Before Boole: Reform, Leibniz, and Bolzano

The emergence of nineteenth-century mathematical logic followed several earlier attempts to reform reasoning. Some sought a better method of discovery, some a clearer analysis of ideas and judgments, and some a symbolic language in which disputes could be calculated.

These projects did not all aim at the same result. Their differences help explain why Boole's algebra and Frege's logical language were significant without making every earlier author an incomplete version of either thinker.

Bacon and the method of inquiry

Francis Bacon's Novum Organum of 1620 criticized habits of inquiry that accepted familiar classifications and authorities too readily.

His account of idols identified recurring sources of error in human judgment. His proposed method sought a disciplined use of observations, comparisons, and exclusions in investigating nature.

The concern was the discovery and warrant of explanatory knowledge. A valid syllogism can transmit a mistaken starting assumption; formal correctness alone does not supply good scientific premises.

Bacon's project therefore belongs to the history of reasoning, while differing from the later construction of an algebraic calculus of consequence.

The historical comparison is more useful when it identifies that difference in aim rather than treating induction as a proposed replacement for every deductive inference.

Port-Royal and the organization of thought

Antoine Arnauld and Pierre Nicole's Logic, or the Art of Thinking, first published in 1662, became an influential presentation of logic in relation to ideas, judgment, reasoning, and method.

Its Cartesian setting gave clear conception and the disciplined organization of thought an important role. It also retained and revised elements of the syllogistic tradition.

The discussion of an idea's comprehension and extension distinguishes its included attributes from the things to which it applies.

For example, adding a condition to the concept of a triangle can narrow the range of objects under discussion. “Equilateral triangle” adds a requirement and applies to fewer figures than “triangle.”

The relationship is not a complete modern semantics, but it gives a useful account of how conceptual qualification affects classification.

Port-Royal's influence also shows why the history cannot jump from medieval textbooks directly to symbolic logic without considering changing educational and philosophical aims.

Leibniz's two connected ambitions

Gottfried Wilhelm Leibniz explored a universal characteristic: a symbolic language capable of representing concepts and their combinations with systematic precision.

He also pursued calculi of reasoning through which relationships among those representations could be determined by formal operations.

The language and the calculus perform different jobs. A notation must represent the intended content appropriately before a calculation over it can resolve a substantive question.

Leibniz's projects included combinatorial methods, analysis of concepts, arithmetic representations, and logical calculi. They were not one completed universal system.

His 1666 Dissertatio de arte combinatoria belongs to the early development; many more technical logical manuscripts remained unpublished during his lifetime.

A small conceptual calculation

Suppose a concept is represented by a collection of attributes. Let the concept of a square include the attributes quadrilateral, equal-sided, and right-angled.

Then the concept contains enough information to support the predication that a square is a quadrilateral.

At this schematic level, reasoning can be understood as checking a relationship among conceptual components.

The example also exposes a difficulty. Mathematical concepts are not always built from an agreed finite list of independent attributes, and relations between objects can resist a simple subject–predicate decomposition.

A successful universal language would need to solve those representational problems. A calculation cannot make them disappear merely by assigning symbols to the terms.

Leibniz's work anticipated important possibilities while leaving several different unfinished approaches. The history should preserve that exploratory character.

Publication and influence

Resemblance to a later system is not enough to establish influence.

A manuscript can anticipate an idea while remaining unavailable to the author who later developed a related one. Conversely, a less technically advanced published text can have a greater historical effect through teaching and circulation.

Louis Couturat's work at the beginning of the twentieth century helped make Leibniz's logical manuscripts more widely known. That later recovery shaped retrospective assessments of his place in logic.

The distinction between discovery, publication, reception, and influence is especially important when comparing Leibniz with Boole and other nineteenth-century authors.

Kant's historical judgment

Immanuel Kant famously described logic as having made little fundamental progress since Aristotle, within the account of general logic presented in his critical philosophy.

The claim reflects a particular conception of logic and its limits. Kant's general logic abstracts from the content of cognition, while his transcendental logic investigates different questions concerning the conditions of objective knowledge.

