implicational propositional calculus

implicational propositional calculus (uncountable)

(logic) A minimalist version of propositional calculus which uses only the logical connectives → ("implies") and ⊥ ("false").


intuitionistic logic

en-noun

intuitionistic logic (plural intuitionistic logics)

law of excluded middle
law of excluded middle

law of double negation
law of double negation

(mathematics, logic) A type of logic which rejects the axiom law of excluded middle or, equivalently, the law of double negation and/or Peirce's law. It is the foundation of intuitionism.

intuïtionistische logica

a type of logic which rejects the axiom "law of excluded middle"


proof by contradiction

proof by contradiction (countable and uncountable, plural proofs by contradiction)

(mathematics, logic) The proof of a statement adduced by deriving a contradiction from the statement's negation.


law of double negation

The law of double negation is not valid intuitionistically. To show this with Heyting algebra semantics, let A=(0,1)∪(1,2). Then ¬A=(-∞,0)∪(2,∞), ¬¬A=(0,2), ¬¬A→A=(-∞,1)∪(1,∞) ne ℝ.

(logic) The statement that the negation of the negation of A implies A, for any proposition A. Stated symbolically: ¬¬A→A.


proof system

(logic) A set of axioms and a set of inference rules which are jointly used to deduce tautologies, thereby providing proofs of them.


valuation function

valuation function (plural valuation functions)

Say that a model's interpration assigns true to A and false to B. Then the valuation function assigns false to "not A", true to "not B", false to "A implies B", false to "A and B", and true to "A or B".

(logic) A function which assigns a truth value to every well-formed formula, which is identical to the model's interpretation function when applied to atomic propositions, and which otherwise assigns a truth value recursively depending on the formula's top logical connective and the truth values of the subformulae surrounding that logical connective.


law of excluded middle

law of the excluded middle (alternative)

(logic) A logical principle which states all statements must be either true or false, i.e. in symbols: P∨¬P.


Kleene star

After Stephen Cole Kleene (1909-1994), American mathematician.

(logic, computing theory) The asterisk, *, used as an operator to concatenate zero or more strings from a given set, widely used in regular expressions.


intensionality

intensionality (countable and uncountable, plural intensionalities)

(logic, mathematics) The condition of having an intension


dialetheism

dialetheism (usually uncountable, plural dialetheisms)

Coined by Graham Priest and Richard Routley, from di- + Ancient Greek ἀλήθεια (alḗtheia, “truth”), in 1981.

der

[It] is important to point out that endorsing dialetheism is not the same as rejecting logic or rational argumentation.

2005, Laurence Goldstein, Logic: key concepts in philosophy, page 158:

(logic) The theory that statements can be both true and false at the same time and in the same sense. The opposite of the law of noncontradiction.


falsum

(logic) An arbitrary contradiction, denoted ⊥.


summum genus

Summum genus, in its strictest sense, is that all-extensive term under which every object of whatever kind may be classed, and of every one of which it may be affirmatively predicated: the word which is generally used to denote such genus is substance, or, as some call it, being. ... ; thus by ornithologist, bird would be regarded as the summum genus under which he would arrange the different subdivisions of birds ...

1827, John Huyshe, A Treatise on Logic, on The Basis of Aldrich:

(logic, semantics) The highest, most all-inclusive genus; the one that is not a species of any other genus.


axiom schema

en-noun

axiom schema (plural axiom schemas or axiom schemata)

axiom schemas (plural)
axiom schemata (plural)

(logic) A formula in the language of an axiomatic system, in which one or more schematic variables appear, which stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions.


subformula

subformula (plural subformulas or subformulae)

subformulas (plural)
subformulae (plural)

(logic) A formula that is part of another formula.


schematic variable

schematic variable (plural schematic variables)

universe of discourse
universe of discourse

(logic) A variable that appears in an axiom schema and ranges over formulas, distinguished from the variables of the axiom schema that are quantified over and that range over the individuals of the universe of discourse.


nonliteral

(programming, logic) That which is not a literal.


tetrachotomy

tĕtrəkŏʹtəmĭ

/tɛtɹəˈkɒtəmɪ/

From τέτραχα (tétrakha, “quadripartite”) + τόμος (tómos, “slice, piece”) (from τέμνω (témnō, “I cut”)).


pons asinorum

From New Latin pons asinorum, from Latin pōns (“bridge”) + genitive plural of asinus (“donkey”). Literally, “bridge of donkeys”.

(logic) A method for finding the middle term of a syllogism in Aristotlean analytics.


dormitive virtue

dormitive virtue (plural dormitive virtues)

A calque translation of Latin, virtus dormitiva, coined by Molière in The Imaginary Invalid. In the play, he lampoons a group of physicians providing an explanation in macaronic Latin of the sleep-inducing properties of opium as stemming from its "virtus dormitiva".

The inevitable next suggestion -— that aesthetic experience is distinguished not by pleasure at all but by a special aesthetic emotion can be dropped on the waste-pile of "dormitive virtue” explanations

1968, Proceedings and Addresses of the American Philosophical Association, American Philosophical Association, page 6:

Solidity now has all the defects of a dormitive virtue, obscurity, unobservability, something-we-know-notwhattery — and none of the advantages of a dormitive virtue

1998, Rae Langton, Kantian Humility: Our Ignorance of Things in Themselves, page 176:

Because (1) is postulated and at the same time used as a covering law, the putative explanations are vacuous in the sense in which explanatory appeals to dormitive virtue are.

2003, Henry E. Kyburg, Mariam Thalos, Probability is the Very Guide of Life, page 66:

Now admittedly, this has the look of a dormitive virtue argument: the reason the intellect can recollect its own activity is that it possesses just such a recollective capacity.

2013, Charles Bolyard, Rondo Keele, Later Medieval Metaphysics: Ontology, Language, and Logic, page 147:

But if talk of such facts just arises via hypostatizations out of the relevant truths, then the facts posited can't explain the truths except via a blatant dormitive virtue explanation.

2021, Karen Bennett, Dean W. Zimmerman, Oxford Studies in Metaphysics, volume 12, page 29:

Fitness is not the empty idea of a dormitive virtue. The point is that although fitness is explanatory, it seems to be a placeholder for a deeper account that dispenses with the concept of fitness [again the propensity concept].

2021, Richard E. Michod, Darwinian Dynamics: Evolutionary Transitions in Fitness and Individuality, page 186:

(idiomatic, rhetoric, logic, linguistics) A type of tautology in which an item is explained in terms of the item itself, only put in different (usually more abstract) words.


precisive

en-adj

precisive (comparative more precisive, superlative most precisive)

more precisive (comparative)
most precisive (superlative)

precisive censure

precisive abstraction

(logic) Exactly limiting by cutting off all that is not absolutely relevant.


conjunctive normal form

en-noun

conjunctive normal form (countable and uncountable, plural conjunctive normal forms)

(logic) The form of a Boolean formula that the formula has if the formula is a conjunction of disjunctions of literals, such as “(A or B or C) and (D or E or not F)”.


disjunctive normal form

(logic) The form of a boolean formula that the formula has if the formula is a disjunction of conjunctions of literals, such as “(A and B and C) or (D and E and not F)”.

disjunktiivinen normaalimuoto

the form of a boolean formula that the formula has if the formula is a disjunction of conjunctions of literals, such as “(A and B and C) or (D and E and not F)”


LEM

What constructive mathematicians know is that there are mathematical universes in which sets are like topological spaces and properties are like open sets. In fact, these universes are well-known to classical mathematicians (they are called toposes), but they look at them from “the outside”. When we consider what mathematicians who live in such a universe see, we discover many fascinating kinds of mathematics, which tend to be constructive. The universe of classical mathematics is special because in it all sets are like discrete topological spaces. In fact, one way of understanding LEM is “all spaces/sets are discrete”. Is this really such a smart thing to assume? If for no other reason, LEM should be abandoned because it is quite customary to consider “continuous” and “discrete” domains in applications in computer science and physics. So what gives mathematicians the idea that all domains are discrete?

2005, Andrej Bauer, Mathematics and Computation: The Law of Excluded Middle:

Law of excluded middle
law of excluded middle#English


infinitary

(mathematics, logic) Of or pertaining to expressions of infinite length


paralogy

paralogy (countable and uncountable, plural paralogies)


superaltern

(logic) A universal proposition from which can be inferred a corresponding subaltern


FO

first-order logic
first-order logic#English

(logic, computer science) Initialism of first-order logic.


finitary

(logic) Pertaining to finite-length proofs, each using a finite set of axioms.


succedent

(logic) The formulas on the right-hand side of a sequent.


accidens

An Accidens (or accident) of an individual, genus, or species is any attribute which is possessed by it, and which does not follow from, or form a part of, its comprehension. […] If, on the other hand, an accidens is sometimes present and sometimes absent in an individual, […]

1886, Prasanna K. Roy, A Text-book of Deductive Logic: For the Use of Students, page 83:

But, where there is no apparent reason why the attribute should always be found in the individuals of a class, it is called an Accidens.

