\documentstyle[11pt]{book}
\begin{document}
\pagestyle{myheadings}
\setcounter{page}{13}
\chapter*{{\LARGE Chapter 2\\ BOLZANO, TARSKI, AND THE LIMITS OF LOGIC}}
{\small Both Bolzano and Tarski were unsure what counts as logic. This means
that Bolzano's concept of logical analyticity, like Tarski's of logical
consequence, is not completely determinate. In a posthumously published
paper, Tarski offers a proposal for demarcating the logical objects in a
type-hierarchy, based on the idea of invariance under arbitrary permutations
of the domain of individuals. In this paper I comment on and extend Tarski's
proposal and show how to combine it with Bolzano's procedure of variation
among concepts, to obtain a definition of logical constants in a logically
significant fragment of a purported Bolzanian realm of meanings in
themselves. I conclude with doubts about the propriety and utility of such a
realm.
\begin{quote}
[S]ometimes it seems to me convenient to include mathematical terms, like
the $\in$-relation, in the class of logical ones, and sometimes I prefer to
restrict myself to terms of `elementary logic'. Is any problem involved
here?\\ Alfred Tarski, letter to Morton White (Tarski 1987, 29).
\end{quote} }
\section*{1. Historical prelude}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{1. Historical prelude}
Since Scholz first pointed out the similarity in 1936,\footnote{Scholz
1937.} it has become commonplace to note that Tarski's concept of logical
consequence was anticipated by Bolzano, to such an extent that one recent
survey speaks of ``the Bolzano-Tarski definition''.\footnote{Hodges 1983,
56. For a useful comparison cf. Berg 1962, 116--8, which also cautions
against drawing the parallels too closely.} I want to focus in this paper on
another concern which Bolzano and Tarski shared. Both were unsure about what
counts as logic. But whereas Bolzano brushed past the issue, Tarski devoted
more thought to it. Although I am interested in question for its own sake, I
think it is instructive to combine Tarski's approach to the question with a
Bolzanian approach, that is, one which, although Bolzano did not follow it,
is in his spirit. The question of the limits of logic has been revived in
recent years. As long as most philosophers took logic for all practical
purposes to be first-order predicate logic, the question hardly arose; the
logical constants could be simply listed. But a number of developments ---
modal logic, relevance logic, intensional logic, higher-order logic --- have
challenged the hegemony of first-order logic, and the old question has been
taken up again in a number of publications.\footnote{Cf. Hacking 1976, 1979,
Peacocke 1976, McCarthy 1981, Barwise and Cooper 1981,
Westersta\hspace{-.06in}$^{\circ}$hl 1985.}
When we mention Bolzano and Tarski together, an intriguing historical
question arises which I shall mention but not attempt to answer. While it is
clear enough that Tarski's major advances in metalogic were made
independently of Bolzano,\footnote{See Tarski 1956 ($^{2}1983$) (hereafter
referred to as $LSM$), 417 n.} the question may be raised whether aspects of
Bolzano's work did not influence Tarski indirectly. Such indirect influence
(which may escape the awareness of the one who is influenced) is of course
extremely difficult to establish, but consider briefly the circumstantial
evidence. Tarski's teachers in philosophy and logic in Warsaw included \L
ukasiewicz, Le\'{s}niewski, and Kotarbi\'{n}ski, all of whom had studied
under Twardowski in Lw\'{o}w before the First World War.\footnote{On the
origins of the Lw\'{o}w-Warsaw School see Skolimowski 1967 and Wole\'{n}ski
1988.} Twardowski, who had studied in Vienna under Brentano, had received
his doctorate under Robert Zimmermann, himself a former pupil of Bolzano,
and from this source, perhaps among others,\footnote{Another possible source
is Benno Kerry. See Haller 1982, vii.} he had picked up a fair amount of
knowledge about Bolzano's logic.\footnote{Bolzano's name occurs more often
than any other in Twardowski 1894.} Some of this was in turn passed on to
his students in Lw\'{o}w.\footnote{D\c{a}mbska 1978, 123 reports that
Twardowski regularly held a course in Lw\'{o}w on ``The Attempts to Reform
Traditional Logic'', which dealt with the work of Bolzano, Brentano, Boole,
and Schr\"{o}der.} Both \L ukasiewicz and Le\'{s}niewski were influenced by
Husserl's {\em Logical Investigations\/}, which also mentions Bolzano very
positively.\footnote{\L ukasiewicz 1913a devotes a section to comparison
with Bolzano. It is also possible that \L ukasiewicz, like others who went
to Austrian {\em Gymnasium\/}, first came into contact with Bolzano's ideas
in the thin disguise of Zimmermann's {\em Philosophische Propaedeutik\/}.
For intriguing speculations on the possible extent of this subterranean
influence, cf. Sebestik 1985, 102.} Bolzano's objectivistic,
anti-psychologistic approach to logic is very much in the spirit of the
Lw\'{o}w-Warsaw School. In particular, Bolzano's conception of truth as
absolute is one which was passed down from Twardowski to
Tarski,\footnote{See Wole\'{n}ski and Simons 1989.} and there is evidence
that Twardowski was directly influenced by Bolzano on the
issue.\footnote{Both Bolzano and Twardowski argue that the occurrence of
indexical expressions in sentences of ordinary language does not entail the
relativity of truth, since both deny that sentences are the primary
truth-bearers (for Bolzano it is propositions, for Twardowski judgements).
In discussing the point, Twardowski uses an example which is also found in
Bolzano's discussion. Bolzano ($WL$ \S 147) has ``Der Duft dieser Blume ist
angenehm''; Twardowski 1902, 416 (in German translation) has ``Diese Blume
riecht angenehm'', but the Polish original, ``Wo\'{n} tego kwiatu jest
przyjemna'', is closer to Bolzano.} Could other specific fragments of
Bolzanian lore have passed to Tarski and influenced or at least confirmed
his opinions? Answers on a postcard, please.
\section*{2. Bolzano on logical analyticity}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{2. Bolzano on logical analyticity}
In \S 148 of the {\em Wissenschaftslehre\/}, Bolzano presents an original definition of analyticity. A proposition\footnote{`Proposition' translates `Satz an sich'.} is analytic iff it contains at least one denoting concept\footnote{`Concept' translates `Vorstellung an sich', and `denoting' translates `gegenst\"{a}ndlich'. A denoting concept is one under which at least one object falls.} such that all propositions which arise by substituting another denoting concept for this one have the same truth-value
as the original.\footnote{The requirement that the concepts substituted all be denoting results from Bolzano's view that all propositions with a non-denoting subject concept are false. Without this restriction the theory is uninteresting. While it is worth considering relaxing this condition, the language B below in effect follows Bolzano's lead by requiring in classical fashion that every name denotes an individual.} If they are all true, we have an analytic truth, and if they are all false, we have an analytic falsehood.
For example, the proposition [A depraved man does not deserve happiness]\footnote{Square brackets are used to form names of propositions and concepts from sentences and phrases respectively. While such a notation has its problems, we cannot go into them here.} is analytic with respect to the concept [man], whereas Bolzano claims that neither [God is omniscient] nor [Any triangle has two right angles] is analytic, although he presumably thought both necessarily true. No proposition is analytic with respect to all constituent concepts,
since Bolzano regards all propositions as obtainable from one another via substitution of concepts. But such known logical principles as `An $A$ which is a $B$ is an $A$' and `Every object is either $B$ or non-$B$' belong to an important subclass of analytic propositions:\footnote{Bolzano {\em Wissenschaftslehre\/}, referred to throughout as `$WL$', \S 148. Reference is made by section number to enable any of the various editions and translations to be used.}
\begin{quote}
The difference between the last mentioned analytic propositions and [those mentioned above] lies in the following: In order to appraise the analytic nature of the [latter] propositions [\ldots], no other than logical knowledge is necessary, since the concepts which form the invariable part of these propositions all belong to logic. On the other hand, for the appraisal of the truth and falsity of propositions like those given [first] [\ldots] a wholly different kind of knowledge is required, since concepts alien to logic intrude.
