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Extensionality

Extensionality is a foundational principle in mathematics, logic, and philosophy that equates entities based on their extensions—the collections of objects they contain or denote—rather than their intensions or modes of presentation. In set theory, it asserts that two sets are identical if and only if they share the same elements, ensuring that sets are uniquely determined by their membership.^[1] This axiom, often formalized as ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y), underpins the Zermelo–Fraenkel axioms and eliminates redundant distinctions between sets with identical contents.^[1]^[2] In logic, extensionality distinguishes contexts where substitution of co-referential terms preserves truth value and where existential generalization holds, contrasting with intensional contexts like belief reports where such substitutions may fail.^[3] For example, in the extensional sentence "Sandra threw the ball to the boy across the street," replacing "Sandra" with a co-referential term like "Suzanne’s older sister" maintains truth, but in the intensional "Bruce believes that Apollo is an admirable god," substituting for "Apollo" (e.g., with a non-referring term) can alter truth value.^[3] This property ensures classical logic's adherence to referential transparency in non-modal settings.^[3] Philosophically, extensionality supports a semantics focused on reference and denotation, influencing debates on meaning and identity, such as Frege's distinction between sense and reference, while extensional approaches prioritize empirical or structural equivalence over conceptual differences.^[2] In type theory and computer science, variants like function extensionality equate functions if they agree on all inputs, aiding formal verification and programming languages.^[2] Overall, extensionality promotes a uniform treatment of equality across domains, facilitating rigorous reasoning while challenging analyses of opaque contexts.

Philosophical and Logical Foundations

Core Definition and Principles

Extensionality is a foundational principle in philosophy and logic that holds two expressions or predicates are equivalent if they possess the same extension, meaning they apply to or denote precisely the same set of objects.^[4] This equivalence relies on the extension—the collection of entities satisfying the predicate—rather than any intrinsic conceptual differences between the expressions.^[4] In extensional contexts, the semantic evaluation of statements depends solely on these references, ensuring that the truth value remains unaffected by variations in how the extensions are described.^[4] The historical roots of extensionality trace to Aristotelian logic in the fourth century BCE, where syllogistic inferences treat terms extensionally, focusing on the classes of individuals they encompass rather than their definitions or essences.^[5] This approach laid early groundwork for viewing logical relations through membership in such classes.^[5] In the late 17th century, Gottfried Wilhelm Leibniz formulated the closely related principle of the identity of indiscernibles, asserting that two distinct objects cannot share all properties, thereby linking numerical identity to complete qualitative overlap in a manner aligned with extensional determination.^[6] Gottlob Frege advanced the concept significantly in his 1892 essay "On Sense and Reference," introducing the distinction between an expression's sense (its mode of presentation or intension) and its reference (its extension or denotation).^[7] Frege argued that while senses may differ, co-referring terms share the same extension, which governs truth in referential (extensional) settings.^[8] At its core, the logical principle of extensionality can be expressed as follows: if two predicates P and Q are satisfied by exactly the same objects, then they define identical extensions.

\forall x \, (P(x) \leftrightarrow Q(x)) \implies \{ x \mid P(x) \} = \{ x \mid Q(x) \}

This biconditional ensures that equality between extensions arises purely from shared membership or satisfaction, abstracting away from any differing conceptual or intensional content. Philosophically, extensionality emphasizes substitutivity: in such contexts, replacing one term with another that has the same reference preserves the sentence's truth value, promoting a referential semantics over one driven by meaning alone.^[8]