1896, James Welton, A Manual of Logic, volume 1, page 86:

(logic) Synonym of accident (“a property, fact, or relation that is the result of chance or is nonessential or nonsubstantive”).


categorematic

From Ancient Greek κατηγόρημα (katēgórēma, “a predicate”) + -atic.

uder

The word "man" is categorematic; the word "many" is not.

(logic, of a word) Capable of being employed alone as a term.


shallow embedding

en-noun

shallow embedding (plural shallow embeddings)

(logic, uncountable) The act of representing one logic or language with another by providing a syntactic translation.

(logic, countable) A specific mechanism for such a syntactic translation.


deep embedding

(logic, uncountable) The act of representing one language, typically a logic or programming language, with another by modeling expressions in the former as data in the latter.

(logic, countable) A specific such representation.


subcontrary

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(logic) Denoting the relation of opposition between the particular affirmative and particular negative. Of these both may be true and only one can be false.


subcontrary

(logic) Either of a pair of propositions at least one of which must be true


ignotum per ignotius

From Latin ignōtum per ignōtius, “[to explain] the unknown by the more unknown”.

(logic) A type of fallacious argument in which one attempts to prove something unknown by relying upon an assumption that is also unknown.


minterm

A Boolean function can be expressed, canonically, as a sum of minterms, where each minterm corresponds to a row (of the function's truth table) whose output value is 1.

If a product term includes all of the variables exactly once, either complemented or not complemented, this product term is called a minterm.

2014 February 17, Linda Null, Julia Lobur, Essentials of Computer Organization and Architecture, Jones & Bartlett Publishers, →ISBN, page 199:

In Boolean algebra, a product term, with a value of 1, in which each variable appears once (in either its complemented or uncomplemented form, so that the value of the product term becomes 1).


discursus

Borrowed from Latin discursus. Doublet of discourse.

(logic) argumentation; ratiocination; discursive reasoning


obvert

en-verb

obvert (third-person singular simple present obverts, present participle obverting, simple past and past participle obverted)

obverts (present singular third-person)
obverting (participle present)
obverted (participle past)
obverted (past)

(transitive, logic) To infer by obversion; too convert a proposition to its obverse.


explicandum

(logic) That which is to be explained or explicated


implicator

Residuated implicators are quite important since a t-norm T and an implicator I can only satisfy the residuation principle T(x,y) ≤_L z ⇔ y ≤_L I(x,z) if and only if the implicator I equals the residuated implicator I_T.

2008, M. Nachtegael, P. Sussner, T. Mélange, E.E. Kerre, “An Interval-Valued Fuzzy Morphological Model Based on Lukasiewicz-Operators”, in Jacques Blanc-Talon, Salah Bourennane, Wilfried Philips, Dan Popescu, Paul Scheunders, editors, Advanced Concepts for Intelligent Vision Systems: 10th International Conference, ACIVS 2008, Juan-les-Pins, France, October 2008, Proceedings (Lecture Notes in Computer Science; 5259), Springer, →ISBN, page 605:

(logic) The antecedent of an implication.


hyperjump

(mathematics, logic) A form of Turing jump in hyperarithmetic.


graphoid

(logic) A set of statements of the form, "X is irrelevant to Y given that we know Z" where X, Y and Z are sets of variables, which can be manipulated by a set of axioms concerning informational irrelevance and its graphical representation.


counterproof

It seems that ghosts and legendary encounters in general thrive under the protection of law because their extranormal statements and implications require proof and provoke the less convincing counterproof.

2001, Linda Dégh, Legend and Belief: Dialectics of a Folklore Genre, page 4:

(logic) A proof that contradicts a given statement.


apodictically

en-adv

apodictically (comparative more apodictically, superlative most apodictically)

more apodictically (comparative)
most apodictically (superlative)


positive logic

(logic) A logical system or fragment thereof not containing a negation operator.


classical logic

en-noun

classical logic (countable and uncountable, plural classical logics)

The Lindenbaum-Tarski algebra of propositional classical logic is a Boolean algebra.

(logic) A kind of logic based on the principles that each proposition has a truth value of either "true" or "false", but not both, and that if a proposition were to be both true and false or neither true nor false then a result would be that all propositions would be both true and false.


De Morgan's law

De Morgan's law (plural De Morgan's laws)

De Morgan's laws (plural)
DM (alternative)

/dɨˈmɔɹɡɪnzˌlɔ/

Named after the British mathematician and logician Augustus De Morgan (1806–1871), who first formulated the laws in formal propositional logic.

Let ‘F’ stand for the state of affairs that consists in finite persons possessing and exercising free will. Let ‘p’ stand for ‘God exists’; ‘q’ for ‘F obtains’; ‘r’ for ‘F poses a serious risk of evil’; and ‘s’ for ‘There is no option available to God that counters F.’ With this in place, the argument may be formalized as follows:
(1) [(p & q) & r] → s Premiss
(2) ~s Premiss
(3) ~[(p & q) & r] 1, 2 MT
(4) ~(p & q) v ~r 3 DM
(5) r Premiss
(6) ~(p & q) 4, 5 DS
(7) ~p v ~q 6 DM
(3) follows from the conjunction of (1) and (2) by modus tollens; De Morgan’s law applied to (3) yields (4); (4) and (5) together lead to (6) by disjunctive syllogism; and another application of De Morgan’s law takes us from (6) to the final conclusion, according to which either God exists or there is free will (but not both).

2004 August, J. L. Schellenberg, “The Atheist’s Free Will Offence”, in International Journal for Philosophy of Religion, volume 56, № 1, pages 11-12:

(mathematics, logic) Either of two laws in formal logic which state that:

The negation of a conjunction is the disjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∧ 𝑞) ⇔ (¬ 𝑝) ∨ (¬ 𝑞)

The negation of a disjunction is the conjunction of the negations; expressed in propositional logic as: ¬ (𝑝 ∨ 𝑞) ⇔ (¬ 𝑝) ∧ (¬ 𝑞)


axiom of extensionality

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that two sets are equal if and only if they contain the same elements.


supervaluationism

(logic) A semantics for dealing with irreferential singular terms and vagueness.


supertruth

LL-Q1860 (eng)-Flame, not lame-supertruth.wav

(logic) In supervaluationism, the quality of being supertrue.


supertrue

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"I am happy or I am not happy" is supertrue, because I must be one or the other.

(logic) In supervaluationism, necessarily true regardless of the values of variables.


superfalse

LL-Q1860 (eng)-Flame, not lame-superfalse.wav

"I am asleep and I am not asleep" is superfalse, because I can never be both.

(logic) In supervaluationism, necessarily false regardless of the values of variables.


Sheffer stroke

Sheffer strokes (plural)
Sheffer's stroke (alternative)

(logic) The vertical bar symbol ( | ), denoting the NAND operation.


SLD

selective linear definite
w:SLD resolution

(logic) Initialism of selective linear definite.


abstract model

(engineering, logic, mathematics) A model that contains an abstraction or abstractions of the reality.


definite clause

definite clause (plural definite clauses)

The clauses #92;negp#92;veeq, #92;negp#92;vee#92;negq#92;veer, and #92;negp#92;vee#92;negq#92;vee#92;negr#92;vees are examples of definite clauses. At least in classical logic these would be equivalent to the formulae p#92;rightarrowq, (p#92;wedgeq)#92;rightarrowr, and (p#92;wedgeq#92;wedger)#92;rightarrows, respectively.

(logic) A Horn clause with exactly one positive literal.


Horn clause

Named after American logician Alfred Horn (1918–2001).

named-after

(logic) A clause (disjunction of literals) with at most one positive literal.


constructive logic

en-noun

constructive logic (countable and uncountable, plural constructive logics)

(logic) Any kind of logic in which any proof of existence can be converted into an algorithm that constructs the mathematical object which it claims to exist.


prime formula

prime formula (plural prime formulas or prime formulae)

prime formulas (plural)
prime formulae (plural)

(logic) A well-formed formula which has not been made by combining simpler well-formed formulas by means of logical connectives.

peruslause

well-formed formula that is not combined from other, simpler formulas


logical connective

logical connective (plural logical connectives)

material implication
material implication

(logic) A truth-valued function such as (the usual suspects): conjunction, disjunction, negation, and material implication.


formation rule

(logic) Any of a set of rules used to construct well-formed formulas of a formal language.


inference rule

(logic, proof theory) A rule for combining (or modifying) well-formed formulas of a formal language in a truth-preserving manner (to yield new well-formed formulas).


existential instantiation

existential instantiation (plural existential instantiations)

(logic) In predicate logic, an inference rule of the form ∃x P(x) ⊢ P(c), where c is a new symbol (not part of the original domain of discourse, but which can stand for an element of it (as in Skolemization)).


conditional proof

conditional proof (plural conditional proofs)

A conditional proof corresponds to the application of what Douglas Hofstadter calls the "Fantasy Rule" in his masterwork "GEB:EGB". Surprisingly, the theory of propositional calculus which he develops in GEB:EGB employs several inference rules but no axioms: the need for any axioms is done away with through application of the Fantasy Rule.