\end{quote}
Now comes the decisive sentence:
\begin{quote}
This distinction, I admit, is rather unstable [{\em hat sein Schwankendes\/}], as the whole domain of concepts belonging to logic is not circumscribed to the extent that controversies could not arise at times.
\end{quote}
Bolzano calls the more restricted class `logically analytic'. There follow notes to the effect that analyticity cannot always be discerned by simple inspection of sentences, historical remarks on antecedents, notably Locke and Crusius, and a moderately famous critique of Kant.
The question of hidden analyticity opens a Pandora's Box of problems which have been tossed back and forth in the discussion following Quine's critique of analyticity, and I do not want to be drawn into it here. The historical comparisons Bolzano makes also call for more comment, and entail revising the oversimplified picture of the history of the concept of analyticity bequeathed to us by the logical empiricists.\footnote{For part of this revisionist picture, see Chapter 15 below.}
But this will not be discussed here.
The important points to come out of our discussion of Bolzano are two: (1) Bolzano distinguished between logical and non-logical concepts, but he did not say how such a distinction was to be made. (2) His discussion of analyticity employs the notions of proposition, concept, truth-value, and variation, which are fundamental parts of his metalogical apparatus. Ad (1) It is not clear from what Bolzano ways whether he thinks that no such distinction can be made, or whether it can be made, but he is not able to see how to do it.
At any rate, he is claiming that there is no present (1837) criterion for distinguishing the logical from the non-logical which will decide all disputes in advance. For `1837' read `1837 + 150' and I think the same still holds. Ad (2) Bolzano's is a World 3 approach, like e.g. those of Frege and Church: the constituents involved, propositions and concepts, are inhabitants of World 3; the properties of truth and falsity are (primarily) properties of propositions, and variation is (again primarily) variation among concepts.
The whole discussion abstracts from peculiarities of and differences among real languages or thoughts, and involves ontological commitment to the inhabitants of Bolzano's realm of entities {\em an sich\/}. The distinction between logical and non-logical concepts, if there is such a distinction, will therefore also be primarily a distinction among World 3 inhabitants. Whoever wants to do things in Bolzano's way must be aware that heavy ontological commitments and epistemological difficulties are involved.
\section*{3. Historical interlude I}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{3. Historical interlude I}
Tarski was very reticent about expressing his philosophical opinions in print, though this does not mean he had none. One such opinion about which we do know something is that he did not believe there to be a sharp distinction between logical and non-logical concepts, or logical and non-logical truths. In the monograph on truth, no distinction is made between logical and non-logical truth. In his 1936 paper on logical consequence, Tarski writes:\footnote{$LSM$ 418--420.}
\begin{quote}
[T]he division of all terms of the language discussed into logical and non-logical [\ldots] is certainly not quite arbitrary. [\ldots] On the other hand, no objective grounds are known to me which permit us to draw a sharp boundary between the two groups of terms. [\ldots] Perhaps it will be possible to find important objective arguments which will enable us to justify the traditional boundary between logical and extra-logical expressions. But I also consider it to be quite possible that investigations will bring no positive results in this direction [.]
\end{quote}
Like Bolzano, Tarski speaks of a fluctuation or instability [{\em Schwanken}] in ordinary usage; the somewhat unusual German word is the same in each case. Tarski took part in discussions in Harvard in 1940 in which he and Quine sided against Carnap on the issue. Both Quine and Morton White, who also attacked the analytic-synthetic distinction, acknowledge their debt to Tarski.\footnote{See the note by M. White to Tarski 1987.} It seems likely that Tarski had developed his sceptical position earlier, perhaps in opposition to Carnap. His position puts him at any rate close not only to Quine but also to other
opponents of a sharp dichotomy such as Neurath.\footnote{I have heard it said that Tarski, Carnap, and Neurath disputed the question in Vienna in the early 'thirties, but I have been unable to find a reference to such discussions.}
\section*{4. Tarski on logical notions}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{4. Tarski on logical notions}
Matter are made more interesting by the posthumous publication of a manu\-script by Tarski with the title ``What are Logical Notion?''\footnote{Tarski 1986. The same idea is found (independently) in Mautner 1946.} This, as its editor John Corcoran explains, began life in 1966 as a lecture at Bedford College, University of London. A typescript prepared from this lecture was read by Tarski in Buffalo, New York in 1973, and the manuscript was finally prepared for publication after Tarski's death.
The main point which Tarski wants to get across is that, just as mathematical theories such as Euclidean, affine, or projective geometry can be viewed, following Felix Klein, as concerned with invariants under structural automorphisms, so logic can be seen as taking this procedure to its conclusion, and understanding `automorphism' so weakly that the only `structure' left to preserve is cardinality: an automorphism is then just any {\em permutation\/} of the domain.\footnote{A permutation may be looked on as a bijective function of the domain onto itself,
or expressed as a one-one relation on the domain. Tarski used the latter conception in $LSM$, we shall use the former. Theoretically, they are equivalent, but the functional notation is perhaps easier to calculate with.} In this way Tarski stresses both the continuity between logic and mathematics and the greater generality of logic. More importantly, he has a way of deciding which ``notions'' are logical and which are not. Tarski's use of the term `notion' is here something of a stop-gap:
he intends it to refer to any object which can be found in a simple type hierarchy: individuals, classes of individuals, relations among individuals, classes of classes, classes of relations, relations among classes, and so on upwards. Corcoran has suggested replacing `notion' by `constant', so Tarski's question becomes `What are logical constants?' There are points both for and against this suggestion. It reminds us that we are looking for invariants, but the term `constant' is usually used today to refer to expressions,
and Tarski is most emphatically {\em not\/} talking about expressions. So, when considering Tarski's own proposal and the extension of it discussed below, I shall talk about logical {\em objects\/}, it being understood then than `object' subsumes more than just individuals. This way of speaking has a precedent in Tarski, who used it in earlier papers.\footnote{See $LSM$ 156 n., 189, 214, 385.} When talking about expressions or the senses of expressions (such as Bolzano's propositions and concepts) I shall however follow Corcoran's suggestion and speak of logical constants.
Tarski then lists a number of logical objects: among individuals there are none; among classes there are two: the empty class and the universal class; there are four logical binary relations, the empty and the universal, identity and diversity.\footnote{These were first considered all together by Schr\"{o}der.} Similarly, for ternary relations and those of higher arity there are finitely many logical relations, all of which are definable in terms of identity and the sentential connectives alone.\footnote{$LSM$, 387. Cf. also Westersta\hspace{-.06in}$^{\circ}$hl 1985, 395.}
As we ascend the type hierarchy the number of logical objects increases and they become more interesting: Tarski mentions cardinality properties of classes as invariant, as well as the relations among classes found in the elementary algebra of classes. Finally, Tarski considers the import of his view for the thesis of logicism: this turns out to be no open-and-shut affair. In view of the near-universality of set theory as a vehicle for representing mathematical theories,
Tarski reduces the question to whether the concept of membership is logical or not. If one takes type theory as the medium within which to formulate one's mathematics, the answer is `yes';\footnote{Tarski 1986, 152.} if one adopts what Tarski calls ``the first-order method'', \footnote{{\em Ibid.} Of course it is also possible to give a second-order set theory. The sets themselves remain individuals, so in this sense the method is still ``first-order''.}
by which he means some set theory like ZF expressed as a first-order theory, so that sets are individuals, then the answer is `no', since membership is not one of the four logical relations among individuals. On this characteristically ambivalent note, the paper ends.
