Language was also important for another of Hegel’s critics, Otto Friedrich Gruppe (1804–76), though for him it was natural language that was important. Gruppe rejected all a priori philosophy entirely; science had shown that naturalism was the path to progress. This did not mean that logic had to be given up, but it had to be approached in a novel way, via how people use natural language. In his Turning Point of Philosophy in the Nineteenth Century,16 Gruppe argued as follows. Traditionally, logicians had taken concepts to be foundational, and judgments to be made up thereof. However, this gets things the wrong way around: it is judgments that are primary; concepts are abstracted from these. And what is one to make of the inferences which comprise judgments? An answer to that was given by another naturalist, Heinrich Czolbe (1819–73). In his New Account of Sensualism 17 Czolbe argued that inference (like other facets of language use) were simply matters of empirical psychology—and in the last instance, the laws of physiology.
The philosophical naturalism of writers such as Gruppe and Czolbe generated a reaction, a resurgence of Kantianism. The most important of the Neo-Kantians, and arguably the most influential of the writers on logic in these interregnum years, was Rudolph Hermann Lotze (1817–81). In his two books called Logic18 Lotze defended Aristotelian logic on a priori grounds. However, he insisted on the distinction between psychological acts of thought, and their objective contents.19 Logic concerns the latter.20
20.6 BOLZANO, A LONE VOICE
None of these post-Hegelian developments produced any really novel developments in logic itself, though they certainty created an atmosphere of uncertainty in which new ideas could flourish. And flourish they did. In fact, even in the earlier part of the century such ideas were developing.
Perhaps the most important person in the early such development was Bernard Bolzano (1781–1848). Bolzano was a remarkable person. Working almost entirely in isolation, he developed notably new ideas in logic, mathematics, and philosophy. As far as logic goes, his most significant publication was his Theory of Science.21 As the title of the book indicates, Bolzano was interested in knowledge quite generally, its constitution, ground, and structure. But logic plays the core role in this.
Knowledge is expressed in propositions. But these are not subjective judgments. Rather, propositions are, essentially, the sorts of things that can be the objective contents of declarative statements. And a proposition is true or false, also objectively, depending on whether the world is as it says it to be. Thus, both propositions and their truth depend in no way on actual thinkers, though thinkers may understand them and grasp their truth.
Propositions are made up of ideas. But the ideas are just as objective as propositions. In particular, they are nothing to do with particular thinkers—so concept might be a better word for what is intended here. Concepts are the sort of things that apply to the objects in their extensions. (So city applies to New York, Melbourne, Berlin, and so on.) We are still working, note, within an Aristotelian framework, so that, for example, Aristotle the Stagyrite is a concept that applies to just one object.
Using the notion of extension, Bolzano characterized a number of important logical relations between concepts. For example:
•A is compatible with B just if there are objects which are in the extension of both A and B.
•A is included in B iff A and B are compatible, and the extension of A is contained in the extension of B.
It is worth noting that one might expect a modal element to be present in some of these relations. Thus, one might expect: A is compatible with B if it is possible that there are objects which are. …Such an element is, however, absent in Bolzano.
Arguably, Bolzano’s most novel contribution to logic was his definition of logical consequence. First, given any proposition, P, fix on some of the concepts which occur in it, Call these parameters. Let be a corresponding string of concepts, where each bi is of the same kind as the corresponding parameter . We can form the proposition which is obtained by replacing each parameter, in P with the corresponding bi. Relative to a bunch of parameters, we can now mirror the logical relations between concepts with relations between propositions. Thus:
•P is compatible with Q just if there is a such that and are both true.
•Q is deducible from P iff P and Q are compatible, and for every such that is true, is true.