(logic) A proof showing that if an assumption A is true then a consequent statement or conclusion B must also be true, i.e. B is true on the condition that A is true.


deduction theorem

deduction theorem (plural deduction theorems)

The deduction theorem reveals the relationship between logical entailment and material implication: it allows to one to "pack" or "record" an inference into a tautology, and conversely, to "unpack" or "play back" a tautology as an inference process.

(logic) A procedure for "discharging" assumptions from an inference, causing them to become antecedents of the conclusion; or vice versa. Symbolically, the conversion of an inference of the form P,A⊢C to an inference of the form P⊢A→C or vice versa, where ⊢ is the turnstile symbol. The validity of the procedure is a metatheorem of the given logical theory.


consensus theorem

(logic) The following theorem of Boolean algebra: XY+X'Z+YZ=XY+X'Z where YZ, the algebraically redundant term, is called the "consensus term", or its dual form (X+Y)(X'+Z)(Y+Z)=(X+Y)(X'+Z), in which case Y+Z is the consensus term. (Note: X+Y,X'+Z⊢Y+Z is an example of the resolution inference rule (replacing the + with ∨ and the prime with prefix ¬ might make this more evident).)


collectional

Such quantifiers are characterized as dispensive(or distributive), and are distinguished from collective reference (e.g. "All S, taken together, are P") and collectional reference (e.g. "All S, separately but simultaneously, are P").

The aggregation relationship appears like a family of four different kinds of relationships. We note Aₑ the set relationship, Aₛ the segmentational one, A_f the functional one, and A_c, the collectional one.

1996, John E.E. Sharpe, AI System Support for Conceptual Design, page 324:

For an infinite of finite order multiplied into itself gives a higher order. That is not the case with collectional infinities. On the contrary, each is the exponential of the preceding.

2022, Stetson J. Robinson, The Correspondence of Charles S. Peirce and the Open Court Publishing Company, 1890–1913, page 493:

(logic) Pertaining to all elements of a collection simultaneously, rather than to the individual elements of the collection or to the collection as a whole.


epichirema

epichirema (plural epichiremas or epichiremata)

epichiremas (plural)
epichiremata (plural)
epicheirema (alternative)

Borrowed from Latin, itself a borrowing from Ancient Greek ἐπιχείρημα (epikheírēma).

(rhetoric, logic) A syllogism in which the proof of the major or minor premise, or both, is introduced with the premises themselves, and the conclusion is derived in the ordinary manner.


prosyllogism

(logic) A syllogism that is preliminary or logically essential to another syllogism.

(logic) The conclusion of such a syllogism, which becomes a premise of the following syllogism.


obversion

obversion (countable and uncountable, plural obversions)

All men are mortal, so, by obversion, no men are immortal.

From 'warmth is agreeable' we can affirm, by formal obversion, 'warmth is not disagreeable, and not indifferent.' We cannot affirm, without an examination of the subject matter, 'cold is disagreeable.'

(logic) An immediate inference that denies the opposite of something previously affirmed.


inconcludent

en-adj

inconcludent (comparative more inconcludent, superlative most inconcludent)

more inconcludent (comparative)
most inconcludent (superlative)

But it is wrong to say that these propositions can be deduced from, or proved by, the premisses associated with them, and that the "clearest sign" that the pair is inconcludent is the fact that "the conclusions deduced from them destroy one another": the conclusion simply cannot be 'inferred' from the given propositions —and in any case they do not contradict one another since they are about different terms.

2013, G. Patzig, Aristotle’s Theory of the Syllogism, page 175:


trivialism

en-noun

trivialism (countable and uncountable, plural trivialisms)

If it is possible for trivialism to be true, it may be false as well as true that all humans are mammals […]

2004, Graham Priest, J. C. Beall, Bradley Armour-Garb, The Law of Non-contradiction: New Philosophical Essays, page 252:

This charge of trivialism is also called the principle of explosion […]

2008, Peter Baofu, The future of post-human mathematical logic, page 97:

Trivialism is pretty hopeless as a philosophy, although it is very easy to defend/maintain verbally!

2012, Koji Tanaka, Francesco Berto, Edwin Mares, Paraconsistency: Logic and Applications, page 296:

(logic) The theory that every proposition and its negation is true.


retroduction

retroduction (countable and uncountable, plural retroductions)

From Latin retroducere, retroductum (“to lead or bring back”), from retro (“backward”) + ducere (“to lead”).

Whereas retroduction involves one step, and deduction involves two, induction, as the third stage, involves three: classification, probationary argument, and appraisal of the probations.

1995, Douglas R. Anderson, Strands of System: The Philosophy of Charles Peirce, page 161:

Note that deduction and retroduction employ the same form of logical statement to determine either the post- or precondition, respectively.

2008, Michael L. Morrison, William M. Block, M. Dale Strickland, Wildlife Study Design, page 11:

What distinguishes retroduction from the other two main forms of inference employed in inquiry into nature, deduction and induction (inductive generalization), is its reliance on explanatory quality for its epistemic force (McMullin 1992).

2010, Janet Broughton, John Carriero, A Companion to Descartes, page 86:


episyllogism

(logic) A syllogism which assumes as one of its premises a proposition which was the conclusion of a preceding syllogism (the prosyllogism).


apagoge

apagoge (countable and uncountable, plural apagoges)

/ˈæ.pə.ɡəʊ.d͡ʒi/

Borrowed from Ancient Greek ἀπαγωγή (apagōgḗ).

(logic) An indirect argument which proves a thing by showing the impossibility or absurdity of the contrary.

apagogie

an indirect argument which proves a thing by showing the impossibility or absurdity of the contrary


crocodility

(logic, archaic) A sophistical mode of arguing.


discretive

«But» is a particle, none more familiar in our language: and he that says it is a discretive conjunction, and that it answers to sed Latin, or mais in French, thinks he has sufficiently explained it.

1689 (indicated as 1690), [John Locke], An Essay Concerning Humane Understanding. […], London: […] Eliz[abeth] Holt, for Thomas Basset, […], →OCLC:

Δαì has generally a discretive or distinctive, often and adversative power.

1829, Hendrik Hoogeveen, Hoogeveen's Greek Particles, page 40:

The coherencey of a discretive proposition depends on the connecton of both parts and their relevancy to one another.

1837, S. E. Parker, Logic, Or, The Art of Reasoning Simplified, page 92:

(grammar, logic) Disjunctive with the additional characteristic that the conjoined clauses, while exhibiting a disjunct, are both asserted as true.


discretive

discretive proposition
discretive proposition


antepredicament

(logic) A prerequisite to a clear understanding of the predicaments and categories, such as definitions of common terms.

antepredicamento

prerequisite to a clear understanding of a predicament


consecution

consecution (countable and uncountable, plural consecutions)

From Middle English consecucioun (“attainment”), from Latin cōnsecūtiō (“effect, proper sequence, attainment”), from past participle of cōnsequor (“to follow, result, reach”).

der

(logic) The relation of consequent to antecedent.

consécution

(logic) relation of consequent to antecedent


subjunction

The primary form of subjunction is 'If P, then Q', which tells us that '{P and nonQ} is logically impossible'.

In the case of subjunction, the reading of →as 'if ... then' is actually strongly misleading.

2013, Sebastian Loebner, Understanding Semantics:

Since supervaluationists, as we have seen, do accept some disjunctions (e.g. A V ~A) while rejecting each disjunct, they must, as we have also seen, fail subjunction.

2016, Dominic Hyde, Vagueness, Logic and Ontology, page 92:

The subjunction ' A B ' corresponds to a proposition of the form ' not A , unless B ' rather than a proposition of the form ' if A , then B ' .

2023, Michael Wolff, Essay on the Principles of Logic: A Defense of Logical Monism:

(logic) An implication or entailment; A proposition that links two states such that one is a logical consequence of the other.


subalternant

Of these [oppositions], the Universal is the Subalternant , and the Particular the Subalternate

1826, Richard Whately, Elements of Logic:


Kripke frame

(logic) A set, whose elements are called nodes or worlds, together with a preordering relation for that set, called its accessibility relation.


Kripke model

A formula is intuitionistically valid iff it is forced true by every world of every Kripke model.

A terminal world of a Kripke model (for intuitionistic logic) has a forcing relation equivalent to a classical model/interpretation.