\section*{5. Historical Interlude II}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{5. Historical Interlude II}
Tarski's paper is in many ways one of the most stimulating contributions to the philosophy of logic I have come across for years. At the same time it leaves many important questions open. But before I come to discuss some of them, I cannot resist another historical comparison. I recalled above that Tarski studied under Le\'{s}niewski, indeed the latter was Tarski's dissertation supervisor.\footnote{Tarski's dissertation under Le\'{s}niewski (the only one the latter supervised --- good quality control) is reprinted in $LSM$ 1--23.}
Without being able to put my finger exactly on what it is about it that makes this impression, the whole tenor of Tarski's paper reminds me of Le\'{s}niewski. No doubt they had their differences, and Tarski moved a long way away from Le\'{s}niewski's rather puritanical views about logic,\footnote{Cf. the remarks on their relationship in Wole\'{n}ski \& Simons 1989.} but the influence still permeates Tarski's thinking in various respects. For one thing,
Tarski appears to be generally more at home when thinking in terms of simple type theory than in terms of a set theory like ZF. Simple type theory, or rather a more general hierarchy containing something very like it, namely the hierarchy of semantic categories, formed the basis of Le\'{s}niewski's Ontology. It was only with some time-lag that Tarski learnt to see beyond the bounds set by this conception.\footnote{See Section 7 of the long truth paper, which Tarski added for the 1935 German translation.}
The logical objects Tarski mentions are among the first one defines when pursuing Le\'{s}niewski's Ontology, allowing for the differences between the language of Ontology and of type theory.\footnote{On how to bridge the gap between these different languages, see Chapter 11.} Another feature of Tarski's logic which he shares with Le\'{s}niewski is an uncompromising extensionalism. Unlike Quine and others, Tarski did not to my knowledge commit his reasons for this position to print. More points of affinity are mentioned below.
Tarski's thoughts on the matter of the 1986 paper go back a long time and evolved slowly: he refers to a paper he and Lindenbaum published in 1936, whose unassuming title (in English) is ``On the Limitations of the Means of Expression of Deductive Theories''.\footnote{$LSM$, 384--392.} Here the basic result which the later lecture mentions is stated without proof:\footnote{{\em Ibid.}, 385.}
\begin{quote}
Roughly speaking, [\ldots] every relation between objects (individuals, classes, relations, etc.) which can be expressed by purely logical means is invariant with respect to every one--one mapping of the `world' (i.e. the class of all individuals) onto itself and this invariance is logically provable.
\end{quote}
Tarski and Lindenbaum go on to present, in somewhat more detail, the logical theorems showing there are no logical individuals, precisely two logical classes, four logical binary relations, and so on, and then consider the import of their result for theories such as Euclidean and projective geometry. The idea of and notation for a permutation of the `world' go back to 1934, when Tarski put Padoa's informal test for independence of primitive concepts of a theory on a firm methodological basis.\footnote{{\em Ibid.}, 296--319.}
The reader of the 1936 paper and the later lecture will be struck by how little Tarski's interests changed in the interim. The logical language with which Tarski and Lindenbaum work is worth noting: it is simple type theory, with extensionality, without defined symbols, and containing only closed formulas as sentences. All good clean Le\'{s}niewskian stuff. In particular, a formula like `$Fx \vee \neg Fx$' or even `$\forall x~(Fx~ \vee~ \neg~ Fx)$' is {\em not\/} a theorem or a logical truth. While students of predicate logic
recover their poise and before they start accusing their logic teachers of pulling the wool over their eyes, the reason is simply that both formulas contain unbound variables (or, as the old and suggestive terminology had it, {\em real\/} variables), and so cannot be true or false, since only a sentence can be that. What they can be is universally satisfiable, which both are, but this translates into truth of the universal closure `$\forall F \forall x(Fx \vee \neg Fx)$'.
Since the variable `$F$' ``goes apparent'' we get no counterexample to the Tarski--Lindenbaum theorem. In effect, the sentence states that the universal property of properties holds of all properties (or the class of all classes of individuals contains all classes of individuals),\footnote{When dealing with extensional concepts, `class' is perhaps more appropriate than `property', since the latter is normally construed intensionally. But `class' is reminiscent of `set', and hence of abstract individuals \`{a} la ZF. Neither term is optimal.}
and this is quite along Tarski--Lindenbaum lines.
\section*{6. Comments on Tarski's proposal}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{6. Comments on Tarski's proposal}
Perhaps the first thing that needs mentioning when we consider the 1936 and 1986 papers together is that the latter is a sort of converse to the former. Tarski and Lindenbaum take for granted that a certain kind of theory is to be regarded as {\em logic\/},\footnote{$LSM$ 384. Tarski and Lindenbaum admit that a range of systems may count as `logic'; theirs is chosen to be representative.} and then they go on to raise the question as to what objects (one is tempted here to say `concepts')
are definable in these terms. The tool for deciding what is logically definable is the permutation of the `world', or domain of all individuals. This is used to {\em induce\/} corresponding permutations among all higher types. The invariants (constants) at each type are precisely the logical objects. By contrast, the 1986 paper starts by considering this tool as offering a possible elucidation of the idea of a logical object, and then goes on to assure us that what we hold dear as logical
objects --- namely those considered above --- are, by the 1936 result, indeed invariant, so our intuitions as to what counts as a logical object coincide to a fair degree with the objects selected by the proposed elucidation. The methodological status of Tarski's suggestion is thus a little obscure: he himself seeds the point and modestly says it is ``in agreement, if not with all prevailing usage of the term `logical notion', at least with one usage which is actually encountered in practice.''\footnote{Tarski 1986, 145.}
This slightly curious status --- half report, half stipulation --- is also to be found in the big truth paper, where Tarski in effect makes a suggestion (which not everyone has accepted) as to what conditions the predicate `is a true sentence of the language $L$' should satisfy, inviting us to compare his suggestion with our intuitive notion of truth. The comparison between formal theory and informal notion is found elsewhere at crucial junctures in modern logic, for example Frege's logicism and Church's thesis.
It remains unclear whether Tarski viewed his suggestion as providing the key test for distinguishing the logical from the non-logical, and so solving the problem he posed in 1936, or whether he is offering it, in relativistic spirit, as one possibility to be considered among others.
Tarski's proposal says nothing about language. The concept of a logical object is not (contrast that of truth) relativized to a language. It may be considered however whether there is not a wandering free variable in the proposal: that of `world' or `domain'. Now Tarski was in fact rather reticent about relativizing concepts such as truth to a domain or a structure. The big truth paper for instance defines truth in $L$, not truth in $L$ relative to $M$.\footnote{See Hodges 1986, Wole\'{n}ski \& Simons 1989.}
It is quite possible that when Tarski spoke of the `world' he indeed meant the term as a constant, standing for all actually existing individuals. This position will be adopted below, because it accords well with a Bolzanian approach. But it is a reasonable assumption that logical constants should be in some sense constant across different possible domains, independent of such matters of fact as how many individuals there happen to be.
Tarski's proposal tells us how to discern the logical objects, {\em given\/} a domain, but not how to make cross-domain comparisons. We can easily see why not: most logical objects will be different objects for different domains, and some objects will be logical in one domain and not logical in another. The simplest example is the universal class of individuals, which is a different object for each domain (by definition of `domain'). And if one domain $D$ is properly included in another $D'$,
then with respect to $D$, $D$ is a logical object, and with respect to $D'$, it is not. Another example is the property of being a five-membered class, which is the null property of classes on domains of less than five objects, is the property of classes having just the universal class in its extension on five-membered domains, and on all larger domains has more than one class in its extension. The only logical objects which are literally the same in any domain are the null objects for each type (except individuals).