Extensional Versus Intensional Contexts

Intensional contexts are linguistic or logical environments in which the substitution of co-referring terms does not preserve the truth value of the sentence, a phenomenon known as referential opacity or failure of substitutivity salva veritate.^[9] This contrasts with extensional contexts, where such substitutions do preserve truth values, allowing terms to be replaced by others with the same referent without altering the sentence's truth.^[10] Gottlob Frege introduced this distinction in his analysis of sense and reference, arguing that while proper names like "Hesperus" and "Phosphorus"—both referring to the planet Venus—can be substituted in extensional contexts (e.g., "Hesperus is a planet" remains true when replaced with "Phosphorus is a planet"), they cannot in intensional contexts such as belief reports: "The ancients believed Hesperus was a god" is true, but "The ancients believed Phosphorus was a god" may be false if the ancients did not know the two names referred to the same object.^[9] Key examples of intensional contexts include modal operators and propositional attitudes. In modal logic, necessity (denoted □) creates opacity: if a = b, then □P(a) does not necessarily imply □P(b), as seen in cases where two entities are identical but their properties hold necessarily only under specific descriptions, for example, letting n be the number of planets in the solar system (currently 8), the sentence "9 > 8" is necessarily true, but "9 > n" is not, since n is contingent.^[11] Similarly, propositional attitudes like "knows" or "believes" exhibit failure of substitutivity: "John knows that 9 = 9" is true, but "John knows that 9 = √81" may be false if John lacks knowledge of the square root equivalence, despite the mathematical identity.^[10] Philosophical debates on intensionality intensified with W.V.O. Quine's critique in the 1950s, where he argued that intensional constructions, particularly those involving propositional attitudes, resist regimentation into extensional predicate logic due to their referential opacity, labeling them as semantically obscure and challenging the clarity of analytic philosophy.^[10] Responses emerged in the 1970s through Saul Kripke's possible worlds semantics in Naming and Necessity, which treated names as rigid designators and modalities as evaluating across counterfactual worlds, thereby accommodating some intensional phenomena by distinguishing necessary identities from a posteriori discoveries, though this framework still collapses certain distinctions.^[11] This led to discussions of hyperintensionality, where even necessarily equivalent contents differ in contexts like attitudes or conditionals, extending beyond possible worlds semantics by requiring finer-grained distinctions, such as impossible worlds or structured propositions, to capture attitudes toward synonymous but non-identical representations.^[12] Criteria for extensionality center on truth-value preservation under substitution: a context is extensional if, for any sentence S(t) containing term t, and any t' with the same extension as t (i.e., referring to the same entity), S(t) and S(t') have identical truth values across all interpretations.^[13] Extensional languages, such as first-order predicate logic without modal or attitudinal operators, satisfy this by treating terms purely by their denotations, ensuring that equivalence in reference guarantees substitutivity without regard to sense or mode of presentation.^[9]

Mathematical Formulations

Axiom of Extensionality in Set Theory

The axiom of extensionality is formally stated as

\forall x \, \forall y \left( \forall z (z \in x \leftrightarrow z \in y) \to x = y \right).

This formulation asserts that two sets are equal precisely when they share all members, ensuring that sets are uniquely identified by their membership relation alone, without regard to internal structure or order.^[14] To derive this, suppose two sets x and y satisfy the antecedent \forall z (z \in x \leftrightarrow z \in y); then any property definable via membership holds equally for x and y, yielding x = y under the equality predicate of first-order logic.^[15] In Zermelo-Fraenkel set theory with the axiom of choice (ZFC), the axiom of extensionality serves as the foundational principle, typically listed first among the axioms, by which sets are defined extensionally through their elements, thereby excluding distinct but membership-indistinguishable objects.^[16] It establishes that equality of sets is governed solely by the \in relation, preventing paradoxes arising from non-unique representations in naive set theory.^[14] Historically, Ernst Zermelo introduced this axiom in his 1908 paper as part of the first axiomatic system for set theory, motivated by the need to resolve foundational crises like Russell's paradox while enabling proofs such as the well-ordering theorem. Key implications include the uniqueness of the empty set \emptyset, since any set lacking elements satisfies the same membership condition as \emptyset, forcing equality by extensionality; a direct proof proceeds by assuming two such sets x and y, noting \forall z (z \notin x \land z \notin y), and applying the axiom.^[14] Similarly, for any set x, its power set \mathcal{P}(x) is unique, as distinct candidates would differ in membership. The axiom interacts with others, such as union (which collects elements into a single set uniquely by extensionality) and replacement (which maps elements while preserving unique outputs), to construct the cumulative hierarchy V = \bigcup_{\alpha \in \mathrm{Ord}} V_\alpha, where each V_{\alpha+1} = \mathcal{P}(V_\alpha) yields sets determined extensionally from prior levels, forming the universe of all sets.^[16] Variations on extensional set theory include non-extensional approaches, which omit or modify the axiom to allow sets with identical members yet distinct identities, often motivated by applications requiring multiplicity (e.g., multisets) or indistinguishable urelements; for instance, in multiset theory, the axiom is replaced by one accounting for element cardinalities, enabling non-unique "bags" while avoiding collapse to standard sets.^[17] Hyperset theory, as formalized by Peter Aczel, preserves the standard axiom of extensionality but relaxes well-foundedness to admit infinite descending membership chains (e.g., sets like x = \{x\}), motivated by modeling circular structures in logic and computer science; unlike ZFC, it uses the anti-foundation axiom to ensure unique decorations of graphs, maintaining extensional equality amid non-well-foundedness.^[18] In contrast, structural set theories like the Elementary Theory of the Category of Sets (ETCS) recast extensionality structurally via functions—stating that two functions f, g: A \to B are equal if \forall a \in A, f(a) = g(a)—prioritizing relational isomorphisms over material membership; this differs from ZFC's element-focused version by treating sets as points in a category without intrinsic \in, facilitating categorical foundations while achieving equivalent expressive power for well-founded sets.

