Logic on emsenn.net

Contradiction

Thu, 02 Apr 2026 00:00:00 +0000

In formal logic, contradiction typically means that two claims cannot be true together. In dialectical analysis, contradiction also names a structured tension internal to a form, where the conditions of that form’s reproduction generate forces that destabilize it.

Dialectical contradiction is not a permission to ignore rigor. It is a methodological signal: identify how incompatibility is produced, mediated, and historically transformed.

Determinate Negation

Thu, 02 Apr 2026 00:00:00 +0000

Determinate negation is negation with content. A form is not merely rejected; it is transformed according to its own internal limits, producing a successor form that preserves certain determinations while reconfiguring their role [@hegel-dialectics-sep-2019].

This distinguishes determinate negation from abstract negation:

Abstract negation: simple opposition (not-X).
Determinate negation: structured transition (X becomes re-specified through its contradiction).

The concept explains how dialectical development can involve real rupture without becoming arbitrary. What follows is constrained by what is negated.

Mediation

Thu, 02 Apr 2026 00:00:00 +0000

Mediation is the principle that determinations do not stand alone or move in isolation. They are connected through intermediate relations, processes, and forms.

In dialectics, mediation opposes immediacy. An immediate account says A produces B directly. A mediated account asks which structures, temporal phases, and relation chains connect A to B.

In social analysis, mediation helps distinguish structural explanation from narrative sequence. Saying “crisis produced reform” is incomplete unless the institutions, conflicts, and mechanisms that carried the transition are specified.

Sublation

Thu, 02 Apr 2026 00:00:00 +0000

Sublation (German: Aufhebung) names the dialectical movement in which something is negated without being simply destroyed. A prior form is cancelled as self-sufficient, preserved in transformed form, and raised into a richer determination [@hegel-dialectics-sep-2019].

Sublation is a key reason dialectical development is neither pure continuity nor pure rupture.

Abduction

Fri, 06 Mar 2026 00:00:00 +0000

Abduction is Charles Sanders Peirce’s name for the mode of inference that generates explanatory hypotheses. Given a surprising observation, abduction asks: what would make this unsurprising? It proposes a possible explanation — not as proven fact but as a conjecture worth investigating. Peirce also called it retroduction and hypothesis; in contemporary philosophy of science, the closest descendant is inference to the best explanation.

¶The three modes of inference

Peirce distinguished three modes of reasoning, each with a different logical form and a different role in inquiry:

Bivalence

Thu, 05 Mar 2026 00:00:00 +0000

Bivalence is the semantic thesis that every declarative statement is either true or false — there are exactly two truth values and every proposition receives exactly one of them. It is a foundational commitment of classical logic and the philosophical ground on which truth-functionality, truth tables, and model-theoretic semantics rest.

Bivalence should be distinguished from the law of excluded middle, though the two are often conflated. Excluded middle is a logical law: the formula A ∨ ¬A is valid. Bivalence is a semantic principle: every proposition has one of two truth values. In classical logic they are equivalent, but they come apart in other settings. A three-valued logic might validate excluded middle (by treating the third value as “designated”) while violating bivalence, or it might preserve bivalence in a restricted domain while rejecting excluded middle for a broader class of statements.

Buddhist Logic Overview

Thu, 05 Mar 2026 00:00:00 +0000

Audience: readers familiar with classical logic who want to understand a non-Western logical tradition that developed independently and reached different conclusions about truth, inference, and the limits of predication.

Learning goal: understand the structure of Buddhist formal inference, the catuṣkoṭi, and their relationship to Western non-classical logics.

¶A different starting point

Western logic, from Aristotle through Frege, begins with the structure of propositions and asks: which inferences preserve truth? Indian logic, in the pramāṇa tradition, begins with the structure of knowledge and asks: which cognitive processes produce valid cognition? This difference in starting point produces a logic that is epistemological from the ground up — not a formal system applied to knowledge but a theory of knowledge that generates formal constraints.

Classical Logic Overview

Thu, 05 Mar 2026 00:00:00 +0000

Audience: readers familiar with propositional logic who want to understand its philosophical commitments.

Learning goal: articulate the foundational principles of classical logic and their philosophical significance.

Classical logic is not merely a set of inference rules. It is a philosophical position about the nature of truth and reasoning. The technical apparatus — truth tables, natural deduction, model theory — rests on commitments that are substantive and contestable.

¶The core commitments

Classical logic rests on three interlocking principles:

Constructive Proof

Thu, 05 Mar 2026 00:00:00 +0000

A constructive proof is a proof that establishes the truth of a proposition by directly exhibiting the required mathematical object or method, rather than by showing that its non-existence leads to contradiction. In intuitionistic logic, all proofs are constructive: to prove that something exists, you must produce it; to prove a disjunction, you must prove one of the disjuncts and say which.