If a given world (in a Kripke model (for intuitionistic logic)) forces neither A nor ¬A then there is some possible "future" world (accessible from the "present" one) in which A is forced true. In particular, if A eventually becomes forced somewhere along any possible time thread (towards the "future"), then the present world would force ¬¬A to be true, but if this does not happen there there exists some possible future world in which ¬A becomes true.

intuitionistic logic
intuitionistic logic

(logic) A Kripke frame together with any one of the following: (1) a function associating each of the frame's worlds to a set of prime formulae which are "true" for the given world, (2) a function associating each prime formula to a set of worlds for which the prime formula is "true", (3) a forcing relation between worlds and prime formulae. Additionally, there is a set of rules for deducing (from the given function or relation) what formulae are forced to be true by a given world. (The set of rules depends on which logic the Kripke model is being applied to, whether one of several modal logics or intuitionistic logic). For intuitionistic logic the forcing relation satisfies a persistence relation, namely, if world w forces proposition p, then all worlds accessible from w also force p.


convertend

From Latin convertendus (“to be converted”). Compare subtrahend.

(logic) Any proposition which is subject to the process of conversion; so called in its relation to itself as converted, after which process it is termed the converse.


preindesignate

(archaic, logic) Having no sign expressing quantity; indefinite.


superordination

en-noun

superordination (countable and uncountable, plural superordinations)

LL-Q1860 (eng)-Flame, not lame-superordination.wav

From super- + ordination: compare Latin superordinatio.

(logic) The relation of a universal proposition to a particular proposition in the same terms.


argument-form

(logic) Alternative form of argument form.


formal validity

formal validity (countable and uncountable, plural formal validities)

(logic) An argument's property of being valid due to being an instance of an argument form.


logical constant

logical constant (plural logical constants)

The logical connective OR can be seen to correspond to the logical constant "unless".

(logic) A word or phrase which has an invariant logical meaning and which is useful for forming argument forms.


sentence connective

sentence connective (plural sentence connectives)

Standard ("classical") propositional logic deals with sentence connectives having a special property: they are truth functional. The terminology derives from the mathematical notion of a function, and one can use this to give a mathematically precise definition of truth functionality.

2001, Mark Sainsbury, chapter 2, in Logical Forms — An Introduction to Philosophical Logic, 2nd edition, Blackwell Publishing, →ISBN, §2, page 60:


elench

LL-Q1860 (eng)-Naomi Persephone Amethyst (NaomiAmethyst)-elench.wav

Latin elenchus, from Ancient Greek ἔλεγχος (élenkhos, “refutation, scrutiny”); compare Old French elenche. Doublet of elenchus.

der

Reprehension or Elench […] is a Syllogisme, which gathereth a conclusion contrary to the assertion of the respondent, as if a man would defend Medea not to love her childe, because she killed it, another might reason against him in this manner: every Mother loveth her child: but Medea is a Mother: Ergo, Medea loveth her child: the Conclusion of this Syllogisme is contrarie to the first Assertion […]

1599, Thomas Blundeville, The Art of Logicke, London: Matthew Lownes, published 1619, Book 6, Chapter 2, p. 186:

(logic) That part of an argument on which its conclusiveness depends; that which convinces or refutes an antagonist; a refutation.


subalternate

(logic) A particular proposition, as opposed to a universal one.


epagoge

/ɛpəˈɡəʊɡiː/

From Latin epagōgē, from Ancient Greek ἐπᾰγωγή (epăgōgḗ, “a bringing in”). See epact.

uder

(logic) The adducing of particular examples so as to lead to a universal conclusion; the argument by induction.


exclusive disjunction

en-noun

exclusive disjunction (plural exclusive disjunctions)

(logic) A disjunction which is true if only one, but not both, of its disjuncts is true.


inclusive disjunction

inclusive disjunction (plural inclusive disjunctions)

(logic) A disjunction which is true if at least one of its disjuncts is true.


analogism

analogism (countable and uncountable, plural analogisms)

From Ancient Greek ἀναλογία (analogía) (from ἀνά (aná) + λόγος (lógos, “speech, reckoning”)) + -ism.

all Judications and Analogisms may faile

1657, Henry Pinnell, Oswald Croll, Philosophy Reformed & Improved in Four Profound Tractates:

Epilogism as opposed to analogism proceeds from what is seen.

2012, P. Nicolacopoulos, Greek Studies in the Philosophy and History of Science, page 106:

The inclination toward simplified and/or distorted (but often useful) reasoning in the form of analogism and a propensity to organise events into teleological narratives to which causality is attributed, is of particular interest to this discussion.

2017, Jacqueline Hill, Mary Ann Lyons, Representing Irish Religious Histories, page 105:

(logic) An argument from cause to effect; an a priori argument.


nonimplication

nonimplication (countable and uncountable, plural nonimplications)

Material nonimplication
w:Material nonimplication

(chiefly logic) That which is not implied; the opposite of an implication.


Hegelianism

Hegelianism (countable and uncountable, plural Hegelianisms)

Hegelianisms (plural)
Hegelism (alternative)

The system of logic and philosophy set forth by G. W. F. Hegel, which can be summed up by the dictum that "the rational alone is real", i.e. all reality is capable of being expressed in rational categories.


elenctical

en-adj

elenctical (comparative more elenctical, superlative most elenctical)

more elenctical (comparative)
most elenctical (superlative)


Baroko

The moods Baroko and Bokardo do not admit of reduction to the first figure, by any fair use of the phrase[…]

1847, Augustus De Morgan, Formal logic: or, The Calculus of inference, necessary and probable, page 132:

But this Conclusion is false, consequently the Minor Premise of the first Syllogism, Baroko, its contradictory, is true.

1870, William Dexter Wilson, An elementary treatise on logic, page 129:

The foregoing list of moods in the imperfect Figures II and III does not contain Baroko or Bokardo.

2005, Charles Gray Shaw, Logic in Theory and Practice, page 161:

(logic, obsolete) A form or mode of syllogism in which the first proposition is a universal affirmative and the other two are particular negative.


Bokardo

Bokardo is converted in a similar manner, only that in this case we substitute the contradictory of the conclusion for the major premiss[…]

1870, H. Coleman, Notes on logic, page 81:

(logic, obsolete) A form or mode of syllogism in which the first and third propositions are particular negatives and the second a universal affirmative.


sentential logic

en-noun

sentential logic (plural sentential logics)

The Leibniz operator and the study of various of its properties that may or may not be satisfied for particular sentential logics have given rise to what is now known as the abstract algebraic hierarchy or Leibniz hierarchy of sentential logics.ᵂᴾ


Skolemization

Named after Thoralf Skolem (1887–1963), Norwegian mathematician.

existential quantifier
existential quantifier

universal quantifier
universal quantifier

(logic) The conversion of a formula of first-order logic which involves the following steps: (1) replace free variables with constants, (2) replace any variable bound by an existential quantifier which lies in the scope of an even number of logical negations with a Skolem function, and (3) replace any variable bound by a universal quantifier which lies in the scope of an odd number of negations with a Skolem function.


Herbrandization

From Herbrand + -ization, after Jacques Herbrand (1908–1931), French mathematician.

(logic) The conversion of a formula of first-order logic which involves the following steps: (1) replace free variables with constants, (2) replace any variable bound by a universal quantifier which lies in the scope of an even number of logical negations with a Herbrand function, and (3) replace any variable bound by an existential quantifier which lies in the scope of an odd number of negations with a Herbrand function.


double turnstile

double turnstile (plural double turnstiles)

(logic, model theory) The ⊨ symbol used to denote semantic consequence, or the ⊧ symbol used to denote the fact that the model to the left of it satisfies the set of sentences to its right.


stoichiology

Ancient Greek στοιχεῖον (stoikheîon, “first element, fundamental principle”) + -logy.

uder

(logic) The doctrine of the elementary requisites of mere thought .


dialetheia

en-noun

dialetheia (plural dialetheias or dialetheia)

dialetheias (plural)
dialetheia (plural)

Coined by Graham Priest and Richard Routley, from di- + Ancient Greek ἀλήθεια (alḗtheia, “truth”), in 1981.

uder

A statement that is both true and false; a true contradiction.


quantum logic

(logic, quantum mechanics) A form of non-classical logic which has been abstracted out of quantum mechanics.

The representation theorem (for quantum logic) shows that these quantum logics form a lattice which is isomorphic to the lattice of subspaces of a vector space with a scalar product.ᵂᴾ

(logic, quantum mechanics) A set of events that is closed under a countable disjunction of countably many mutually exclusive events.ᵂᴾ


apodeictical

en-adj

apodeictical (comparative more apodeictical, superlative most apodeictical)

more apodeictical (comparative)
most apodeictical (superlative)


completory

completory (comparative more completory, superlative most completory)

more completory (comparative)
most completory (superlative)

completory of ancient presignifications

a. 1678 (date written), Isaac Barrow, “(please specify the chapter name or sermon number). Suffered under Pontius Pilate”, in The Works of Dr. Isaac Barrow. […], volume (please specify |volume=I to VII), London: A[braham] J[ohn] Valpy, […], published 1830–1831, →OCLC:


eigenvariable

(mathematics, logic) A kind of term in natural deduction.


neutrosophic

In a neutrosophic triplet set, there is a neutral element and antielement for each element.