A concept of logical objects which was so restricted that only null objects quality would be too trivial to consider, although it shows that Tarski's scheme is not the most puritanical one possible.
Corcoran has suggested that in such cases we say that each logical constant has a {\em branch in \/} a given domain.\footnote{Suggested at a lecture given at the University of Geneva in June 1987.} He then proposes as a simple generalization of Tarski's view that a logical constant is one which has in each domain a branch which is a logical object in Tarski's sense. By quantifying over domains, this eliminates the relativization. This suggestion is eminently reasonable. At the same time, it goes beyond Tarski's view,
since logical constants in this sense are no longer objects in a type hierarchy. One possible suggestion is that a logical constant be the set of its branches. However, this might well be a very big collection, not a set, and it inserts set (or set-and-class) theory into the metatheory. A suggestion more Bolzanian in spirit would be to take a logical constant to be a concept whose extension in each domain is the appropriate Tarskian logical object. How do we know which object is ``appropriate''?
Presumably because we can in principle write down a definition of the concept, e.g. `$x = x$' will do for the universal property. A more Tarskian solution would be to relativize the concept of logical constant to a language. It would then turn out that not all languages have the means to express all logical constants (languages of $n$-th order cannot be used to define logical constants of order higher than $n+1$). We should also need to take care of the multiplicity of logical equivalents of any given object,
either by abstracting or by picking a canonical representative. All of these directions move away from Tarski's original idea.
While the basic idea of Tarski's proposal seems to me to be completely right within its limits, it is surprising what it leaves out. One would have thought that the truth-functions and the quantifiers were logical constants {\em par excellence\/}, but they find no place in Tarski's scheme. Roughly speaking, one may say the truth-values and truth-functions are too {\em basic\/} to find a place in the scheme, while the quantifiers are too {\em general\/}. Once again, I find a parallel to Le\'{s}niewski. His {\em protothetic\/}
(`science of first principles') is a system of propositional types with quantifiers. All grammatical categories in protothetic are derived from that of {\sc sentence} (S) alone; I call them {\em purely propositive\/} categories. Because no names, nominal functions, predicates etc. occur in protothetic, a sentence of protothetic can have no subject-matter in the usual sense, and in particular cannot determine an object in the usual type-theoretic hierarchy. The quantifiers on the other hand are not tied to any particular category;
as they can bind variables of any category.\footnote{Cf. $LSM$ 218, n.2.} Hence they cannot be assigned any position or object in the type-theoretic hierarchy, and so cannot be logical objects in Tarski's sense.
Of course one can {\em find a place\/} for protothetical constants and quantifiers in Tarski's scheme: in discussion in Buffalo, Tarski suggested taking the truth-values to be the universal and null classes, so the truth-functions are functions of higher type.\footnote{Cf. Tarski 1986, 150, n.6.} The quantifiers can be dealt with by splitting them into infinitely many sorts, one for each semantic category. The universal quantifier binding a single variable of category $X$ will then be the functor taking
the universal object whose type corresponds to category S$(X)$\footnote{The category of sentence-forming functors taking expressions of category $X$ as inputs. For the notation, see below.} to the True, and all other objects of this type to the False. There is a minor technical problem: the particular identification suggested only works if the domain is non-empty (otherwise the True and the False coincide), so if one is prepared to admit the empty domain another identification is required. This can be solved by moving the truth-values to a higher type which always contains at least two logical objects.
As a formal trick such an identification of course works, but is it satisfactory as part of an {\em explication\/} of the notion of logical constant? I think not: it is a fudge. In an issue like this, every thing should be painfully literal and above-board. Firstly, the proposal is conventional: any other two objects would have done for the truth-values, as far as their structural role is concerned. So why were logical objects chosen? Presumably because we want to say that the True and the False are in some sense independent of matters of fact, themselves logical. But then we have no explanation within Tarski's framework for why we are so convinced of this:
either this framework then builds on intuitions which it cannot account for, or it must presuppose a more basic account for these constants, and hence not be the full story. Contrast the case with those logical objects which Tarski's proposal does comprehend: here we {\em do\/} get an insight into why e.g. no individual is logical: the idea of an arbitrary permutation of the domain embodies at least in part the intuition that logic is topic-neutral. The idea found its way into Mostowski's seminal paper on generalized quantifiers and is now well-established in the literature
as part of what distinguishes logical from non-logical determiners.\footnote{Mostowski 1957. Cf. Barwise \& Cooper 1981, Westersta\hspace{-.06in}$^{\circ}$hl 1985.}
Recall that Frege was quite unabashed about taking the truth-values to be individuals, something which is not compatible with Tarski's procedure if the truth-values are logical constants. To accommodate Frege's view, one would have to restrict permutations of the individuals to those mapping each truth-value into itself. The whole Warsaw School, Tarski included, adopted the standpoint, in opposition to Frege,\footnote{But in agreement (for once) with Wittgenstein.} that the truth-values are not individuals,
that being true or false is something distinct from denoting this or that individual or any semantic role presupposing denoting. I am convinced this view is correct. Since the semantic categories of the types in which Tarski's logical objects are to be found all presuppose the role of denoting, any attempt to find a place for the truth-values in the hierarchy of Tarskian logical objects is in my view mistaken.
For want of a {\em theory\/} explaining why at least the truth-values and truth-functions (so-called) should be so obviously logical constants I can only offer informal ruminations. Somewhere around here, if not just here, we reach bedrock, and run out of reasons. Since logic is traditionally the science of valid argumentation, and no argument is valid which fails to preserve truth, we cannot except truth from the logical constants, otherwise we should have no fixed notion of validity. Since we also know there are sentences which are false,
and invalid arguments which go from truths to falsehoods, we are led to accept falsehood as a logical constant too. No characterization of what is logical can allow `false' to be substituted for `true', whatever further variation is admitted. Allowing `true' and `false' to exchange roles does indeed leave some things invariant, as we note below, but truth and validity are not among them. I doubt however whether any characterization of logic at such a basic level can tell us whether truth and falsity are the only possible values for sentences.
Logic is not by definition bivalent. Similarly, while concentration on extensional values of sentences, such as `true' and `false', is understandable, the characterization of logic as the science of valid argument does not of itself exclude other {\em modi significandi\/} of\linebreak sentences\footnote{Ways of meaning are the basis of the semantics for Le\'{s}niewski's Ontology in Chapter 12. No claim is made that this concept coincides with the medieval one of this name, but the medieval notions of {\em modus\/} and {\em functio\/} are worth considering as possible starting points for modern semantic theories.}
from consideration (such as their Fregean sense, or their modality), nor does it exclude logical connections among sentences which are not extensional, such as connections of relevance. Here, too, as Bolzano would say, controversies could arise at times.
In view of these imponderables, we concentrate, like Tarski, on a more modest aim, that of delimiting the logical constants for a bivalent, extensional logic, which is nevertheless more powerful than first-order predicate logic. This is a theory of types incorporating also the propositional types, and is thus comparable with Le\'{s}niewski's Ontology.
\section*{7. The hierarchy of logical objects}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{7. The hierarchy of logical objects}
Like Tarski, we stipulate how to determine logical objects of arbitrary type in terms of permutations.
\subsection*{7.1. Types}
There are infinitely many types, which are recursively determined in the usual way:
\begin{itemize}
\item[TP1]The type t is a type.
\item[TP2]The type i is a type distinct from t.