The requirement for constructive proof is formalized in the Brouwer-Heyting-Kolmogorov (BHK) interpretation, which defines the meaning of each logical connective in terms of proof obligations:

Decidability

Thu, 05 Mar 2026 00:00:00 +0000

A proposition P is decidable when there exists a constructive method for determining whether P holds or ¬P holds — that is, when P ∨ ¬P can be proved constructively. In intuitionistic logic, excluded middle does not hold in general, but it does hold for decidable propositions. Decidability marks the boundary between the territory where classical and intuitionistic logic agree and the territory where they diverge.

Finite domains are paradigmatically decidable. If P is a property of natural numbers and the domain is finite, then one can in principle check each element and determine whether P holds or fails. For infinite domains, decidability becomes the exception rather than the rule. The proposition “every even number greater than 2 is the sum of two primes” (Goldbach’s conjecture) is currently neither proved nor refuted. The intuitionist refuses to assert that it is true or false in advance of a proof or refutation; the classicist asserts this on the strength of bivalence alone.

Dialetheia

Thu, 05 Mar 2026 00:00:00 +0000

A dialetheia is a proposition that is both true and false. Dialetheism is the philosophical position that dialetheia exist — that some contradictions are not merely apparent or the result of error but are genuine features of reality or of our best theories about it.

The position is most associated with Graham Priest, whose In Contradiction (1987, 2nd ed. 2006) argues that the liar sentence (“this sentence is not true”) is a dialetheia: it is true if and only if it is not true, and the correct response is to accept that it is both. The liar paradox is not an anomaly to be dissolved by semantic restrictions (Tarski’s hierarchy) or pragmatic dismissal but a window into the structure of truth itself.

Explosion

Thu, 05 Mar 2026 00:00:00 +0000

Explosion (ex falso quodlibet, “from the false, anything follows”) is the principle of classical logic that from a contradiction, any proposition whatsoever can be derived. If both A and ¬A hold, then for any B, the following argument is valid: from A, derive A ∨ B by disjunction introduction; from ¬A and A ∨ B, derive B by disjunctive syllogism. This means that any inconsistent classical theory is trivial — it proves everything.

Intuitionistic Logic Overview

Thu, 05 Mar 2026 00:00:00 +0000

Audience: readers who have encountered the mathematical treatment of intuitionistic logic and want to understand its philosophical significance.

Learning goal: explain why intuitionistic logic is not merely classical logic with fewer axioms, but a fundamentally different conception of truth and proof.

¶The philosophical shift

Classical logic asks: is this proposition true or false? The answer is assumed to exist, independent of anyone’s knowledge. Intuitionistic logic asks: has this proposition been proved or refuted? The answer depends on the state of mathematical knowledge.

Paraconsistent Logic Overview

Thu, 05 Mar 2026 00:00:00 +0000

Audience: readers who have encountered the classical logic principle of explosion and want to understand why some logicians reject it.

Learning goal: explain the philosophical motivations for paraconsistent logic and its relationship to classical and intuitionistic traditions.

¶Why tolerate contradiction?

Classical logic treats contradiction as catastrophe: from A ∧ ¬A, anything follows. This principle — explosion — means that a single contradiction in a theory makes the theory trivial. Every proposition becomes a theorem, and the theory loses all discriminatory power.

Philosophical Foundations of Classical Logic

Thu, 05 Mar 2026 00:00:00 +0000

¶The realism implicit in classical logic

Classical logic encodes a form of realism about truth. When the logician writes a truth table for P ∨ ¬P and observes that it comes out true on every row, the implicit assumption is that every proposition has a row — that reality assigns a truth value to every well-formed claim, whether or not anyone has verified it. This is not a theorem of classical logic but a presupposition of its semantics.

Philosophical Motivations for Intuitionistic Logic

Thu, 05 Mar 2026 00:00:00 +0000

¶Brouwer’s original motivation

L. E. J. Brouwer’s intuitionism, developed from his 1907 doctoral thesis onward, rests on a radical claim about the nature of mathematics: mathematics is not the study of abstract objects or formal systems but a free constructive activity of the human mind. Mathematical objects exist only insofar as they are mentally constructed. Mathematical truths are truths about these constructions.

From this starting point, the law of excluded middle cannot be accepted as a universal principle. To assert P ∨ ¬P is to claim that one can either construct a proof of P or construct a refutation of P. For finite, surveyable domains this is unproblematic — one can check. For infinite domains, the claim outruns the capacity of any finite mind. Brouwer’s rejection of excluded middle is not skepticism about truth but a consequence of taking the constructive nature of mathematical activity seriously.

Truth-Functionality

Thu, 05 Mar 2026 00:00:00 +0000

A connective is truth-functional when the truth value of the compound it forms is determined entirely by the truth values of its components. Classical propositional logic is truth-functional throughout: knowing the truth values of A and B suffices to determine the truth values of A ∧ B, A ∨ B, A → B, and ¬A. This property is what makes truth tables possible and is the semantic backbone of classical propositional logic.