2018, Memet Şahin, Abdullah Kargın, Mehmet Ali Çoban, Fixed Point Theorem for Neutrosophic Triplet Partial Metric Space:

(logic) Of or relating to a general form of logic in which each proposition has separate values for truth, falsehood, and indeterminacy.


infima species

Every infima species of the adaptation can be related unambiguously to a[n] infima species of the standard.

1993, Willem Hirs, “XV: The Use of Terminological Principles and Methods in Medicine”, in Helmi B. Sonneveld, Kurt Loening, editors, Terminology: Applications in Interdisciplinary Communication, page 239:

Combining Leibniz's view about species differentiation with his view that completed individuals constitute infima species, [DM 9; N 326] yields a promising speculation as to why he might hold that any change in property of our Adam would be a different person.

1998, R. O. Savage, Real Alternatives, Leibniz's Metaphysics of Choice, page 95:

So two things are pivotal in the Categories account of primary ousiai: the idea that the things that are ontologically fundamental are basic subjects, and the idea that the ontologically basic entities are instances or members of their infimae species.

2008, Michael J. Loux, Primary Ousia: An Essay on Aristotle's Metaphysics Z and H, page 4:

But when Pratt gives illustrations of his claim, he selects different infimae species under the genus 'material substance'.

2009, James Porter Moreland, The Recalcitrant Imago Dei, page 124:

(logic, philosophy, semantics) The narrowest of species; one that that is not a genus to anything else.


precisification

en-noun

precisification (countable and uncountable, plural precisifications)

(logic) The process of making a statement more precise.


antitheorem

(logic) A syntactically correct expression whose negation is a theorem.


contractionless

a contractionless logic


law of noncontradiction

The statement "this piece of chalk is not a piece of chalk" violates the law of noncontradiction.

(logic) The principle that no statement may be simultaneously true and false at the same time and in the same sense.


principle of explosion

(logic) The principle or axiom of classical logic and some other logical systems stating that if a contradiction or a false proposition is proven to be true, then it proves that everything is true. In symbols: ⊥→P


vacuous truth

(logic) A proposition which is true only because it is a conditional statement with a necessarily false premise, or which asserts a property of all elements of an empty domain of discourse.


doxastic logic

doxastic logic (countable and uncountable, plural doxastic logics)

(logic) A type of modal logic which deals with statements involving beliefs.

doksastinen logiikka

a logic involving statements about beliefs


paraconsistent logic

paraconsistent logic (plural paraconsistent logics)

principle of explosion
principle of explosion

(logic, countable) A particular formal logical system which allows some contradictions to be true without all contradictions or all statements becoming true; i.e. in which the principle of explosion does not hold.

parakonsistentti logiikka

formal logical system without principle of explosion

parakonsistentti logiikka

study of logical systems without principle of explosion

(logic, uncountable) The study of such systems.


disjunctive syllogism

disjunctive syllogism (plural disjunctive syllogisms)

(logic) A logical argument of the form that if there are only two possibilities, and one of them is ruled out, then the other must take place. In symbols: P∨Q,¬P⊢Q


paramodulation

paramodulation (countable and uncountable, plural paramodulations)

(logic) A technique for reasoning on sets of clauses where the predicate symbol is equality.


polysyllogism

(logic) A number of propositions that, together, constitute a sequence of syllogisms

polisillogismo

propositions that together constitute a sequence of syllogisms


categorical proposition

categorical proposition (plural categorical propositions)

(logic) A proposition that asserts or denies that members of one category (the subject term) belong to another (the predicate term). There are four types: "all S are P", "all S are not P", "some S are P", and "some S are not P".

mulisiping palagay

a proposition that asserts or denies that members of one category (the subject term) belong to another (the predicate term)


Stone space

Named after Marshall Harvey Stone (1903–1989), an American mathematician.

For every Boolean algebra there is a Boolean ring corresponding to it. The prime spectrum of a Boolean ring is a Stone space.

topological space
topological space#English

(topology, logic) A topological space constructed from a Boolean algebra, whose points are ultrafilters on the algebra.


closed formula

closed formula (plural closed formulas or closed formulae)

closed formulas (plural)
closed formulae (plural)

(logic) A formula which has no free occurrences of variables; or equivalently, in which all occurrences of variables are bound.


Herbrand universe

Herbrand universe (plural Herbrand universes)

Named after Jacques Herbrand (1908–1931), French mathematician.

(logic) The set of all ground terms that may be formed (even recursively) using any of the constants and functions which appear in a given set of formulae. (If there is no constant in that given set of formulae, then designate one default constant as being usable for the aforementioned purpose.)


Herbrand function

Herbrand function (plural Herbrand functions)

(logic) A function which replaces a variable bound by a universal quantifier which lies in the scope of an even number of logical negations; such function is a function of the remaining bound variables whose scope contain the given variable (being replaced).


Skolem function

Skolem function (plural Skolem functions)

(logic) A function which replaces a variable bound by an existential quantifier which lies in the scope of an even number of logical negations; such function is a function of the remaining bound variables whose scope contain the given variable (being replaced).


prenex normal form

(logic) A way of expressing a formula of predicate logic such the formula consists of a prenex and a matrix, with the prenex preceding the matrix.


impredicative

(logic, of a proposition) definable only in terms of a totality of which it is itself a part


masked man fallacy

masked man fallacy (plural masked man fallacies)

masked man fallacies (plural)
masked-man fallacy (alternative)

That is to say, one cannot infer from one's subjective state of certainty and uncertainty about two propositions, to the objective connection or lack of connection between them; but it is just such an inference that Descartes seems to be making about himself and his body, and which gives his argument its embarrassing resemblance to the masked man fallacy.

1978, Bernard Williams, Descartes: The Project of Pure Enquiry, New York: Routledge, published 2005, page 97:

Consider the Masked Man fallacy, which shows that I can know something under one description but not under another.

1999, Jonathan E. Pike, From Aristotle to Marx: Aristotelianism in Marxist social ontology, Ashgate, page 176:

[…] that because the per diem payments are compensatory, ipso facto they are not travel expenses. Without further explication, this latter proposition might be thought to exemplify a species of the so-called masked man fallacy ("I know who my father is; I do not know who the masked man is; so, my father is not the masked man.").

2001, Reports of the Tax Court of the United States, volume 117:

(logic, philosophy) A fallacy of inferring that since one knows (or does not know) something by one description, one must know (or not know) it by another, as in "I know who my father is. I do not know who the masked man is. Therefore, my father is not the masked man."


counterextension

counterextension (countable and uncountable, plural counterextensions)

(countable, logic) The set of tuples of values that, used as arguments, do not satisfy the predicate.


destructive sorites

destructive sorites (plural destructive sorites)

In a destructive Sorites, you, of course, go back from the denial of the last consequent to the denial of the first antecedent

1826, Richard Whately, Elements of Logic:

reductio ad absurdum
reductio ad absurdum

(logic) A form of reductio ad absurdum which involves the denial of the first of a series of dependent propositions as a consequence of the denial of the last.


ignoratio elenchi

/ˌɪɡ.nəˈɹeɪ.ʃiˌoʊ ɪˈlɛŋ.kaɪ/

/ˌɪɡ.nəˈɹɑ.tiˌoʊ/

From Latin ignōrātiō elenchī (“ignorance of the refutation”).

bor

The fallacy of presenting an argument that may or may not be logically valid, but fails to address the issue in question.


semifactual

From semi- + factual. In the logic sense, introduced by Nelson Goodman.

(logic) A conditional with a false antecedent and a true consequent.


ratio formalis

en-noun

ratio formalis (plural rationes formales)

From Latin ratiō (“reason”) + fōrmālis (“formal”).

(philosophy, logic) the formal grounds for something; the essential attributes of matter as they appear in the mind or in a definition.


metavariable

The pragmatic process of substitution requires some given context to provide an appropriate substituend for a metavariable projected by a pronoun.

2006 March 13, Lutz Martin, “Relative Clause Construal”, in The Dynamics of Language, Brill, →ISBN, page 85:

If it is a programming language, it will almost always have its own notion of variables, and hence the distinction between object-language variables and metavariables is useful.

2016, Aaron Stump, Verified Functional Programming in Agda, page 198:

(logic) A symbol or string of symbols belonging to a metalanguage and standing for elements of some object language.


conservative extension

conservative extension (plural conservative extensions)

(mathematics, logic) An extension of a logical theory such that every theorem expressible in the original theory is also derivable within the original theory.


one-to-many

"Being a genetic mother of" is a one-to-many relation from the set of cats to the set of cats.