It is to be noted that deducibility holds with respect to a bunch of parameters (so that ‘Fred is red’ is deducible from ‘Fred is coloured’ with respect to the parameter Fred, since if b is any concept referring to a physical object, if it is true that b is red, then it is true that b is coloured). Bolzano does appear to accept the distinction between what would now be called logical constants (like if and not) and non-(logical constants) (like Fred and red)—or to give them their medieval names syncategorematic terms and categorematic terms—though he offers no principled account of the distinction. But given this distinction, he can frame an absolute notion of consequence, namely deducibility where the parameters are the non-(logical constants).
Note also that for Q to be deducible from P, P and Q must be compatible. Now, with respect to the parameters which are the non-(logical constants), P is not compatible with ‘it is not the case that P’. A fortiori, no Q is compatible with ‘P and it is not the case that P’. Hence, according to this conception of consequence, contradictions do not entail everything; in fact they entail nothing. The account of consequence was therefore paraconsistent. In fact, though there is probably no way he could have known this, Bolzano was reinventing the connexive notion of logical consequence endorsed by medieval logicians such as Abelard.22 This account is quite different from contemporary explosive logics, according to which a contradiction entails everything, and even from most contemporary paraconsistent logics, according to which contradictions entail some things but not others.
A pleasing feature of Bolzano’s notion of logical consequence is that it allowed him to extend his account of consequence to a non-inductive one. Fix the parameters, and assume that the possible replacements for each parameter are finite in number. We can define the conditional probability of Q given P, Pr(Q/P), as the number of true things of the form divided by the total number of true things of the form Given Bolzano’s account, if Q is a consequence of P, then Pr(Q/P) = 1. (And this can hold in general only because P and Q are compatible. In particular, then, substituting for some parameters makes P true. Hence, the divisor is non-zero.) But the value Pr(Q/P) can, in principle, be any rational number between 0 and 1. So a proposition P may offer some lesser degree of support (or unsupport) for another.
Because of his isolation, Bolzano’s work had very little immediate effect on the developments in logic. It first appears to have been noticed late in the century by Franz Brentano and his school. When Brentano’s student Kazimierz Twardowski founded what was to become the Lvov-Warsaw school, this knowledge moved there, though developments made by logicians such as Alfred Tarski (né Teitelbaum) were already overtaking it. That story belongs to the history of the twentieth century, however.23
20.7 SCHRÖDER AND THE ALGEBRA OF LOGIC
When one reads Bolzano, it is striking that, though the ideas he is expressing are quite complex, beyond the occasional use of letters for quantities, he makes no use of mathematical symbolism. Matters are quite different with the next two people in our story, Schröder and Frege. Both were professional mathematicians; both used mathematical symbolism freely.
The branch of mathematics called abstract algebra started to blossom towards the end of the eighteenth century, and developed throughout the nineteenth. Ernst Schröder (1841–1902) worked squarely in this tradition. In abstract algebras, we are concerned with a bunch of objects and operations on them. Thus, if a, b, and c are objects of our concern, and + and × are binary operations on the objects, we may form objects such as and 24 Relationships between objects are typically expressed by equations, such as and the algebra seeks to determine which relationships of this kind obtain, vi
a a manipulation of these equations (of a kind now familiar from high school algebra). It is characteristic of an algebra, note, that the objects of the algebra can be thought of as different kinds of things. In other words, the algebra may have more than one natural interpretation. (In the language of modern logic, the algebras are not intended to be categorical.) The point, indeed, is to chart the commonalities of structure between different domains.
Schröder framed the project early in his life of developing an algebra that charted the commonalities of structure between all mathematical quantities, very generally understood—a universal algebra—and applying it to various areas of mathematics and physics. He then came under the influence of two brothers with similar sympathies, Herman Günther Graßman (1809–77) and Robert Graßman (1815–1901). Soon after this, he discovered the work of the English logician and algebraist George Boole (1779–1848), and a little later, that of the polymath from the United States of America, Charles Sanders Peirce (1839–1914), both of whom made significant contribution to the algebraicization of logic.