Extensionality in Predicate and Higher-Order Logic

In predicate logic, extensionality is formalized through the principle of the indiscernibility of identicals, also known as Leibniz's law, which states that two objects are identical if and only if they share all properties: \forall P\ (P(a) \leftrightarrow P(b)) \leftrightarrow a = b. This bidirectional equivalence ensures that identity is fully extensional, meaning objects are indistinguishable precisely when no predicate differentiates them. To prove the equivalence, assume a = b; then by substitution, any predicate P true of a holds for b and vice versa, yielding P(a) \leftrightarrow P(b) for all P. Conversely, if \forall P\ (P(a) \leftrightarrow P(b)), then in particular for the predicate P(x) \equiv (x = a), we have P(a) \leftrightarrow P(b), so a = a \leftrightarrow b = a, and since a = a, it follows that b = a. In first-order theories, this principle underpins the semantics of equality, allowing models to interpret predicates extensionally and ensuring that logical consequence respects indiscernibility, as in the definition of structures where relations are sets of tuples without intensional distinctions. In higher-order logic, extensionality extends to functions and relations, treating them as identical based on their behavior across domains. Function extensionality asserts that two functions f and g of the same type are equal if they agree on all inputs: \forall x_{\beta}\ (f_{\alpha \beta}\ x_{\beta} = g_{\alpha \beta}\ x_{\beta}) \supset f_{\alpha \beta} = g_{\alpha \beta}.^[19] Relation extensionality follows similarly, as relations are encoded as functions to truth values, equating two relations if they hold for exactly the same arguments: \forall x_{\beta}\forall y_{\gamma}\ (R_{\beta\gamma}\ x_{\beta}\ y_{\gamma} \leftrightarrow S_{\beta\gamma}\ x_{\beta}\ y_{\gamma}) \supset R_{\beta\gamma} = S_{\beta\gamma}.^[19] These principles are axiomatized in Alonzo Church's simple type theory (1940), where axioms of extensionality (such as 10^{\alpha\beta}) identify entities by their extensions, preventing non-extensional collapses while maintaining type safety; for instance, propositional extensionality equates logically equivalent formulas as [x_{o} \equiv y_{o}] \supset x_{o} = y_{o}. A key logical consequence of extensionality in these systems is the elimination of non-extensional predicates, where predicates are reduced to their extensions—sets of objects they apply to—ensuring no two distinct predicates share the same extension without being identical.^[19] This facilitates model theory's isomorphism theorems, as extensional interpretations guarantee that isomorphic structures preserve all relations and functions, hence satisfying the same first-order sentences; specifically, an isomorphism f: \mathcal{M} \to \mathcal{N} maps elements such that for any relation R in \mathcal{M}, R^M(a_1, \dots, a_k) holds if and only if R^N(f(a_1), \dots, f(a_k)) holds, making \mathcal{M} and \mathcal{N} elementarily equivalent. Historically, Bertrand Russell advanced extensionality in his 1903 The Principles of Mathematics, where he reduced classes to the extensions of predicates, arguing that classes arise extensionally from propositional functions such that identical extensions imply identical classes, avoiding paradoxes by treating classes as incomplete symbols.^[20] This approach was refined in Principia Mathematica (1910–1913) with Whitehead, using contextual definitions like x \in \hat{z} \phi z \equiv \phi x to eliminate class terms in favor of extensional predicates.[]https://plato.stanford.edu/entries/pm-notation/) Intuitionism, as developed by L. E. J. Brouwer, critiqued this extensional reduction for assuming non-constructive existence in infinite domains, favoring intensional constructions that reject impredicative definitions and the full extensional identification of predicates without explicit mental proofs.