(mathematics, logic, of a relationship between two sets) Having the property that an element of one set may be assigned by the relationship to several elements in the other set, but that a given element in the second set can have only one member of the first set assigned to it.


many-to-one

(mathematics, logic, of a relationship between two sets) Having the property that many elements of one set may be assigned by the relationship to one element in the other set, and that a given element in the first set can be assigned by only one member of the second set.


many-to-many

"Is a fellow citizen with" is a many-to-many relationship on the set of people (from the set of people to the set of people).

(mathematics, logic, of a relationship between two sets) Having the property that many elements of one set may be assigned by the relationship to any element in the other set, and that a given element in the first set can also be assigned more than one member of the second set.


monadic second-order logic

monadic second-order logic (countable and uncountable, plural monadic second-order logics)

(logic) A formal deductive system which extends first-order logic by the ability to quantify over unary predicates over individual members of the universe of discourse.


elementary equivalence

elementary equivalence (plural elementary equivalences)

(logic) A relation between two structures which have the property that every first-order formula is valid in one structure if and only if it is valid in the other.


forgetful functor

forgetful functor (plural forgetful functors)

(category theory) a functor that forgets or drops some or all of its input's structure or properties before mapping to the output

gleymskuvarpi

category theory: a functor that forgets its inputs's structure


liar paradox

liar paradoxes (plural)
liar's paradox (alternative)

Graham Priest has argued the liar paradox is a true dialetheia.

(philosophy, logic) A paradox involving statements such as "this sentence is false", or "the following statement is true: the previous statement is false", which cannot be meaningfully regarded as either true or false.


antilogism

(logic) A false syllogism in which two premises contradict the third


noneism

From none + -ism, coined by Richard Routley.

(logic, metaphysics) The theory that some things do not exist, and it is possible to quantify over non-existent things using the particular or existential quantifier.


switching function

switching function (plural switching functions)

Boolean function
Boolean function#English


contrapose

en-verb

contrapose (third-person singular simple present contraposes, present participle contraposing, simple past and past participle contraposed)

contraposes (present singular third-person)
contraposing (participle present)
contraposed (participle past)
contraposed (past)

We certainly do not want to take our simple categorical statements and contrapose them into cumbersome natural language.

2005, Robert Malcolm Murray, Nebojsa Kujundzic, Critical Reflection: A Textbook for Critical Thinking, →ISBN, page 214:

To contrapose an argument one swaps the conclusion with any one of the premisses and negates each of the swapped statements.

2006, Philosophy of Psychology and Cognitive Science, →ISBN, page 461:

But subjunctive conditionals do not contrapose, and we are misled into accepting a sensitivity condition by confusing it with a safety condition.

2015, Ernest Sosa, Judgment and Agency, →ISBN, page 120:

(transitive, logic) To place in contraposition.


contraposed

Inductive extrapolation, in effect, is grounded in the contraposed universal generalisation 'where the reason, so the target.'

2001, Jonardon Ganeri, Philosophy in Classical India: The Proper Work of Reason, →ISBN, page 118:


law of identity

The law of identity states that an apple is an apple.

(logic) The logical principle that anything equals itself, expressed by the symbolic equation A=A.


law of nontriviality

Aristotle argued the law of non-triviality was an inevitable belief for all people because denying it is not possible if a person is going to behave in a discriminating way.

(logic) The rule that states that not all propositions are true, the opposite of trivialism.


equisatisfiable

The "isomorphic" PPTL formulas can be obtained by first an equisatisfiable translation and then an isomorphic mapping.

2015, Xu Lu, Zhenhua Duan, Cong Tian, “Extending PPTL for Verifying Heap Evolution Properties”, in arXiv:

(logic) Of a pair of formulas, where one formula is satisfiable whenever the other is satisfiable (either both formulas are satisfiable or both are not).


equisatisfiability

(logic, of two formulas) The condition of being equisatisfiable.


trivialist

(logic) A proponent of trivialism, the metaphysical and logical theory that all propositions are true.


linear logic

Gentzen also wrote down inference rules suitable for the intuitionistic propositional and predicate calculi. These rules lack the mirror symmetry of the classical case. But in the 1980s, this symmetry was restored by Girard’s invention of ‘linear logic’ [43].

2009 February 3, John C. Baez with Mike Stay, “Physics, Topology, Logic and Computation: A Rosetta Stone”, in (Please provide the book title or journal name), page 39:

(logic) A logic in which two structural rules are missing from its sequent calculus: those for weakening and contraction; which has some extra logical connectives, so that it has both "additive" and "multiplicative" versions of the typical binary connectives and truth constants; and which has a pair of modal, "exponential" operators for resource management, to help make up for the loss of the two structural rules.


counterpossible

(philosophy, logic) A counterfactual in which the antecedent could not possibly be true.


sequent calculus

en-noun

sequent calculus (plural sequent calculi)

(mathematics, logic) A set of inference rules for deriving true sequents from other true sequents.


atomic formula

atomic formula (plural atomic formulae or atomic formulas)

atomic formulae (plural)
atomic formulas (plural)

In predicate logic, an atomic formula consists of a predicate together with its arguments, its arguments being terms.

(logic) A formula which does not have smaller subformulas.


superpredicate

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(logic) A kind of relation between predicates.


propositional constant

propositional constant (plural propositional constants)

(logic) One of a set of letters or symbols which are assigned fixed truth values independently of any interpretation or valuation, and each of which qualifies as a prime formula of their language.


hypolemma

hypolemma (plural hypolemmas or hypolemmata)

hypolemmas (plural)
hypolemmata (plural)

From Latin hypolēmma, from Ancient Greek ὑπόλημμᾰ (hupólēmmă, “supposition”), from ὑπο- (hupo-, “lesser”) + λῆμμα (lêmma, “assumption”).

affix

(logic, rare) A lesser proposition or subsumption within a larger syllogism.


paraconsistentization

Moreover, we study the case of paraconsistentization of propositional classical logic.

2016, Edelcio G. de Souza, Alexandre Costa-Leite, Diogo H.B. Dias, “On a paraconsistentization functor in the category of consequence structures”, in arXiv:

(logic, rare) Conversion to a paraconsistent form.


superfalsity

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(logic) In supervaluationism, the quality of being superfalse.


superfalsehood

LL-Q1860 (eng)-Flame, not lame-superfalsehood.wav


protothetic

In this paper, we present a system which is a natural deduction (in Fitch-Ja\'skowski's style) version of protothetic.

2015, Pierre Joray, “Teaching Leśniewski's Prothetic with a Natural Deduction System”, in arXiv:

(philosophy, logic) A particular theory of the relationships between propositions


Löb's theorem

Named after Martin Löb, who formulated it in 1955.

(logic) A theorem stating that, in any formal system F with Peano arithmetic, for any formula P, if it is provable in F that "if P is provable in F then P is true", then P is provable in F.


superintuitionistic

The results also apply to other fragments of other superintuitionistic logics under certain conditions. In particular, the exponential lower bounds on the length of intuitionistic EF proofs by Hrube\v{s} \cite{hru:lbint}, generalized to exponential separation between EF and SF systems in superintuitionistic logics of unbounded branching by Je\v{r}\'abek \cite{ej:sfef}, can be realized by implicational tautologies..

2015, Emil Jeřábek, “Proof complexity of intuitionistic implicational formulas”, in arXiv:

(logic) Describing an extended form of intuitionistic logic


functionhood

He did not distill the essential idea of functionhood: that to each x in the domain there corresponds a unique y in the range.

1999, William Dunham, Euler: The Master of Us All, The Mathematical Association of America, published 1999, page 17:

(mathematics, logic, grammar) The state or quality of being a function.


definee

[…] by presenting us with another term (the definer) that has the same meaning as the definee. Moreover, a definition does not merely give us a single word that is a synonym of the definee:
"Liberty" means the same as "freedom."

1976, Monroe C. Beardsley, Writing With Reason: Logic for Composition - Page 37:


predicativism

(philosophy, logic) A philosophical position that definitions of concepts can only use other concepts that don't depend on the concept being defined.


predicativist

(philosophy, logic) An advocate of predicativism.


impredicativity

(mathematics, logic) A self-referencing definition.


gentle murder paradox

A paradox of deontic logic, whereby the plausible statements (i) if you murder, you ought to murder gently, (ii) you do commit murder, and (iii) to murder gently you must murder imply the less plausible statement (iv) you ought to murder.


counterlegal

(logic, philosophy) That goes against a system of laws, such as the laws of nature.


counterlegal

(logic, philosophy) A proposition that goes against a system of laws.


veristic

en-adj

veristic (comparative more veristic, superlative most veristic)

more veristic (comparative)
most veristic (superlative)

veristic variables

(logic) Having multiple valid solutions.


Zhegalkin polynomial

en-noun

Zhegalkin polynomial (plural Zhegalkin polynomials)

Named after the Russian mathematician Ivan Ivanovich Zhegalkin (1869–1947), who introduced them in 1927.

