“Why look back now? Let me start by stating my non-historian’s view of the modem history of logic. Like many scientific disciplines, flourishes while being ill-defined. Despite textbook orthodoxy, the issue what logic should be about is a legitimate topic of discussion, and one to which answers have varied historically. One key topic is reasoning: its valid laws for competent users, and perhaps also its sins: mistakes and fallacies. But the modern core also includes independent concerns such as formal languages, their semantic meaning and expressive power. Moreover, the modem research literature, much of it still in a pre-textbook stage, reveals a wide range of topics beyond reasoning and meaning, dealing with general structures in information, and many-agent activities other than reasoning, such as belief revision or communication. Thus, the agenda of logic keeps evolving, as it should. In this light, going back to the pioneers is not just a matter of piety, but also of self-interest.
One striking feature of older literature is its combination of issues in logic with general methodology of science. One sees this with Bolzano, Mill, or Peirce, but also with major modem authors, such as Tarski, Carnap, or Hintikka. The border line between logic and philosophy of science seems arbitrary. Why have ‘confirmation’, ‘verisimilitude’, or ‘theory structure’ become preserves for philosophers of science, and not for logicians? This separation seems an accidental feature of a historical move, viz. Frege’s ‘contraction of concerns’, which tied up logic closely with the foundations of mathematics, and narrowed the agenda of the field to a point where fundamentalists would say that logic is the mathematics of formal systems. Admittedly, narrowing an agenda and focusing a field may be hugely beneficial. Frege’s move prepared the ground for the golden age of logic in the interbellum, which produced the core logic curriculum we teach today. At the same time, broader interests from traditional logic migrated, and took refuge in other disciplines. But as its scientific environment evolved in the 20th century, logic became subject to other influences than mathematics and philosophy, such as linguistics, computer science, AI, and to a lesser degree, cognitive psychology and other experimental disciplines.
Compared with Frege, Bolzano’s intellectual range is broad, encompassing general philosophy, mathematics, and logic. This intellectual span fits the above picture. Even so, I am not going to make Bolzano a spokesman for any particular modern agenda. The current professional discussion speaks for itself. But I do want to review some of his themes as to contemporary relevance. Incidentally, the main sources for the analysis in my 1985 paper, besides reading Bolzano himself, have been Kneale & Kneale 1962, and Berg 1963. After the Vienna meeting this autumn of 2002, I learnt about Rusnock 2000, whose logic chapters turned out sophisticated and congenial.
A short summary of Bolzanian themes:
We quickly enumerate those points in Bolzano’s logical system that are the most unusual and intriguing to logicians. These will return at lower speed in later sections.
The systematic idea of decomposing propositions into general constituents is linguistically attractive, and reminiscent of abstract analyses of constituent structure in categorial grammars.
In doing so, looking at different ways of setting the boundary between fixed and variable vocabulary in judging the validity of an inference is another innovation, which ties up with the recurrent issue of the boundaries of ‘logicality’.
Moving to logical core business, acknowledging different styles of reasoning: ‘deducibility’, ‘strict deducibility’, or statistical inference, each with their own merits, is a noteworthy enterprise quite superior to unreflected assumptions of uniformity.
As to detailed proposals, consider Bolzano’s central notion of deducibility. It says that an inference from premises f to a conclusion ? is valid, given a variable vocabulary A (written henceforth f ? ?) if (a) every substitution instance of the A‘s which makes all premises true also makes the conclusion true, and (b) the premises must be consistent. Clause (a) is like modem validity, modulo the different semantic machinery, but with a proviso (b) turning this into a non-monotonic logic, the hot topic of the 1980s. Moreover, the role of the vocabulary argument A making inference into a ternary relation really, will also turn out significant later.
But also other notions of inference are reminiscent of modem proposals trying to get more diversity into how people deal with large sets of data, such as ‘strict deducibility’: using just the minimal set of premises to get a given conclusion.
Bolzano’s statistical varieties of inference involve counting numbers of substitutions that make a given statement true. Such connections between qualitative logic and quantitative probability were still alive in Carnap’s inductive logic, a fringe topic at the time, but they are coming back in force in modem logic, too.
Very striking to logicians at the interface with AI is Bolzano’s formulation of systematic properties of his notions of inference, such as versions of transitivity or the deduction theorem, some depending on the fixed/variable constituent distinction. No truth tables, model-theoretic semantics, and their ilk, but instead, some of the more sophisticated structural theory of inference that carne in fashion in the 1980s.
All these themes do, or should, occur in modern logic! Let’s take them up now one by one.”
From: Johann van Benthem, Is There Still Logic in Bolzano’s Key?, in: Edgar Morscher (ed.), Bernard Bolzanos Leistungen in Logik, Mathematik und Physik, Sankt Augustin: Academia Verlag, 1999, pp. 12-14.