Applications and Extensions

In Type Theory and Category Theory

In type theory, extensionality manifests through variants of Martin-Löf's intuitionistic type theory, where equality is treated as a primitive judgment rather than a constructed type. In extensional type theory, introduced by Martin-Löf in the 1970s and formalized in his 1982 work, identity proofs are unique, and the reflection rule equates propositional equality with judgmental equality: if p : Id_A(x, y), then x \equiv y : A.^[21] This collapses the distinction between intensional and extensional equality, making identity types mere propositions with at most one inhabitant, but it renders type-checking undecidable due to the potential non-existence of normal forms.^[22] In contrast, intensional type theory, as in Martin-Löf's 1972 notes, maintains multiple proofs for identities and decidable judgments, preserving computational content without enforcing extensional collapse.^[21] Quotient types provide a mechanism for extensionality in such systems by redefining equality via user-specified equivalence relations. In extensional type theory extensions like quotient Minimal Type Theory (qmTT), a quotient Q of a set A by relation R forms a new type where elements are equal if related by R, with rules ensuring reflexivity, symmetry, and transitivity while collapsing propositions to proof-irrelevant monos.^[23] This supports extensional collapse by interpreting quotients in setoid models, balancing constructivity with equality coercion, as in Maietti's 2005 framework for two-level type systems.^[23] In category theory, extensionality equates objects up to isomorphism, emphasizing structural equivalence over strict identity. Objects A and B are considered equal if they share the same hom-sets, formalized by the Yoneda lemma: for a locally small category \mathcal{C}, the natural isomorphism \hom_{[ \mathcal{C}^{op}, \Set ]}(y(c), X) \cong X(c) implies that isomorphic representables y(c) \cong y(d) yield isomorphic objects c \cong d, determining objects extensionally via their morphisms.^[24] Skeletal categories embody this reduction, where isomorphic objects are identical, serving as equivalence-preserving subcategories that eliminate redundant isomorphisms without altering categorical structure.^[25] Function extensionality in λ-calculi aligns these ideas by equating functions based on input-output behavior, as in the η-conversion rule: \lambda x. M x \equiv M if x is not free in M.^[26] In typed variants like PCF, extensional equality extends this to computable functions, treating them as continuous maps in cpos where equivalence holds if outputs match for all inputs, though undecidable in full generality.^[26] This contrasts with intensional models like homotopy type theory (HoTT), developed in 2013, where identity types form higher groupoids with path induction and univalence—equating type identity with equivalence (A = B) \simeq (A \simeq B)—rejecting extensional reflection to preserve homotopy levels and multiple proofs.^[27] Recent work has revisited the relationship between these paradigms, proving Morita equivalence between extensional type theory and intensional type theory extended by functional extensionality and uniqueness of identity proofs, bridging their logical structures.^[28] Categorical interconnections arise in topos theory, where limits preserve extensionality by maintaining equalities in subobjects. Every elementary topos, like

\Set

, has finite limits (products, pullbacks) that respect isomorphisms, and its extensivity ensures coproducts preserve equalities, as in Giraud's theorem for Grothendieck toposes.^[29] For instance, fiber products in a topos \mathcal{E} equate morphisms extensionally, mirroring set-theoretic behavior while supporting sheaf models of type theories.^[29]