(logic, algebra) A Boolean-valued polynomial formed as a XOR-sum (or "ring sum") of conjunctions.

Ivan Ivanovich Zhegalkin
Zhegalkin polynomial


ring sum normal form

"Ring sum" refers to a XOR-sum of monomials, where XOR is the additive operator of a Boolean ring.

(algebra, logic) The form of a Boolean formula expressed using only the operators and constants of a Boolean ring (XOR, AND, 0, 1), some variables and possibly also coefficients, and without using any parentheses.


algebraic normal form

"Algebraic" would refer to the fact that a two-element Galois field, 𝔽₂ adjoined with some variables forms a polynomial algebra, 𝔽₂[x_1,...,x_n].

It is well known that every (n,1)-function f can be written uniquely as a multilinear polynomial over #92;mathbb#123;F#125;#95;2
#92;qquad#92;qquad#92;qquad#92;qquad#92;qquadf(x#95;1,...,x#95;n)#61;#92;sum#95;#123;X#92;subseteq#91;n#93;#125;#92;alpha#95;X#92;prod#95;#123;i#92;inX#125;x#95;i.
This polynomial is called the Zhegalkin polynomial or the algebraic normal form (ANF) of f.

2015 September 22, Magnus Gausdal Find, Joan Boyar, “Multiplicative Complexity of Vector Valued Boolean Functions”, in arxiv.org, retrieved 2017-08-10:

ring sum normal form
ring sum normal form


clausal normal form

clausal normal form (countable and uncountable, plural clausal normal forms)

conjunctive normal form
conjunctive normal form#English

(logic) Synonym of conjunctive normal form.


truth tree

(logic) A variation of a truth table in which any node (representing a statement) has branches if and only if other statements (true or false) may be derived from it


relevance conditional

relevance conditional (plural relevance conditionals)

To be more precise, the speaker of a relevance conditional assumes a mutual understanding on the parts of the speaker and the addressee[s] that the actualization of P is a sufficient condition for the relevance of the Q-utterance.

2001, Renaat Declerck, Susan Reed, Conditionals: A Comprehensive Empirical Analysis, page 425:

For example, relevance conditionals are constructions in conditional shape that do not seem to express a clear conditional meaning at first glance. This has led semanticists to introduce analyses of relevance conditionals that capture many of their properties but are very specific to this construction.

2013, Tatjana Scheffler, Two-dimensional Semantics: Clausal Adjuncts and Complements, Walter de Gruyter, page 124:

A known grammatical marker of relevance conditionals is their resistance to linking their antecedent and consequent clauses with 'then': consider the mild oddity of 'If you want biscuits then there are some on the table.'[…]This peculiarity of relevance conditionals is straightforwardly explained by the ellipsis theory, according to which the consequent clause doesn't in general express a real (logical) consequent; 'There are some biscuits on the table' is just asserted as true.

2016, Stephen Finlay, Confusion of Tongues: A Theory of Normative Language, Oxford University Press, page 71:

(logic, grammar) A subordinate clause, usually introduced by if, that asserts the relevance of the clause to the main clause of the sentence, but not that it entails the main clause; a construction involving such a clause.

zdanie podrzędne związku

antecedent clause that asserts a relevance relationship, not a consequence relationship, to its associated consequent clause; construction comprising such a clause and its consequent


modus ponendo ponens

Borrowed from Latin modus ponendo ponens (roughly, “mode where affirming affirms”).

1. If P, then Q.

2. P.

3. Therefore, Q.

(philosophy, logic) A valid form of argument in which the antecedent of a conditional proposition is affirmed, thereby entailing the affirmation of the consequent. Modus ponendo ponens has the form:

肯定前件

a valid form of argument in which the antecedent of a conditional proposition is affirmed, thereby entailing the affirmation of the consequent


modus ponendo tollens

Borrowed from Latin modus pōnendō tollens (roughly, “mode where affirming infirms”).

1. If P, then non-Q.

2. P.

3. Therefore, non-Q.

(philosophy, logic) A valid form of argument in which the antecedent of a conditional proposition is affirmed, thereby entailing the infirmation of the consequent. Modus ponendo tollens has the form:


modus tollendo ponens

Borrowed from Latin modus tollendō pōnēns (roughly, “mode where negating confirms”).

1. Either P or Q.

2. non-P.

3. Therefore, Q.

(philosophy, logic) A valid form of argument in which the antecedent of a conditional proposition is negated, thereby entailing the affirmation of the consequent. Modus ponendo tollens has the form:


modus tollendo tollens

Borrowed from Latin modus tollendō tollēns (roughly, “mode where infirming infirms”).

1. If non-P, then non-Q.

2. non-P.

3. Therefore, non-Q.

(philosophy, logic) A valid form of argument in which the antecedent of a conditional proposition is infirmed, thereby entailing the infirmation of the consequent. Modus tollendo tollens has the form:


negand

A negation is true if and only if the negand is not true.

2006, Dov M. Gabbay, John Woods, Logic and the Modalities in the Twentieth Century, page 345:

"Maisie is a rottweiler" is the negand of "Maisie is not a rottweiler."

2012, Nicholas Jeremy Josef Smith, Logic: The Laws of Truth, page 25:

In a negation, the formula being negated is called the negand.

2015, Howard Gregory, Language and Logics:

(logic) The proposition that is to be negated.


indesignate

(logic) Without any indication of quantity.


semantic tableau

semantic tableau (plural semantic tableaus or semantic tableaux)

semantic tableaus (plural)
semantic tableaux (plural)


contradictionism

(logic, philosophy) A worldview according to which contradictory statements can both be true.


prejacent

Multiple origins. Borrowed from Middle French prejacent (“previously existing”) during the sixteenth century. Also attested in twelfth century British sources, from post-classical Latin praeiacens (“situated before”).

bor

So, every exception set S that is not a subset of C(={b}) makes the prejacent statement false, i.e. makes it so that some (NPᵂ–S) (VPᵂ)=0.

2013, John Gajewski, “An analogy between a connected exceptive phrase and polarity items”, in Eva Csipak, Regine Eckardt, Mingya Liu, editors, Beyond ‘Any’ and ‘Ever’: New Explorations in Negative Polarity, page 196:

(logic, of a proposition) Laid out earlier, prior.


prejacent

The subject of the prejacent and its exclusive must be one and the same (I grant that), but the subject of the prejacent ‘every man is a man’ is just ‘man’

1979 [c. 1499], Norman Kretzman, Logica Magna, translation of original by Pauli Veneti, pages 27–29:

I have suggested elsewhere (Horn 1992: 182-83) that this case requires an invocation of the notion of explicature or pragmatic enrichment […] so that the prejacent, while not constituting part of the linguistic meaning contributed by only, enters into the determination of what is said and hence into the enriched propositional content.

1997, Laurence Horn, “Negative polarity and the dynamics of vertical inference”, in Danielle Forget, Paul Hirschbühler, France Martineau, María Luisa Rivero, editors, Negation and Polarity, page 176:

(philosophy, logic, linguistics) A proposition laid out previously; a proposition from which another proposition is inferred.


middle term

(logic) The term of a syllogism that appears in both the major premise and the minor premise, but not in the conclusion.


counterinduction

(logic) The opposite process to induction (creation of a specific example of a general principle)


axiom of regularity

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that every non-empty set contains a member that is disjoint from that set.


axiom of pairing

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that if two sets exist, there exists a set with those two sets as its sole elements.


axiom of union

(set theory) One of the axioms in axiomatic set theory, equivalent to the statement that a union exists for any set, containing exactly all the elements contained within the sets within that set.


axiom of infinity

(set theory) One of the axioms in axiomatic set theory that guarantees the existence of an infinite set.


many-valued logic

en-noun

many-valued logic (plural many-valued logics)

propositional calculus
propositional calculus

(logic) A kind of non-classical propositional calculus whose semantics makes use of more than two truth values.


horned syllogism

horned syllogism (plural horned syllogisms)


canonical disjunctive normal form

canonical disjunctive normal form (uncountable)

Meronym: minterm

Disjunctive normal form
disjunctive normal form

(logic) Disjunctive normal form with the additional property that all of the terms of the sum contain the same literals, so that the terms differ from each other only in their patterns of complementation (of their literals).

kanoninen disjunktiivinen normaalimuoto

disjunctive normal form with the additional property that all of the terms of the sum contain the same literals, so that the terms differ from each other only in their patterns of complementation (of their literals)

but see kanoniczna alternatywna postać normalna


canonical conjunctive normal form

canonical conjunctive normal form (plural canonical conjunctive normal forms)

canonical conjunctive normal forms (plural)

Conjunctive normal form
conjunctive normal form

(logic) Conjunctive normal form with the additional property that all of the terms of the product contain the same literals, so that the terms differ from each other only in their patterns of complementation (of their literals).

but see kanoniczna koniunkcyjna postać normalna


Boolean atom

This is a consequence of a fact established earlier, namely, that any set C formed by choosing one formula from each Sⱼ must contain some formula πB and its conjugate, where B is a Boolean atom of X.