\item[TP3]If $a,b_{1},\ldots,b_{n}$ are any types, then there is a unique type $a(b_{1}\ldots b_{n})$, distinct from each of $a,b_{1},\ldots,b_{n}$.
\item[TP4]That is all the types there are.
\end{itemize}
The types t and i are {\em basic\/} types: all others are {\em functor\/} types: the types $b_{1},\ldots,b_{n}$ are {\em input\/} types and $a$ is the {\em output\/} type with respect to $a(b_{1} \ldots b_{n})$. Note that the order of inputs is relevant. This is a somewhat questionable feature and will be discussed below.
\subsection*{7.2. Objects}
We have said nothing so far about the occupants of each type. In general we assume the following principles:
\begin{itemize}
\item[]Inclusion: Every object belongs to some type.
\item[]Exclusion: No object belongs to more than one type.
\end{itemize}
For starters, we fill up the two basic types. The type t is that of {\em
truth-values\/}. Since we are assuming bivalence, t has precisely two
occupants: the True (T) and the False (F). The type i is the type of {\em individuals\/}. In keeping with his indifference to matters of fact, a logician is not usually interested in what sorts of individuals there are or how many there are, but he {\em is\/} concerned to deal with all individuals there are. So we shall assume that i includes all actual individuals
but nothing which is not an actual individual. By Exclusion, no individual is a truth-value.
As the choice of occupants so far will have made clear, the types we are considering are thoroughly extensional in nature. This holds for the functor as well. The identity of functors, or rather, the analogue of identity which holds in each case, namely coextensionality of the appropriate type, is determined solely by the identity of the inputs and outputs of the functor. In this functors form the obvious extension of the concept of function to the case where inputs and output may be of any type, not just the type of individuals.
In accordance with the usual requirements on functions, we further specify for every functor:
\begin{itemize}
\item[]Totality: Every combination of suitable inputs has an output.
\item[]Uniqueness: No combination of inputs has more than one output.
\end{itemize}
If $A$ is a functor of type $a(b_{1} \ldots b_{n})$ and $B_{1},\ldots,B_{n}$ are objects of types $b_{1},\ldots,b_{n}$ respectively, we denote the output of $A$ for $B_{1},\ldots,B_{n}$ (in order) as inputs by `$A(B_{1} \ldots B_{n})$'. It has type $a$. By Totality there is such an output, and by Uniqueness there is no more than one. Finally, we require that every functor that could occur in our hierarchy does occur:
\begin{itemize}
\item[]Plenitude: In each functor type there is a functor corresponding to every combination of inputs and outputs.
\end{itemize}
The types t(t\ldots t) are those of what are usually called truth-functions. Since truth-values are not individuals,\footnote{The very use of the nominal term `truth-value' is rather unfortunate, deriving as it does from Frege. If however we understand `value' as a type-neutral metalogical term, the usage is acceptable.} this is a strained usage of the word `function', and we may call them `{\em truth-value-functors\/}'. Any functor, such as there, which has a `t' as the first letter in its index,
we may call a `t-functor', and one with no i's in its index a {\em pure\/} t-functor. Similar terminology may be applied, {\em mutatis mutandis\/}, to i-functors. The types t(i \ldots i) are those of {\em relations\/} (one-place relations may be called {\em properties\/} or {\em classes\/}, although neither term coincides in this usage with its usual intuitive meaning).\footnote{Cf. note 35.} The types i(i \ldots i) are those of {\em functions\/} in the usual sense. The usual semantic treatment of functors of higher types is to see them as functions,
the relevant arguments and values being certain sets. If we view sets as abstract individuals (Tarski's ``first-order'' view), then in view of Exclusion, this embodies an infinite sheaf of category-mistakes.
\subsection*{7.3. Coextensionality}
Coextensionality is an analogous or typically ambiguous concept, defined recursively over types as follows:
\begin{itemize}
\item[CX1]Coextensionality for truth-values is just material equivalence.
\item[CX2]Coextensionality for individuals is just identity.
\item[CX3]Given coextensionality for type $a$, coextensionality for type $a(b_{1} \ldots b_{n})$ is given by:\\
for all $A$ and $A'$: $A$ coex $A'$ iff for all $B_{1},\ldots,B_{n}$:
\[A(B_{1} \ldots B_{n})\; \mbox{coex}\; A'(B_{1}\ldots B_{n}),\]
where the coextensionality in the {\em definiens\/} is that for type $a$.
\end{itemize}
\subsection*{7.4. Permutations}
Suppose $D$ is the domain of all individuals, where $D$ may be empty, and let $P$ be a permutation of $D$. Then the inverse $P^{-1}$ is of course also a permutation of $D$. We extend $P$ analogously to other types as follows:
\begin{itemize}
\item[PM1]$P$(T) $\equiv$ T; $P$(F) $\equiv$ F (i.e. $P$ leaves the truth-values alone).
\item[PM2]Suppose we have defined the effect of permutations for types $a, b_{1},\ldots,$ $b_{n}$. Then we define the effect of $P$ for the functor type $a(b_{1}\ldots b_{n})$ as follows:\\
for all $A$ in $a(b_{1}\ldots b_{n}), B_{1}$ in $b_{1},\ldots ,B_{n}$ in $b_{n}$:
\[P(A)(B_{1}\ldots B_{n}) \; \mbox{coex}\; P(A(P^{-1}(B_{1})\ldots P^{-1}(B_{n}))).\]
\end{itemize}
Note that the symbols `$P$' and `$P^{-1}$' are here typically ambiguous: on the left-hand side the $P$ is the $P$ to be defined, of type
\[a(b_{1}\ldots b_{n})(a(b_{1}\ldots b_{n})),\]
that on the right-hand side is of type $a(a)$, assumed already given, and the $P^{-1}$'s are of types $b_{1}(b_{1}),\ldots,b_{n}(b_{n})$, again assumed already given. The coextensionality signified is that appropriate to type $a$. Note also that for any permutation $P$, neither $P$ nor $P^{-1}$ has any effect on members of t or any pure t-functors.
It might be wondered whether we could weaken PM1 to the extent of allowing permutations which exchange the truth-values. Such permutations are indeed admissible in themselves, but if we allow them, we capture not the logical constants but only the {\em self-dual\/} logical constants: the assertion and negation connectives, monadic ``identity'' and negation (complementation) functors for all t-functors, as well as all logical constants which are pure i-functors. Truth and falsity themselves, tautology and contradiction,
conjunction, disjunction, implication, equivalence, universal and particular quantifiers, identity and difference, etc., etc., all well-known and well-loved logical constants, are not self-dual.\footnote{For a clear account of duality as involving the exchange of T and F, see Quine 1974, \S 12.} So PM1 must stay as it is.
\subsection*{7.5. Logical objects over a domain}
The logical objects over a domain $D$ are those which are invariant under all permutations of $D$: $A$ is a logical object of type $a$ over $D$ iff
\[\mbox{for all permutations}\; P\; \mbox{of}\; D: P(A) \;\mbox{coex}\;A\]
where the coextensionality is that appropriate to $a$ (i.e. has type t($aa$)). It follows by the way in which $P$ is defined that the truth-values and all pure t-functors are logical objects over any domain. Their status as such is indeed completely independent of the domain, which is as we should expect: the considerations motivating their status as logical precede those surrounding Tarski's procedure, which comes as a distinct ``second stage'', like predicate logic is to sentential logic, or Ontology to protothetic.