In Computer Science and Linguistics

In functional programming languages such as Haskell, extensional equality for functions stipulates that two functions are equivalent if they yield identical outputs for all possible inputs, focusing on behavioral equivalence rather than internal implementation.^[30] This principle underpins verified generic programming, where higher-order functions preserve extensional equality of their arguments, ensuring that transformations like mapping over data structures maintain equivalence based on element content rather than structural details.^[31] For data types like lists or trees, while standard equality operators are structural (order-sensitive), extensional variants treat structures as equivalent if they contain the same elements, ignoring representation differences to support reasoning about program correctness.^[32] Denotational semantics in computer science formalizes this extensional view by mapping programs to mathematical objects, such as functions from inputs to outputs, where two programs are equal precisely if their denotations coincide on all inputs.^[33] This approach, foundational since the 1970s, enables equational reasoning about program behavior, treating computational artifacts extensionally to abstract away from intensional details like execution steps.^[33] In linguistics, Montague grammar from the 1970s employs extensional semantics to achieve compositionality, assigning denotations to linguistic expressions as sets: noun phrases denote sets of entities satisfying their descriptive content, while sentences denote truth values in a model.^[34] This set-theoretic interpretation allows systematic combination of meanings, mirroring syntactic structure to derive sentence truth conditions from constituent extensions.^[34] However, extensionality encounters limitations with intensional verbs like "seek," where the semantics requires reference to concepts or intentions beyond mere entity sets, prompting extensions to intensional logics.^[35] Practical applications include database query optimization, where extensional equivalence enables rewriting SQL expressions—such as reordering joins or pushing selections—into forms that denote the same relational extension but reduce computational cost.^[36] In AI-driven natural language processing, extensional models support entailment recognition by evaluating whether one text's denotation (e.g., a set of entailed propositions) includes another's, as in few-shot classification tasks using extensional definitions to refine semantic alignment.^[37] Modern extensions in machine learning approximate extensional equivalence for probabilistic models, providing denotational semantics where higher-order probabilistic programs are equal if they induce identical distributions over observations, facilitating optimization in probabilistic programming languages developed post-2010.^[38]

Examples and Case Studies

Illustrative Examples

A classic illustration of extensionality in philosophy arises from Gottlob Frege's distinction between the sense (Sinn) and reference (Bedeutung) of expressions. The proper names "the Morning Star" and "the Evening Star" both refer to the planet Venus, making them extensionally identical since they share the same referent in the actual world. However, they differ in sense, as "Morning Star" evokes the idea of the bright object seen at dawn, while "Evening Star" suggests the one visible at dusk; this difference explains why the identity statement "The Morning Star is the Evening Star" conveys new information, unlike the tautological "The Morning Star is the Morning Star." Frege uses this example to argue that extensional equivalence concerns only references, not the cognitive content or modes of presentation provided by senses.^[9] In mathematics, particularly set theory, extensionality equates sets based solely on their members, regardless of presentation order or notation. Consider the collections denoted as {1, 2} and {2, 1}: both comprise exactly the elements 1 and 2. To confirm equality under extensionality, verify membership step by step—1 belongs to {1, 2} (as the first listed element) and to {2, 1} (as the second); 2 belongs to {1, 2} (as the second) and to {2, 1} (as the first); no other numbers belong to either. Thus, since every element of one is an element of the other and vice versa, the sets are identical.^[39] Linguistic applications of extensionality emphasize substitutivity of expressions with the same denotation in truth-preserving contexts. The definite description "the set of even primes" denotes {2}, as 2 is the sole even prime number—any larger even number is divisible by 2 and hence composite.^[40] This set is extensionally identical to "the singleton of 2," also {2}, because both have precisely the member 2. In a sentence such as "The set of even primes has one element," replacing the description with "the singleton of 2" yields "The singleton of 2 has one element," which remains true, demonstrating how extensional equality allows seamless substitution without altering semantic value.^[39] In computer science, extensionality treats functions as equal if they map every input to the same output, focusing on observable behavior rather than internal implementation. For example, define f(x) = x² and g(x) = x × x over the real numbers; these are extensionally equal because, for any x, the multiplication yields the square. Verification via input-output pairs confirms this: f(0) = 0 and g(0) = 0 × 0 = 0; f(3) = 9 and g(3) = 3 × 3 = 9; f(-2) = 4 and g(-2) = (-2) × (-2) = 4. This equivalence holds universally, as the definitions compute identically for all inputs.^[41]