His assessment is evidence about how logic was understood in that setting. It should not be adopted as this series' verdict on the intervening Arabic, medieval, early-modern, Indian, or Chinese developments.

The preceding companions have shown why a broader history needs to examine the actual research questions and textual traditions rather than repeat a single retrospective judgment.

Bolzano's propositions in themselves

Bernard Bolzano's four-volume Wissenschaftslehre appeared in 1837. It developed a systematic account of propositions, truth, consequence, and the organization of scientific knowledge.

Bolzano distinguished propositions in themselves from spoken or written sentences and from particular acts of thinking.

A proposition's truth does not depend on whether anyone happens to assert or recognize it. This gives logical analysis an objective subject distinct from psychological episodes.

The position belongs to a larger philosophical theory. It should not be reduced to the claim that Bolzano merely invented a convenient notation for modern formulas.

His work in analysis and mathematical rigor is important, but the Wissenschaftslehre deserves treatment as a major logical project in its own right.

Variation and consequence

Bolzano investigates relationships among propositions by varying selected component ideas while keeping others fixed.

In a schematic example, consider:

Every $A$ is $B$. The object $c$ is $A$. Therefore $c$ is $B$.

Vary the ideas represented by $A$, $B$, and $c$ together, retaining the relevant logical form. Whenever the resulting premises are true, the resulting conclusion is true.

Bolzano's account of deducibility also includes conditions concerning the compatibility of the premises under the selected variations. It is not simply the unrestricted modern definition that makes every conclusion follow vacuously from an unsatisfiable premise set.

The choice of which ideas may vary is part of the analysis. Holding more content fixed can change which consequence relationship is being tested.

The scholarly account of Bolzano explains these distinctions and their relation to his broader theory.

A failed variation

Now consider:

The object $c$ is $A$. Therefore $c$ is $B$.

Choose an interpretation in which $A$ applies to $c$ and $B$ does not. For example, let $c$ be a red wooden block, $A$ mean red, and $B$ mean metal.

The premise is true and the conclusion false. This variation shows that the displayed pattern does not preserve truth.

Modern model theory uses formally specified structures and interpretations to conduct related tests. Bolzano's method concerns variation of ideas within his theory of propositions.

The resemblance is historically important, but identifying the two accounts without qualification would hide differences in ontology, admissible variation, and logical consequence.

Consequence and grounding

Bolzano also distinguished deducibility from objective grounding: the explanatory relationship by which some truths account for others.

Two propositions can be mutually derivable in a setting without being equally explanatory. A useful proof may show that a statement follows, while a better explanatory organization shows why it holds.

This concern connects with the older project of demonstrative science while taking a different systematic form.

The distinction also remains relevant to formalized mathematics. A checker can establish a derivation without deciding which derivation best explains the theorem to a reader.

Toward the modern series

Bolzano's reception was limited and uneven during his lifetime. His later importance should not be used to invent direct lines of influence where the documentary evidence is lacking.

The early-modern projects nevertheless clarify several questions inherited by the nineteenth century: how to represent logical form, how to calculate consequences, how to justify scientific starting points, and how to distinguish logical relations from psychological processes.

Boole's algebraic approach and Frege's analysis of quantification answered parts of this problem in distinct ways. Their innovations begin the main route of this series, while the earlier traditions retain their own historical importance.

Sources and further reading

• Francis Bacon, Novum Organum (1620).
• Antoine Arnauld and Pierre Nicole, Logic, or the Art of Thinking (1662 and subsequent editions).
• G. W. Leibniz, Dissertatio de arte combinatoria (1666) and logical manuscripts; Louis Couturat, La logique de Leibniz (1901) and Opuscules et fragments inédits de Leibniz (1903).
• Immanuel Kant, Critique of Pure Reason, preface to the second edition and discussions of general and transcendental logic (1787).
• Bernard Bolzano, Wissenschaftslehre (1837), translated as Theory of Science.
• Bernard Bolzano, Stanford Encyclopedia of Philosophy, for the theory of propositions, variation, grounding, and reception.