The reckoning

Reasoning

ratiocinatio · ‘the act of reckoning, of reasoning’

The operation that turns thoughts and observations into beliefs. Logic is the discipline that studies it. What follows is a map of the field, then a small set of reference pages.

The map

Three axes, not one hierarchy

Most explanations of logic conflate three independent classifications. Once they are separated, the territory resolves. A concept has a position on all three axes at once — propositional logic is Inference · Formal · Deductive; a Bayesian update is Inference · Formal · Inductive.

ObjectReasoningArgumentInferenceMethodInformalFormalModeDeductiveInductiveAbductive
  1. 01Axis

    Object

    What is being studied. Reasoning is the cognitive act; argument is its linguistic expression; inference is the move from premise to conclusion within it.

    • Reasoningthe cognitive activity of moving from one judgement to another.
    • Argumenta set of premises offered in support of a conclusion.
    • Inferencethe individual move from premises to conclusion within an argument.
  2. 02Axis

    Method

    How reasoning is analysed. Formal logic considers the structure of arguments and ignores their content; informal logic examines arguments as they appear in natural language.

    • Formalvalidity is decided by structure alone — symbols, connectives, quantifiers.
    • Informalvalidity depends on content, context, and the relationship between the parties to the argument.
  3. 03Axis

    Mode

    The character of the inferential move. Deduction preserves necessity; induction extends from observed cases; abduction selects the explanation that best accounts for the data.

    • Deductivethe conclusion follows from the premises with necessity. Truth-preserving.
    • Inductivethe conclusion is supported by the premises but not entailed. Ampliative.
    • Abductivethe conclusion is the best available explanation of the premises. Selective.

Worked examples

Reading positions in the space

Each concept is given by its three coordinates. Concepts that differ on one axis can share the other two — which is what makes them independent dimensions rather than levels of a single hierarchy.

A propositional-logic proof
InferenceFormalDeductive
Hume's sunrise inference
InferenceInformalInductive
Bayesian updating from evidence
InferenceFormalInductive
A Toulmin-model analysis
ArgumentInformalDeductive
Medical diagnosis from a symptom set
InferenceInformalAbductive

The territory

Where each subject lives

Rules of inference live inside the formal system they belong to; fallacies live in informal logic; truth is the semantic anchor that all the rest hangs on. Each sub-page below is a reference, not an essay.

  1. I

    Argument

    The anatomy of an argument — premise, conclusion, inferential move; validity, soundness, cogency; the enthymeme and the Toulmin model.

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  2. II

    Modes

    Deductive, inductive, abductive. The character of an inference, independent of how it is presented. The probability calculus as the formal articulation of induction.

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  3. III

    Truth

    What premises and conclusions can have. Correspondence, coherence, pragmatic, deflationary, and semantic theories — and how truth functions inside formal systems.

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  4. IV

    Formal logic

    Propositional, predicate, modal, and non-classical systems. Rules of inference, logical equivalences, truth-tables, semantics.

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  5. V

    Informal logic

    Argumentation in natural language. The Toulmin model, dialectic and rhetoric, and the catalogue of fallacies — formal and informal.

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