It can also be shown that the logical objects mentioned by Tarski are in the sense here defined logical objects over any domain. We show this explicitly for just one case: the class-membership or `falling-under' functor $\epsilon$. It is of type t(it(i)) and is introduced via the usual comprehension principle
\[\forall F \forall x(\epsilon(xF) \equiv F(x)).\]
So let $P$ be any permutation of $D$, extended as defined above. Then we have, for all $F$ and $x$:
\begin{center}
\begin{description}
\item[$P(\epsilon)(xF)\;$] $\;\equiv \; P(\epsilon(P^{-1}(x)P^{-1}(F)))$\\
$\makebox[.35in]{}\equiv \epsilon(P^{-1}(x)P^{-1}(F))$, since the left $P$ has type t(t)\\
$\makebox[.35in]{}\equiv P^{-1}(F)(P^{-1}(x))$, by definition of $\epsilon$\\
$\makebox[.35in]{}\equiv P^{-1}(F(P(P^{-1}(x))))$, since $(P^{-1})^{-1}$ coex $P$\\
$\makebox[.35in]{}\equiv P^{-1}(F(x))$, since $P(P^{-1}(x)) = x$\\
$\makebox[.35in]{}\equiv F(x)$, since the left $P^{-1}$ has type t(t)\\
$\makebox[.35in]{}\equiv \epsilon(xF)$, definition of $\epsilon$.
\end{description}
\end{center}
Hence we have
\[P(\epsilon)\; \mbox{coex}\; \epsilon\]
for any permutation $P$, i.e. $\epsilon$ is a logical object over $D$. And this holds for any $D$.
Other logical objects are the functors we assign as suitable for quantifiers to stand for. Let $\langle a_{1},\ldots,a_{n}\rangle$ be any sequence of types. Then since the universal and null objects in the category t$(a_{1} \ldots a_{n})$ are logical objects, and so are the two truth-values, then so are there four objects of type t(t($a_{1} \ldots a_{n}$)):
\begin{itemize}
\item[A:] takes the universal object of type t$(a_{1} \ldots a_{n})$ as input to T as output, and all other objects of this type to F,
\item[E:] takes the null to T, all others to F,
\item[I:] takes the null to F, all others to T,
\item[O:] takes the universal to F, all others to T.
\end{itemize}
We know of course that any of these may be taken as primitive and the others defined in terms of it using negation. But other Boolean operators may be used to yield new quantifiers: often overlooked for instance are the two trivial quantifiers of a given type, those taking all objects to T and all to F respectively.
A number of other examples of logical objects may be mentioned, many having to do with cardinality. The English expression in each case is only a rough indication of the object meant.\\
\begin{center}
\begin{tabular}{ll}
Type & Examples \\ \hline
t(t(i)) & there are finitely many but not three\\
t(t(i))(t(i)) & most, all but three\\
t(t(i)t(i)) & there are more --- than \ldots\\
t(t(i))(t(i)t(i)) & more --- than \ldots \\
t($a$ \ldots t($a$ \ldots)) & generalized membership or falling-under\\
\end{tabular}
\end{center}
\noindent They may be illustrated by examples:\\
There are finitely many but not three apostles
Most birds fly
There are more apostles than planets
More men than women smoke
John falls under the concept man
John stands in the relation of loving to Mary\\
\noindent The membership or falling-under functors are interesting because of their connection with set theory via type-restricted comprehension principles. The simplest one was mentioned above. It is historically not uninteresting to note that in the 1930s Tarski preferred to understand membership not as a single first-order relation but in the manner just indicated, as a sheaf of typically analogous functors, but the success of type-free set theory encouraged him to consider taking
the symbol `$\epsilon$' as designating not just one but any of these functors according to context.\footnote{$LSM$, 241 f., 270 f.} While the 1986 paper does not indicate any clear preference for this view or the more strictly regulated {\em Principia\/} view of membership over the ``first-order view'', his concern to mention this alternative at a time when most logicians and mathematicians would hardly have considered an alternative to set theory is an indication that he had not forgotten his earlier outlook.
Our characterization of logical objects, though it gives us a test for deciding whether a given object over a given domain, does not constitute a recursive method for determining exhaustively, for every type, which are the logical objects of that type. Intuitively, we should expect there to be such a method for determining at least some of the logical objects, and it is desirable to find one. Inspection of some of the examples above shows that one obvious way of obtaining new logical objects from old is to form Boolean compounds, at least for functors involving a `t' somewhere in their index.
Investigation of such methods must be left over for another occasion.
It must be recalled that the number and identity of the logical objects for a given type (except for the truth-values and pure t-functors) will depend on the size of the domain $D$. For instance, if $D$ is finite, there will be only finitely many cardinality properties, one of which will coincide with the universal quantifier for individual variables, whereas for denumerably infinite $D$ there will be infinitely many finite cardinalities and one infinite cardinality.
Our specifications of logical objects unavoidably had a certain awkwardness or unfamiliarity, because we were working directly with them, rather than taking the more usual route via a language. This language-independent account is in the spirit of Tarski. But for a {\em logic\/} we need more than just logical objects.
\section*{8. A Bolzanian language}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{8. A Bolzanian language}
One primary motivation for delimiting the logical constants is that we are thereby given a means for delimiting logical from non-logical truths (and falsehoods, but we shall stick to truths). The method employed is essentially that due to Bolzano, although the immediate source for most modern accounts is Quine.\footnote{Quine 1936.}
Tarski's proposal alone, even with the additions we have proposed, gives us no way to distinguish logical from non-logical truths and falsehoods, for the simple reason that the meaning values available for sentences are still only the two truth-values. To distinguish some truths from others, we need a multiplicity of truth-bearers. Bolzano took the primary truth-bearers to be propositions, whereas Tarski and other ontologically more cautious logicians eschewed these in favour of sentences of a language. Tarski's own original
approach to truth however involved an {\em already interpreted\/} language,\footnote{$LSM$ 166. This view is inherited from Le\'{s}niewski.} so that, in contrast to most modern treatments, he presupposed interpretation and did not define it. It is this which, although he did not accept abstract meanings in the sense of Frege or Bolzano, nevertheless renders his views not dissimilar to theirs, since the variables of domain and interpretation are not present in his truth-definition. Both Bolzano and Frege understood their World 3 entities as explicating
the traditional notion of intension, and regarded intension as determining extension without any outside help from empirical facts of the matter.\footnote{As a view about Frege's philosophy, this is controversial. It is upheld by Baker \& Hacker 1984, and I think they are (for once) right. No other view properly explains Frege's principle that sense determines reference, much as one may wish Frege had thought otherwise.}
It is notable that the views of Bolzano, Frege, and Tarski all have little or no room for the explication of modal notions in terms of possible worlds or situations, and so stand in stark contrast to more usual modern approaches to truth and modality. This similarity allows us to bring Tarskian and Bolzanian ideas together without doing excessive violence to either. So we take as truth-bearers not expressions of an actual language, but propositions in Bolzano's sense, and work entirely with World 3 entities and their extensions.
The advantage is that the entities {\em an sich\/} making up the Bolzanian realm are inherently symbolic, and are excepted from such untidy phenomena of natural languages as ambiguity. Also we need not worry about the meaning (in the sense of `sense') of such entities, since they are themselves the sense of ordinary mortal expressions. And we are not plagued by accidental limitations. For instance, we may happily suppose that every individual has a proper name sitting up in Bolzano's heaven, indeed that every item in the type hierarchy is signified by a simple concept.
This lush perspective has its dangers however, which we consider in the final section.
For ease of expression we shall speak of a Bolzanian {\em language\/} B, although of course it is not a language in the usual sense. In its construction it corresponds to the type hierarchy of the previous section. The language is categorial: every expression (concept or proposition) in it belongs to exactly one category. The categories form a recursive hierarchy in the usual way:
\begin{itemize}
\item[CT1]The categories S ({\sc sentence}) and N ({\sc name}) are distinct {\em basic\/} categories.