Limitations and Counterexamples

In philosophical contexts, extensionality fails in belief attitudes due to referential opacity, where substitutivity of coreferential terms does not preserve truth. A classic counterexample involves Lois Lane, who believes that Superman can fly but does not believe that Clark Kent can fly, despite Superman and Clark Kent being the same individual.^[42] This opacity arises because belief contexts are sensitive to modes of presentation rather than mere extensions, violating the principle that coextensive expressions are interchangeable. In non-wellfounded set theories, the axiom of extensionality requires modification to accommodate circular structures, such as sets that contain themselves (e.g., Ω = {Ω}). The Anti-Foundation Axiom (AFA), introduced by Forti and Honsell, replaces traditional extensionality with bisimulation equivalence, where sets are considered equal if their graph representations are bisimilar, allowing non-wellfounded sets while preserving a form of structural identity.^[43] This adaptation fails under the standard ZFC extensionality, as infinite descending membership chains violate well-foundedness without ensuring unique extensional equality.^[43] Intensional type theories, such as Homotopy Type Theory (HoTT), treat equalities as paths rather than extensional propositions, leading to non-extensional behavior where isomorphic types may not be judgmentally equal. In HoTT, identity types represent paths in a space-like structure, allowing multiple proofs of equality that are not collapsed, contrasting with extensional type theories where equality reflection enforces propositional and judgmental equality to coincide. This path-based equality introduces higher-dimensional distinctions, failing strict extensionality by distinguishing between transportable equalities.^[27] In category theory, non-skeletal categories illustrate limitations of extensionality, as isomorphic objects are not required to be equal, permitting multiple distinct objects with the same universal properties. For instance, the category of sets is non-skeletal, where two sets can be isomorphic (bijectively equivalent) without being identical, unlike skeletal categories that quotient by isomorphisms to enforce equality.^[25] This structure highlights how extensional identification via isomorphisms does not imply strict equality, complicating applications where object identity matters beyond structural similarity.^[44] To address these limitations, post-2000 developments in hyperintensional semantics have introduced finer-grained distinctions beyond extensional and possible-worlds approaches, such as impossible worlds and structured propositions. For example, Berto and Jago propose impossible worlds semantics to differentiate necessarily equivalent contents in belief reports, resolving opacity issues like the Lois Lane case by allowing non-normal worlds where coreferents behave differently.^[45] Similarly, paraconsistent logics respond by tolerating inconsistencies without explosion, incorporating intensional negations in relevant variants that avoid extensional substitutivity failures in belief revision contexts.^[46] These alternatives, including Yablo's aboutness-based propositions, enable handling of hyperintensional phenomena while mitigating the breakdowns in standard extensional frameworks.^[45]

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Apr 13, 2021 · We describe a mathematical structure that can give extensional denotational semantics to higher-order probabilistic programs.Missing: extensionality | Show results with:extensionality
[38]
[PDF] Section 1. The Axiom of Extension
Jan 2, 2023 · Axiom of Extension. Two sets are equal if and only if they have the same elements. Note. Halmos illustrates the Axiom of Extension by ...
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2 -- from Wolfram MathWorld
The number two (2) is the second positive integer and the first prime number. It is even, and is the only even prime (the primes other than 2 are called the ...
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The Lambda Calculus - Stanford Encyclopedia of Philosophy
Dec 12, 2012 · What is the value of this expression when $x = 2$? We compute ... lambda x[x]\) and $\lambda y[y]$. 3. Brief history of \(\lambda ...
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Frege's Problem: Referential Opacity
The problem of referential opacity is to explain why a certain inference rule of classical logic sometimes produces invalid-seeming inferences.<|separator|>
[42]
Gödel's Incompleteness Theorems
Nov 11, 2013 · Gödel's two incompleteness theorems are among the most important results in modern logic, and have deep implications for various issues.
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Non-wellfounded Set Theory - Stanford Encyclopedia of Philosophy
Apr 16, 2008 · A hyperset or non-wellfounded set is a set that is obtained by decorating an arbitrary graph. Another way of thinking about hypersets is in ...Missing: extensional | Show results with:extensional
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[PDF] Category Theory in Context Emily Riehl
Mar 1, 2014 · The aim of theory really is, to a great extent, that of systematically organizing past experience in such a way that the next generation, ...
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Hyperintensionality - Stanford Encyclopedia of Philosophy
Feb 8, 2021 · A hyperintensional concept draws a distinction between necessarily equivalent contents. If the concept is expressed by an operator, $H$, then $H$ is ...
[46]
Paraconsistent Logic - Stanford Encyclopedia of Philosophy
Sep 24, 1996 · If LLL is extended with the requirement that no abnormality is logically possible, one obtains the upper limit logic (ULL).