Untyped Lambda Calculus

Thu, 05 Mar 2026 00:00:00 +0000

¶Entry conditions

Use this lesson when you want to understand computation at its most fundamental level — before types, before programming languages, before machines. The lambda calculus is a formal system where every computation is expressed using only three constructs: variables, function definitions, and function application.

No prior knowledge of logic, type theory, or programming is assumed. If you have written functions in any programming language, the intuitions will be familiar.

Witness

Thu, 05 Mar 2026 00:00:00 +0000

A witness is a specific object that satisfies an existential claim. In intuitionistic logic, to prove ∃x.P(x) one must produce a particular object a together with a proof that P(a) holds. The object a is the witness. This contrasts with classical logic, where ∃x.P(x) can be proved indirectly — by assuming ∀x.¬P(x) and deriving a contradiction — without ever identifying the object in question.

The demand for witnesses is a direct consequence of the BHK interpretation: a proof of ∃x.P(x) is a pair (a, p) where a is an element of the domain and p is a proof that P(a). Without a, there is no proof, no matter how compelling the indirect argument.

Dependent Types

Sun, 01 Mar 2026 00:00:00 +0000

¶Entry conditions

Use dependent types when you need:

Types that vary according to a value — a vector whose type includes its length, a matrix whose type includes its dimensions, a proof whose type is the proposition proved.
A single framework where programs, specifications, and proofs coexist — where writing code and proving theorems are the same activity.
A formal language strong enough to state and verify mathematical theorems, not just check that functions return the right kind of output.

¶From simple types to dependent types

The simply typed lambda calculus assigns every term a type, and function types have the form $A \to B$ : the input has type $A$ , the output has type $B$ . The output type does not depend on the input value — a function of type $\mathbb{N} \to \mathbb{N}$ returns a natural number regardless of which natural number it receives.

Heyting Algebras

Sun, 01 Mar 2026 00:00:00 +0000

¶Entry conditions

Use Heyting algebras when you can:

Start from a lattice with top $\top$ and bottom $\bot$ .
Define an implication operation $a \Rightarrow b$ for every pair of elements.
Accept that some elements may lack complements — excluded middle may fail.

¶Definitions

A bounded lattice is a lattice with a greatest element $\top$ and a least element $\bot$ .

A Heyting algebra [@esakia_HeytingAlgebras_2019] is a bounded lattice $(H, \leq, \wedge, \vee, \bot, \top)$ equipped with a binary operation $\Rightarrow$ (called Heyting implication) satisfying:

Intuitionistic Logic

Sun, 01 Mar 2026 00:00:00 +0000

¶Entry conditions

Use intuitionistic logic when:

You need to reason about evidence, construction, or partial information rather than absolute truth and falsity.
You want logical operations that have computational content — where a proof of “there exists an $x$ ” must exhibit the $x$ .
You are working in a Heyting algebra rather than a Boolean algebra.

¶The key difference from classical logic

Classical propositional logic assumes the law of excluded middle: for every proposition $P$ , either $P$ or $\neg P$ holds. Intuitionistic logic drops this assumption. A proposition is true when you have a construction (a proof, a witness, an exhibit) that establishes it. A proposition is false when you have a construction showing it leads to contradiction. A proposition for which you have neither is simply undecided — it is not forced into one bin or the other.

Typed Lambda Calculus

Sun, 01 Mar 2026 00:00:00 +0000

¶Entry conditions

Use typed lambda calculus when you need:

A formal language for defining functions, applying them to arguments, and composing them.
A type system that prevents ill-formed combinations (applying a number to a number, for instance).
A bridge between logic and computation, where types correspond to propositions and terms correspond to proofs.

¶The untyped lambda calculus

The untyped lambda calculus begins with three operations [@barendregt_LambdaCalculus_1984]:

Logic on emsenn.net

Contradiction

¶Related terms

Determinate Negation

Mediation

Sublation

¶Related terms

Abduction

¶The three modes of inference

Bivalence

Buddhist Logic Overview

¶A different starting point

Classical Logic Overview

¶The core commitments

Constructive Proof

Decidability

Dialetheia

Explosion

Intuitionistic Logic Overview

¶The philosophical shift

Paraconsistent Logic Overview

¶Why tolerate contradiction?

Philosophical Foundations of Classical Logic

¶The realism implicit in classical logic

Philosophical Motivations for Intuitionistic Logic

¶Brouwer’s original motivation

Truth-Functionality

Untyped Lambda Calculus

¶Entry conditions

Witness

Dependent Types

¶Entry conditions

¶From simple types to dependent types

Heyting Algebras

¶Entry conditions

¶Definitions

Intuitionistic Logic

¶Entry conditions

¶The key difference from classical logic

Typed Lambda Calculus

¶Entry conditions

¶The untyped lambda calculus