\item[CT2]If $x$ and $y_{1},\ldots,y_{n}$ are categories, where $n \geq 1$, then there is a unique {\em functor\/} category $x(y_{1} \ldots y_{n})$, different from $x$ and the $y_{i}$, of expressions taking expressions from categories $y_{1},\ldots,y_{n}$ (in order) as {\em inputs\/} and yielding an expression of category $x$ as {\em output\/}. The categories $y_{i}$ need not be all different.
\item[CT3]That is all the categories there are.
\end{itemize}
A category whose first letter in our notation is an S is a {\em propositive\/} category, one whose letters are all S's is {\em purely propositive\/}. One whose first letter in an N is a {\em nominative\/} category. By construction, all categories are either nominative or propositive, and none is both.
What reasonable assumptions can we make about the expressions in these categories? One thing we assume, with Bolzano, is that some expressions are absolutely {\em simple\/} or unanalysable. We assume that B contains simple propositions Verum and Falsum signifying the True and the False respectively in all circumstances, and for every pure t-functor, a simple concept signifying that functor. This does not rule out the possibility that for practical purposes only a few of these concepts need be actually used. One of Tarski's early
achievements was to show in effect that protothetic may be based on expressions for material equivalence and the pure t-functor quantifiers alone, expressions for the truth-values and the other logical functors being definable in terms of these.\footnote{Cf. note 28. Note that the sense of `definition' relevant here is not one to do with human linguistic conventions, but concerns the coextensionality or cointensionality of a simple concept with a complex one.}
Bolzano himself regarded some logical constants as simple concepts.\footnote{$WL$, \S 78.}
Before deciding what expressions we need in the other categories, we should know what there is in the `world'. For reasons to be discussed below, this will here be understood to contain no individuals from the language {\em an sich\/}. Apart from that, we may safely follow Bolzano in assuming the world contains all the real individuals there actually are, together with such other non-real individuals (e.g. numbers) as our favoured metaphysics sees fit to assume. The world is symbolized `I' (for `individuals').
We may now assume, with Bolzano, that every individual in I has a proper name, that is, a simple concept denoting it and nothing else.\footnote{These are the true ``logically proper names''. Their sense or intension consists solely in their designating the individual they do. Such atomic intensions form the basis of a possible identity criterion for intensions. Cf. Morscher 1981, 111.} No other proper names will be admitted. We know that numerous individuals actually exist, and there is a bijection between proper names and individuals.
This is the sort of assumption which makes substitutional semantics straightforward, and is here bound up with no dubious empirical assumptions, but only with (perhaps equally dubious) metaphysical ones.
Given the world I, the whole hierarchy of objects based on I, T, and F is generated. It would be in accord with Bolzano's metaphysical generosity to suppose there to be a simple concept for each object, including all the logical objects. There is every reason to suppose that no human language would have more than a tiny fragment of such concepts among its intensions.
So far, we have only mentioned simple propositions and concepts, but while such a language would have the resources to express every extension, it would be a strange language which stopped there. So we suppose, again with Bolzano, that B embodies a means for compounding expressions. In accordance with the parsimony of type- and category-construction principles we need only one compounding principle, that of the {\em application\/} of a functor expression to a sequence of categorically suitable expressions. The extension or {\em signification\/}
of the resulting compound expression is determined in the obvious way: if expression $X$ of category $x(y_{1} \ldots y_{n})$ signifies object $C$ of type $a(b_{1} \ldots b_{n})$, and for each $i$, $1 \leq i \leq n$, expression $Y_{i}$ of category $y_{i}$ signifies object $B_{i}$ of type $b_{i}$, and the functor $C$ has output $A$ of type $a$ for inputs $B_{i}$ in order, then the compound expression formed by applying $X$ to the $Y_{i}$ in order, which we call $X(Y_{1} \ldots Y_{n})$, signifies $A$. We assume then generously that every compound expression which can be formed in B according to this principle indeed exists in B.
One of the restrictions we have imposed is that functor types and categories are only ever finitely complex. This may not be in accord with Bolzano's views, since Bolzano thought that some expressions may be infinitely complex. However, this could be catered for within the present framework by allowing infinite repetition of application. For example, given a sign for identity and all our proper names, we can form an expression for the singleton property for each individual, and by disjoining such expressions, if need be infinitely often, we obtain a compound expression for any class of objects. The same goes for any relation.
Bolzano certainly thought along such lines.\footnote{Cf. $WL$, \S 101.} However, for present purposes, which enjoin us to stay fairly close to Tarski's proposal, we shall only consider finitely complex expressions. Every expression of B can thus be assured of a finite analysis terminating in simple expressions. The {\em way\/} in which the simple components are compounded is the {\em form\/} of the expression.\footnote{The form is representable by a labelled tree with cross-referencing showing which positions are ``identified'' and must be uniformly filled.}
What it is for an expression to be a {\em constituent\/} of another is defined as follows:
\begin{itemize}
\item[CN1]Every expression is a constituent of itself.
\item[CN2]For all $Z,X,Y_{1},\ldots,Y_{n}$, if $Z$ is a constituent of $X$, or of $Y_{1}$ or \ldots or of $Y_{n}$, then $Z$ is a constituent of the compound expression $X(Y_{1} \ldots Y_{n})$.
\item[CN3]No expression has any constituents except by virtue of CN1 and CN2.
\end{itemize}
Because complexity is finite, the analysis of expressions into constituents always terminates in finitely many steps (application of K\"{o}nig's Lemma).
\section*{9. Logical constants}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{9. Logical constants}
If $X$ is any expression of B and $Y$ is a constituent of $X$ (where $Y$ need not be simple), then any expression of B obtained from $X$ by replacing $Y$ (uniformly) by another expression of the same category as $Y$ is a $Y$-{\em variant\/} of $X$. $X$ will have as many $Y$-variants as there are distinct expressions in B of the category of $Y$. If the expressions $Y_{1},\ldots,Y_{k}$ are constituents of $X$ such that none is a constituent of another, the result of replacing all the $Y_{i}$ by (not necessarily distinct)
expressions of suitable category is a $Y_{1},\ldots,Y_{k}$-variant of $X$, and $X$ will have a number of such variants. The part of $X$ which remains invariant throughout all such variants (understanding this to include not just the constituent expressions but the form in which they and the varying ones are composed) may be called, following Quine,\footnote{Quine 1936 (1976, 80).} the {\em skeleton\/} of all these variants of $X$, and they may be called {\em completions\/} of the skeleton. In each case, there are two possibilities: either the signification of all the completions of a certain skeleton is the same (coextensional), or it is not.
In the former case we call the skeleton {\em rigid\/}. If an expression has constituents yielding a rigid skeleton we say it is {\em constant\/} with respect to these constituents, otherwise we say it is {\em variable\/} with respect to these constituents.
We may now specify what it is for an expression of B to be a logical constant. Two cases must be distinguished.
\begin{itemize}
\item[LC1]If $X$ is a simple expression signifying a logical object, it is a {\em simple\/} logical constant.
\item[LC2]If $X$ is a complex expression signifying a logical object, and it is constant with respect to the set of all its simple constituents which are not simple logical constants, then it is a {\em compound\/} logical constant.
\item[LC3]There are no other logical constants.
\end{itemize}
A special case is afforded by propositions. All propositions by definition signify logical objects, but not all are thereby logical constants. For example the proposition [John loves Mary] is not, whereas [John loves Mary or John does not love Mary] is, under plausible assumptions about the way in which the linguistic sentence determines a proposition, a logical constant denoting the True. Propositions which are logical constants are {\em logically analytic\/} in Bolzano's sense. A proposition is {\em analytic\/} in Bolzano's sense iff it is constant with respect to one or more constituents.
