Established MMXXVI · Volume I

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論理

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The Logic Foundation

An institution dedicated to the cultivation of clear reasoning, the study of formal systems, and the public stewardship of logical inquiry.

A scholarly imprint — founded in the spirit of the open library

Bound at the Foundation Press · Reading Room III

No. I

A Journal of Reason & Inference

Issued quarterly by the Logic Foundation, comprising essays, definitions, and marginalia for the careful reader.

Article I · pp. 1–14

On the Foundations of Logic

By the Editors of nonri.org · The Foundation Press

From Aristotle's syllogisms to the predicate calculus of Frege and Russell, the discipline of logic provides the scaffolding upon which all rational inquiry is constructed. To organise thought is, in a quiet and exacting sense, to organise the world — for the patterns of valid argument are the patterns by which we distinguish what follows from what merely seems to follow.

The foundations are not laid in haste. They are inherited: from the porticos of the Lyceum, from the marginalia of mediaeval schoolmen, from the slow accretion of definitions written in copperplate by candlelight. Each generation refines the inheritance, returning to the ancient questions with sharper instruments and more patient eyes.

A foundation of logic asks first what an argument is, and only then what makes one valid. The order matters. Many an honest reasoner has begun with a verdict and constructed premises afterwards; the foundation demands the opposite ascent — from datum to inference, from inference to conclusion, with every joint visible to the reader.

Article II · pp. 15–28

The Structure of Arguments

A Treatise on Premises, Inference, and Conclusion

Every argument possesses architecture. Premises support conclusions in the way that columns support a roof, or fluted piers a vaulted ceiling; when the structure is sound, truth descends as a matter of necessity, not of chance. The student of logic learns first to read these structures, and only afterwards to construct them.

Consider the modest modus ponens: if P implies Q, and P is the case, then Q follows. The whole edifice rests on three pieces — an implication, an antecedent, and a permission to conclude. Strip away ornament and this remains. It is the load-bearing wall of countless arguments, classical and contemporary alike.

Yet structure is not sufficient. An argument may be valid in form and unsound in matter; the columns may stand though the bricks be rotten. The careful reader inspects both. She asks of each premise: is this so? And of the inference: does the conclusion follow? Only when both questions earn their yes may the argument be admitted to the record.

Logic is not the art of being right, but the art of being shown to be right when one is.

— Marginalia of an unknown editor, ca. 1907

Three classical forms repay study. The syllogism binds three terms across two premises. The conditional proposes a contingent relation. The reductio assumes the contrary and lets contradiction do the work. Each is a method — and a discipline. To master them is to acquire a draftsman's hand for the architecture of thought.

Article III · pp. 29–42

Formal Systems & the Skeleton of Reason

An Essay on Symbols, Rules, and the Pursuit of Precision

Language alone is imprecise. The natural tongues, however supple and beautiful, carry within them the residue of a thousand contexts, ironies, and ancestral half-meanings. Formal logic strips these away. It leaves only the skeleton of reason — pure, abstract, and, within its proper sphere, universally valid.

A formal system has three parts: a lexicon of symbols, a grammar of well-formed expressions, and a body of rules of inference by which new theorems are derived from old. The whole is a closed garden of reasoning, austere and self-consistent — a machinery of thought that runs without ambiguity and, when properly tended, without error.

Yet austerity is not poverty. Within the formal garden grow theorems of arresting beauty: Gödel's incompleteness, the soundness and completeness of first-order logic, the Curry–Howard correspondence between proofs and programmes. Each is a small monument to what reason can achieve when granted the silence to work.

  • ∀ universal quantifier — for all
  • ∃ existential quantifier — there exists
  • → material conditional — if … then
  • ∧ conjunction — and
  • ∨ disjunction — or
  • ¬ negation — not
  • ∴ therefore — the descent of conclusion

The therefore symbol — three points arranged as a triangle — is the silent hinge of every demonstration. It marks the moment at which the reasoner declines to argue further and lets the conclusion stand on the strength of what came before. We have adopted it as the printer's flower of this volume.

Article IV · pp. 43–56

Applied Reasoning — Logic in the World

A Closing Address to the Reader

Logic is not merely academic, nor the private possession of philosophers. It is the engine of the law, the instrument of the sciences, the substrate of computation, and — perhaps above all — the quiet conscience of ethics. To think clearly, in this view, is itself a small public service; to act on clear thought, a kind of justice.

In the courtroom, the reasoned argument distinguishes evidence from rumour, motive from caprice. In the laboratory, it tells the experimenter what would count as a refutation and what would not. In the engine of computation, it is the very air the machine breathes — for every algorithm is a syllogism in motion. In the council chamber, finally, it asks of each policy: does the conclusion truly follow?

Yet applied reasoning has its limits. Not every disagreement is a disagreement of logic; some are disagreements of value, taste, or conviction, and these the syllogism cannot adjudicate. The wise applier knows where the tools end. She does not press logic into the service of what logic cannot decide — for to do so is to disgrace a noble instrument.

It remains, then, to commend the discipline to the reader. Let her keep a notebook and a sharp pencil; let her rule the margins; let her write down the premises before the conclusion and read them aloud, slowly, and ask of each: is this so? If she does, the rest will follow — in the small, irrefutable way that all good things at last follow.

— The Editors

at the Reading Room, Foundation Press

Appendix α

A Short Index of Definitions

For the convenience of the reader who consults this volume in fragments rather than reading it through. Page numbers refer to the present edition.

Argument
A set of statements, one of which (the conclusion) is alleged to follow from the others (the premises). p. 4
Inference
The mental movement from premise to conclusion. The act, not the record, of reasoning. p. 7
Validity
A property of arguments: the conclusion follows necessarily from the premises, irrespective of their truth. p. 16
Soundness
Validity together with the truth of every premise. The double virtue of arguments. p. 18
Tautology
A proposition true under every assignment of truth-values to its components. p. 31
Contradiction
A proposition false under every assignment. The negation of a tautology. p. 33
Quantifier
A symbol expressing the extent of a claim — universal (∀) or existential (∃). p. 36
Therefore (∴)
The mark of conclusion. Read aloud as therefore; written as a triangle of three points. p. 41