Schröder’s first main foray into the area was his The Circle of Operations of the Logical Calculus.25 This was followed by his mammoth Lectures on the Algebra of Logic, in three volumes.26 The second part of Vol. 3 was published posthumously, edited by Karl Eugen Müller.
Volumes 1 and 2 contain an exposition of what would now be called Boolean algebra. In Volume 1, the objects concerned are thought of as classes; in Volume 2, they are thought of as propositions. (A proposition, Schröder notes, following Boole, may be identified with the set of times at which it is true.) There are two special objects, 1 and 0. 1 represents the set of all the objects (times) in the domain of inquiry; 0 represents the empty collection. There are three main operations, union (disjunction), +; intersection (conjunction), ×; and complementation (negation), now standardly indicated by an overline: is whatever it is that remains when the members of a are taken away from those in 1. Schröder discusses the relations between these various notions, such as a + a = 1. aa = 0. He proves that there is no way of deducing the distribution law, , from other standard principles concerning + and ×, by showing that the other principles, but not distribution, hold in a structure which would now be called a non-distributive lattice. This may be the first appearance of both such a lattice, and an independence proof in logic. (Independence proofs of this kind had been known in geometry for some time.)
Of special importance is the relation of subsethood (subsumption), a ≤ b—which Schröder takes as primitive, but which may be defined as ab = a. Using this, one may algebraicize standard logical reasoning. Thus, take the syllogism (Barbara): All as are bs; all bs are cs; hence all as are cs. The premises may be written as a ≤ b and b ≤ c. Operating on these equations by algebraic rules, one may deduce the conclusion, a ≤ c. Thus, we are given that ab = a and bc = b. Hence, ac = (ab)c = a(bc) = ab = a. That is, a ≤ c.
Schröder departs from Boole in small but significant ways. Notably, he interprets + as inclusive. For Boole, a + b is defined only if a and b are disjoint (that is, ab = 0); this causes a number of unnecessary complexities. Secondly, Boole needed a way to express the thought that a and b are not disjoint. To do this, he introduced a special symbol, ν, where νa is to be interpreted as some non-deterministically determined non-empty subset of b. The fact that a and b overlap can then be expressed by νa=νb. The notion ν is both of dubious intelligibility and complex to operate with. Schröder does not dispense with ν, but does not need it. Unlike Boole, he operates with inequalities as well as equalities. He can therefore express overlap simply as: ab ≠ 0.
There are inelegancies in Schröder’s own system, though. The symbol ‘=’, and so ‘≤’, does duty for more than one thing. Thus, we find him writing things such as: (a ≤ b)(b ≤ c) ≤ (a ≤ c). Here, if the main ‘≤’ is to be interpreted as subsethood, the things on either side of it must be sets. Hence, a ≤ c, for example, must be interpreted This is possible because a ≤ c iff However, the failure to draw this important conceptual distinction betokens an unfortunate confusion.
In Volume 2, and following Peirce, Schröder introduces a notion that may be thought of as quantification. He writes things such as to mean the (possibly infinite) sum of all things of the form ai, where the i can take a value from some predetermined range. Similarly, he writes things such as to mean the (possibly infinite) product of all things of the form ai. If one thinks of i as a free variable, this is some form of quantification. However, in virtue of the algebraic context in which Schröder is working, it arguably makes more sense to take ∑ and ∏ to be the infinitary generalizations of + and ×. If so, the notation is not so much a precursor of the notion of quantification, as that of languages where the formulas can be of infinite length, infinitary logic.27
The following is also worth noting. Modern presentations of algebras are axiomatic. That is, axioms concerning the algebra are laid down, and then theorems of the algebra are deduced. In a posthumously published essay, ‘Outline of the Algebra of Logic’ (also edited by Müller),28 he does offer something like a list of axioms; but in the Lectures the algebra is not developed axiomatically.