It is worth pointing to one or two consequences of this reconstruction of Bolzano's ideas with a Tarskian background. Firstly, no expression of B is itself variable: each has a fixed signification. It might be thought that this cripples the language considerably, since after all, the logical languages we use most often contain variables and operators binding them. But in fact variable expressions are dispensable, and indeed variable-binding operators and variables do not fit into the functor-argument scheme of a simple categorial language.
It has been known since the 1920s that variables can be eliminated from a wide range of languages without loss of expressive power, by the use of combinators. I have shown that by the use of a small number of categorically flexible combinators --- or, as is preferable in the present context, a sufficiently large supply of categorially rigid ones --- a language of the structure we have described can manage perfectly well without variables of any category.\footnote{Simons 1989.}
It is easy to show that the seven basic kinds of combinators involved signify logical objects in Tarski's sense, and since they may plausibly be assumed to be simple, they are logical constants according to LC1. Compound combinators, being compounded by the principle of application from simple ones, are then logical constants according to LC2 (trivially, since they have no simple constituents which are not simple logical constants), and the same goes for any expression compounded only of combinators and logical constants like material equivalence, which indeed was to be hoped.
Since the account proceeds in Bolzanian-Tarskian fashion without varying the domain of individuals, it might be wondered whether we do not end up having expressions as logical constants which would not be so if the domain were allows to vary. Intuitively, we should expect the admission of different domains to reduce the number of logical constants. For example, consider a domain having only three individuals, $a,b$, and $c$,
and take the complex predicate expressible by\footnote{The Greek letter marks places which must be uniformly filled. The ultimate unsuitability of such multiple occurrence in B is discussed below.}
\[\xi = a \vee \xi = b \vee \xi = c.\]
This happens to be universal, but surely it is accidentally so, and is not a logical constant. If we allow different domains, the problem is solved, because on a domain containing a fourth object $d$ the predicate in question no longer signifies a logical object. In fact the Bolzanian approach has a trick up its sleeve which rules this out as a logical constant: one of the variants obtained with respect to `$b$' is
\[\xi = a \vee \xi = a \vee \xi = c,\]
and this is not universal. In this way we can simulate a reduction in size of the domain without actually having to change it. But we cannot simulate an increase in size. In this case the problem is not that the wrong expressions are logical constants, so much as that not all logical constants can be guaranteed to signify the objects we should like them to. For instance, on a three-object domain, the concept [there are at least four] signifies, {\em per accidens\/}, the empty quantifier.
I do not think there is any cure for this problem within this framework. But on the other hand, I do not see that it is a problem only for this approach. Only if one has it in mind to prove some strong existence theses to the effect that there are such and such logical objects and they are all distinct --- as in logicism, where one needs enough objects to guarantee the truth of statements of arithmetic and analysis on logical grounds --- is such independence from fact very important. The axiom of infinity (that there are infinitely many individuals) is not self-evident.
In fact there are a lot of individuals, perhaps infinitely many. So there are a lot of logical objects, even if there might have been more.
\section*{10. Problems with the Bolzanian {\em an sich}}
\markboth{Ch.2 Bolzano, Tarski, and the Limits of Logic}{10. Problems with the Bolzanian {\em an sich}}
Bolzano's realm of concepts and propositions {\em an sich\/} is indeed a handy assumption for certain purposes. But, unlike Bolzano, we cannot be sure there is such a realm. In particular a whole series of epistemological problems arise concerning the nature of our access to such entities. These show themselves most clearly in the {\em aporiai\/} attending on Frege's concept of grasping, {\em erfassen\/}, of senses.\footnote{See Frege's essay ``The Thought'' ({\em Der Gedanke}) Frege 1967, 342--61; Frege 1984, 352--72.}
There are however other general problems with the idea of a realm of
entities suitable to provide the intensions for any linguistic
expressions. The realm turns out to be inconsistent. Morscher has argued
that under assumptions which it is plausible to suppose Bolzano accepted,
one can derive both the Liar and Russell's antinomies.\footnote{Cf.
Morscher 1981, 125. Berg (Berg 198) disagrees with Morscher. The case of sets is admittedly the less clear of the two, but I think Morscher is essentially right.}
The very universality of the realm of meanings {\em an sich\/} makes it plausible that it be semantically closed: Bolzano indeed makes considerable use of concepts denoting concepts, and there is no obvious reason why semantic paradoxes should not follow from his views. But perhaps the simplest proof that the {\em an sich\/} is an inconsistent totality is purely arithmetical: Bolzano expressly affirms that every individual and every class of individuals has its own concept.\footnote{$WL$, \S 101. It is not assumed of course that any human being is ever in mental touch with more than a few of these concepts.}
But since concepts are themselves abstract individuals, there will be at least as many concepts as there are classes of concepts, which by Cantor's diagonalization argument cannot be true. For all these reasons, we restricted the domain of individuals to those outside the realm of the {\em an sich}.
These difficulties can be alleviated, at least in part, by restricting the concepts there can be, or by stratifying them, or by other expedients known from set theory. The wonderful inclusiveness and simplicity of the {\em an sich\/} is thereby lost, but we must expect the antinomies to extract their price in some way or other. However I have other difficulties of principle with the idea of a realm of abstract meanings in Bolzano's fashion.
It is not their abstractness as such which causes so much difficulty, as the requirement that each abstract meaning be the meaning of {\em many\/} token expressions. We naturally allow in natural languages that distinct tokens occur in expressions, and that more than one token of the same type can occur in a given complex expression. But the type, or its abstract meaning, is something unique, and cannot as such occur more than once. So the proposition [John loves Mary and Mary loves John],
which contains the concepts [John], [Mary], [loves], and [and], cannot contain any concepts twice, because each concept is unique. The structure of the proposition must be visibly different from that of the sentence, with its multiple occurrences of the various words. It might look as though we can avoid the multiple occurrences by employing combinators: the combinator {\bf\sf W} is employed to reduce repetitions. But in performing such a reduction, the various combinators may themselves occur more than once, so we are no further forward.
The problem is a variant of the old one--over--many. The only recourse is then to an infinity of different forms, taking account of the ``identification'' of arbitrarily many different occurrences of a concept. There may be nothing in principle against such a multiplicity of terms --- recursive construction principles admit infinitely many in any case --- but the surveyability which characterizes recursive constructions is lost, and it begins to be unclear how the {\em an sich\/} can in any sense provide the ideal model of a language which we could use.
In our account of B we took arguments {\em in order\/}, for the sake of familiarity. So two functors or functor expressions could differ merely in the order of their inputs, like a relation and its converse. At the level of Bolzanian concepts, this is a superfluous duplication. The propositions [Paris is larger than Rome] and [Rome is smaller than Paris] are identical,\footnote{$WL$, \S 148, note 1. This example is probably not representative of Bolzano's considered view on the identity of propositions using converse relations.
He does however consider active/passive sentence--pairs to express the same proposition. Here we see that his work lacks explicit principles for deciding when two linguistic sentences express the same proposition.} which means the the connection between a functor and its inputs is not one given by an extrinsic feature like order so much as an intrinsic one analogous to case in natural language. Thus the input slots to a functor are not just marked for category: they are also distinguished according to the functor's intrinsic meaning. This removes them further still from the mere place--counting of functor--argument constructions.
A defender of the autonomy of meanings {\em an sich\/} such as Frege or Bolzano may well be prepared to accept that they have an intrinsic structure which is further removed from the structure of linguistic expressions than we might imagine. But the more alien such a realm is forced to be, the less use it can be in helping us to understand mundane facts about actual languages.
\end{document}