher set theory, the nat-ural target, one would think, of a suitably modern logistic reduction of mathematics tologic.49

Working in an interpreted second-order theory with extensions might seem to defeat theseclaims against neo-logicism but categoricity does not calibrate the cardinal scale in a way

48 Presumably the space of witnessing functions is ‘all of them’, where ‘all’ is not relativized.49 Antonelli (2010) presents a very clear explication of permutation invariance as a principled way to pick out a distinguished

class of abstraction principles as logical. He distinguishes several ways in which permutation invariance may be applied in

an analysis of abstraction. In his philosophical assessment of his results Antonelli describes abstraction principles as applying

inflationary pressure on the size of the first-order domain but also deflationary pressure (running the abstraction principle in

reverse, metaphorically speaking) on the size of the second-order domain. Finding the right balance (between hyper-fine grained

abstraction like Frege’s inconsistent Law V ) and coarser grained notions (like Hume’s principle) yields an instrumentalist, not

logical, conception of abstraction. The objects yielded by abstraction are simply representatives from some equivalence class –

the choice among representatives is simply an instrument for coordinating between the size of the first- and second-order

domains. We are then free, according to Antonelli, to effect whatever instrumental choices have the desired mathematical

outcomes – there are no worries about things like the ‘Bad Company’ objection in this form of instrumentalism. The so-called

abstracts play no privileged ontological or epistemological role. Antonelli’s commentary liberates abstraction from service

to particular philosophical conceptions of it, including neo-Fregean philosophy of mathematics, a separation perhaps long

overdue. If Antonelli is correct I cannot see neo-logicism continuing as a distinct philosophy of mathematics – it has not been

refuted, of course, but from the mathematical point of view its contribution is absorbed into a much larger mathematical project

classifying consistent abstraction principles with respect to proof-theoretic strength (in the spirit, I take it, of the analysis of

subsystems of second-order arithmetic). This would be a most worthy project.

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that isn’t question begging or involve mysterious appeals to Gödelian intuitions. How doesone argue in pure logic that the cardinality of the continuum just is ℵα , for some ordinalα > 0? The mathematician need hardly care that the English expression ‘The number ofreals’ is a genuinely referential singular term since outside small intuitionist circles this wasnever in doubt. The second-order approach does not have the mathematical tools necessaryfor refining the cardinal scale in contrast to the rich tools developed in contemporary settheory (for example, those used in the elegant apparatus for forcing generic extensions oftransitive models of ZFC). If the latter mathematical constructions are viewed as non-logical,then I think it is safe to say that neo-logicism as a coherent philosophy of mathematics willfalter as it is extended into the higher reaches of subsystems of arithmetic.50

Fitch’s approach to abstraction is happily free from these anxieties. Fitch’s abstracts aremixed in with other logical principles, indeed, it will no doubt be noticed that abstracts arenot part of the primitive notation of basic logic but are derived from the defining clauses ofK itself and have whatever logical status these original principles have. There is no conflictwith contemporary mathematical practices. Fitch’s basic logic is an admirable example ofthe principle of tolerance which was a guiding philosophical, social, and political tenet ofthe now nearly forgotten Unity of Science Movement, a movement which Fitch appearsto have harbored sympathies.51 As a metatheory basic logic provides unlimited abstractionover any theory of interest. The metatheoretical role of abstracts is to yield additionallogical information about the languages of science and mathematics, rooted in the commonframework provided by K .

There are two obvious objections to what has been said so far. First, it seems that Fitchmust plea for the logical status of the combinators before assessing the logical status ofabstraction principles. A way to respond to this is to first note that abstraction as understoodby Fitch is universal and the role of the combinators as universal operators is to facilitatethe definition of abstraction in the theory. Fitch might have simply assumed abstraction asa primitive operation at the beginning though as I have tried to argue above there is somephilosophical cost to bear in doing this. If basic logic is a nominalistic metatheory, thatis, a pre-logic prior to formalization, then the combinators simply re-arrange, duplicate,or eliminate terms as part of a description of syntax. These descriptions are a necessaryconcomitant to the practice of axiomatization. As meta-syntax the combinators are notcontroversial and basic logic may be understood as a nominalist metatheory for the formalsciences.

The second objection is that it is well known that abstraction must be isolated fromthe classical conception of implication because of the Curry paradox. Hence, abstractionmust be elevated, by a special plea, to the status of a logical principle and the theory ofimplication modified in the light of this declaration. One response to the objector is toadopt logical pluralism, in particular, the theory of implication is not developed in isolationbut in conjunction with the theory of inference. The two are connected together by thededuction theorem. If the substructural part of the logic is not classical (for example, insome deductive systems permutation of premises do not always preserve truth) the resultingtheory of implication will be modified accordingly. Since basic logic is a metatheory for all(finitary) formal theories (and recall that these theories, like basic logic, are simply classesof U-expressions) then the lack of an object language sign for implication is necessaryat the level of basic logic since otherwise its implication structure will be mistaken asuniversal. A genuine notion of implication for basic logic which does not have this problem

50 Consistent fragments of Frege’s theory have been calibrated against the analytic hierarchy, see especially the paper by Ferreira

and Wehmeier Ferreira and Wehmeier 2002 ‘On the Consistency of the �11-CA Fragment of Frege’s Grundgesetze’.

51 Fitch’s teacher was F. S. C. Northrop.

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was given in Myhill 1984. Another reason for the priority of abstraction over implicationis that abstraction is a vehicle for the expression of concepts which have semanticallysignificant internal structure as given by a finite specification. These concepts may wellinvolve paradoxes or have ungrounded truth-conditions for their satisfaction by objects. Ifa classical theory of implication is simply adopted in advance then it will, in conjunctionwith the abstraction principles, expose this inner structure to external logical analysis. Thisyields the Curry paradox in a classical theory of implication with abstraction. The priorityof abstraction over implication and the theory of inference is simply the priority of conceptsover the formal machinery, a view found in both Frege and Russell. From this point ofview mathematical progress always involves conceptual change, changes which may evencall for modification of the underlying theory of implication, an idea worked out by Aitkenand Barrett in their program of algorithmic logic in a way that conforms to many Fitcheaninsights about logic.

AcknowledgementsThe present study is the first part of an ongoing research program on Fitch’s basic logic. I thank Aldo Antonelli, JeffBarrett and Kai Wehmeier for their comments on an earlier version of this work, and for their strong encouragementand support for what is sometimes a frustrating endeavor. I must also thank Wayne Aitken for introducing me toFitch’s program (which he re-discovered in the course of developing algorithmic logic, in collaboration with JeffBarrett), and from whom I learned a great deal about abstraction as understood by a native mathematician. Finally,I thank the two anonymous referees for their very helpful suggestions.

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Abstraction in Fitch's Basic Logic

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