Volume 2 of the Lectures is devoted to the topic of the algebra of relations, developed by Peirce. If the objects in Volume 1 can be thought of as sets, the objects in Volume 2 can be thought of as relations (in modern understanding, sets of ordered pairs). Schröder introduces appropriate operations on these, such as converse, and product, a.b (in modern notation, iff29 yax; and x(a.b)y iff ∃z(xaz and zby)), and investigates their properties. It is certainly wrong to take the logic of relations to be unimportant for logic. In a certain sense, traditional logic recognizes only monadic properties, not binary relations (or relations of higher arity). The recognition and incorporation of relations into the syntax of logic was a key feature in increasing the power of logic. However, Schröder’s main concern in this volume is not so much with the application of the algebra of relations to logic, but to areas such as set theory. Hence, we may pass over this topic here.
There is no doubt that Schröder was an original thinker, and that he made important contributions to the nascent discipline of set theory, as it was being developed by the likes of Cantor and Dedekind. He certainly introduced novelties in logic as well, such as algorithms for operating on systems of equations. However, it must be said that both Boole and Peirce were much more original in their thinking about the algebra of logic, and that Schröder’s main contribution to this area was in the systematic exposition and polishing of others’ thought.30
20.8 FREGE AND BEGRIFFSCHRIFT
The same cannot be said of Friedrich Ludwig Gottlob Frege (1848–1925), who must count as one of the most original logicians in its history.
The nineteenth century was not only an epoch in which abstract algebra developed. It was also an epoch of increasing rigour in mathematics. In particular, a whole menagerie of kinds of number was known: natural numbers (0, 1, 2), rational numbers (1/2, 3/5), real numbers (π, 0.1111·), complex numbers , infinitesimals (used in the differential and integral calculus); but how exactly to understand these, and even how to operate with them exactly, was not really clear. (It is worth noting that the only branch of mathematics that had received an axiomatic treatment by this time was geometry.) The nineteenth century organized the zoo. Weierstrass and others showed how to do the calculus without appealing to infinitesimals; and they disappeared from the zoo entirely. Argand showed how complex numbers could be understood as pairs of real numbers. Weierstrass, Dedekind, and Cantor showed how real numbers could be seen as sets of rational numbers; and Tannery showed how rational numbers could be seen as sets of pairs of natural numbers.31 So by the time we arrive at Frege, all the numbers could be seen as set-theoretic constructions out of the natural numbers. But what of the natural numbers themselves? Frege set out to show that they could be seen as constructions out of just sets, and, moreover, that set theory was simply part of logic.
To do this, he
needed a language to express his ideas clearly, and, additionally, to draw inferences employing his concepts in a clear and rigorous way. Traditional logic was up to neither of these tasks. He had read Trendelenberg, Boole, and Lotze, but in none of them did he find what he needed. So he invented it, and called it ‘Begriffsschrift’. This was published in his A Formula Language of Pure Thought Modelled upon the Formula Language of Arithmetic32—known nowadays simply as the Begriffsschrift—a book that barely exceeds 100 pages in modern editions. Two subsequent books made the mathematical application of the language/logic that Frege envisaged; and a number of later essays articulated many of the philosophical ideas underpinning it. Three of the most important of these are ‘Function and Concept’, ‘On Sense and Reference’, and ‘On Concept and Object’.33
The sentences of Frege’s formal language and their component parts were taken to have objective content, as for Lotze (and Bolzano). If A is a formula of the language, Frege writes –A for its content. A vertical line indicates that a content is judged to be true. So A means that the content of A is judged true. Psychology is thus separated from content right at the start. In ‘On Sense and Reference’, and in an attempt to explain why, for example, ‘Hesperus is Hesperus’ has a different content from ‘Hesperus is Phospherus’, even though ‘Hesperus’ and ‘Phosphorus’ refer to the same object, Frege comes to advocate a bicameral theory of content. Sentences and their parts have both a sense (Sinn) and a reference (Bedeutung). This does not play a role in the Begriffsschrift, however, content operating purely on the level of reference.
The Oxford Handbook of German Philosophy in the Nineteenth Century Page 76