1975, Sue Ann Toledo, Tableau systems for first order number theory and certain order theories, page 234:

If C is a Boolean atom, the claim is obviously true.

1982, Krister Segerberg, Classical propositional operators, page 55:

A list is a bit vector if each of its members is a Boolean atom.

1994, Specification and Verification of Gate-level VHDL Models of Synchronous and Asynchronous Circuits, page 4:

(logic) A logical proposition that cannot be derived from other logical propositions by a Boolean operation.


major term

(logic) In a syllogism, the term that is the predicate of the conclusion.


minor term

minor terms (plural)
minor (alternative)

The minor term in John Stuart Mill's famous syllogism is Socrates; the major term is mortal.

(logic) The subject of the conclusion in a categorical syllogism.


metatheorem

From these observations we conclude that we have the following metatheorem. Metatheorem 3: If φ is a theorem of second-order monadic predicate logic, then trs'(φ) is a theorem of our present conceptualist logic.

2007, Nino B. Cocchiarella, Formal Ontology and Conceptual Realism, Springer Science & Business Media, →ISBN, page 224:

The metatheorem of deduction will play an essential role ...

2008, Walter Carnielli, Marcelo Coniglio, Dov M. Gabbay, Paula Gouveia, Cristina Sernadas, Analysis and Synthesis of Logics: How to Cut and Paste Reasoning Systems, Springer Science & Business Media, →ISBN, page 70:

This paper presents a metatheorem with the following property: Given a proven axiom-free lemma, which interrelates some of the elementary properties of a binary relation, the metatheorem mechanically transforms this lemma into its uniquely corresponding complementary lemma.

(logic) A statement about theorems proven in a metalanguage.


meta-induction

(logic, philosophy) The use of inductive reasoning to select the best choice from a number of different methods of prediction.


law of contradiction

law of noncontradiction
law of noncontradiction#English

(logic) Synonym of law of noncontradiction.


monosyllogism

(logic) An argument expressed as a single syllogism


Gödel numbering

Gödel numbering (plural Gödel numberings)

(logic) A function assigning a Gödel number to each well-formed formula of some formal language.


Gödelization

From Kurt Gödel (or Gödel number) and -ization.

2013, Encyclopedia of Mathhttps://encyclopediaofmath.org/wiki/Gödelization
Gödelizations with bijective mapping β are of special interest.


Andersonian

(logic) Of or pertaining to Alan Ross Anderson (1925–1973), American logician, or his work in relevance logic and deontic logic.


clausify

en-verb

clausify (third-person singular simple present clausifies, present participle clausifying, simple past and past participle clausified)

clausifies (present singular third-person)
clausifying (participle present)
clausified (participle past)
clausified (past)

(logic) To convert logic statements into standard clauses


clausification

(logic) The process of converting logic statements into standard clauses.


clausifier

(logic) A method, algorithm or program that converts logic statements into standard clauses.


by extension

He argued that the failure of a democratic government, by extension, meant the failure of the people.

(logic) Used to connect the internal logic of two statements.


substituend

Borrowed from Latin substituendus; compare substituendum.

Each substituend SN, it seems, would amount to the same thing as its corresponding UN in SOME not very strained context.

1972, John King-Farlow, W.N. Christensen, “Probability and the 'Will to Believe'”, in Logic, Identity, and Consistency, D. Reidel, →ISBN, page 177:

Note that a variable is a substituend of a variable only in a vacuous sense , and so we shall be ignoring them while speaking of the substituends of a given variable.

1998, Pranab Kumar Sen, Logic, Identity, and Consistency (Studies in Philosophical and Non-Standard Logic; II), Allied Publishers, →ISBN, page 134:

The pragmatic process of substitution requires some given context to provide an appropriate substituend for a metavariable projected by a pronoun.

2006 March 13, Lutz Martin, “Relative Clause Construal”, in The Dynamics of Language, Brill, →ISBN, page 85:

(logic, linguistics) A substitute; something that substitutes for another.


contrapositively

(logic) By the inverse of the converse of a given proposition.


coalternate

(logic, of a set of propositions) mutually alternate


monosyllogistic

(logic) Relating to, or expressed as a monosyllogism


pseudoproposition

LL-Q1860 (eng)-Flame, not lame-pseudoproposition.wav

(philosophy, logic) A spurious or meaningless proposition


eubouliatic

From Ancient Greek εὐβουλία (euboulía) + -tic.

On the basis of the last square of opposition, Anderson [1968] suggests christening the topic (provided that it is worth pursuing) eubouliatic logic, the logic of prudence and related concepts, just as deontic logic is the logic of obligation and related concepts.

1987, Introduction to Deontic Logic and the Theory of Normative Systems:

(logic, rare) Pertaining to the modality of safety or prudence.


sorites paradox

en-noun

sorites paradox (plural sorites paradoxes)

(logic) A paradox arising from vague predicates and repeatedly applying a small difference. For example, removing one grain from a heap of sand still leaves a heap, yet removing enough grains will eventually not leave a heap.


hyperproperty

(logic, physics) A generalisation of a property (or sets of properties)


paralogistic

(logic) Relating to paralogy; fallacious


modus morons

1. If P, then Q.

2. Q.

3. Therefore P.

(logic) An invalid argument, which has this form:


Buridan formula

Named in honour of the medieval philosopher Jean Buridan.

(logic) In quantified modal logic, the formula ◇∀xFx→∀x◇Fx, meaning "if possibly everything is F, then everything is possibly F".


Barcan formula

Introduced by the logician Ruth Barcan Marcus.

(logic) In quantified modal logic, the formula ∀x□Fx→□∀xFx, meaning "if every x is necessarily F, then it is necessary that every x is F".


fuzzy math

(logic) A form of logic based on the concept of a fuzzy set.


Herbrand's theorem

(logic) A fundamental result of mathematical logic, essentially allowing a certain kind of reduction of first-order logic to propositional logic.


Davis-Putnam algorithm

Developed by Martin Davis and Hilary Putnam.

(logic) An algorithm for checking the validity of a first-order logic formula using a resolution-based decision procedure for propositional logic.


Baroco

(logic, obsolete) Alternative form of Baroko.


Bocardo

(logic, obsolete) Alternative form of Bokardo.


ecthesis

From Ancient Greek ἐκθετικός (ekthetikós).

(logic, historical) An exposition or setting out; used by Aristotle in setting forth various proofs, though the nature of the process is disputed; perhaps simply writing out in letter notation.

The Ecthesis

a letter published in 638 CE by the Byzantine emperor Heraclius which defined Monothelitism as the official imperial form of Christianity.


truthset

truthsets (plural)
truth set (alternative)

(mathematics, logic) A set of all true values


ultratotal

(logic, archaic) That quantifies more than the total quantity.


unverity

unverity (countable and uncountable, plural unverities)

All and only classically valid arguments are those which preserve verity in the following sense: there is no assignment of verities such that the unverity of the conclusion exceeds the sum of the unverities of the premises.

2013, A. Bottani, Massimiliano Carrara, P. Giaretta, Individuals, Essence and Identity, page 311:

(logic) A value equal to 1 minus the truth value of a statement.


self-justify

en-verb

self-justify (third-person singular simple present self-justifies, present participle self-justifying, simple past and past participle self-justified)

self-justifies (present singular third-person)
self-justifying (participle present)
self-justified (participle past)
self-justified (past)

To provide self-justification, to justify or excuse oneself or itself.


hold come what may

Popularized by the logician W.V.O. Quine.

(logic) Any belief that one is unwilling to relinquish, regardless of the evidence with which one might be presented.


toupee fallacy

en-noun

toupee fallacy (plural toupee fallacies)

In reference to the conventional wisdom that all toupees look bad, caused by the fact that good quality toupees are usually not noticed.

Not all toupees are bad, but you only spot the bad toupees. That's the toupee fallacy.

(logic) A form of selection bias in which a thing whose quality is measured in terms of being difficult to detect is wrongly judged to be of poor quality in general, caused by the fact that most people only notice poor quality instances of it.


contraponend

(logic) The original proposition involved in a contraposition.


precompleteness

(logic) The quality of being precomplete.


precomplete

(logic) Having a certain property that is analogous to completeness but weaker.


pseudostatement

LL-Q1860 (eng)-Flame, not lame-pseudostatement.wav

(logic, philosophy) An unverifiable statement.


Peirce arrow

(logic) The downward arrow symbol ↓ denoting the NOR operation.


inference to the best explanation

inference to the best explanation (uncountable)

desc

→ Finnish: päättely parhaaseen selitykseen (calque)

abductive reasoning
abductive reasoning#English

(logic, fixed phrase) Synonym of abductive reasoning.


square of opposition

en-noun

square of opposition (plural squares of opposition)

categorical proposition
categorical proposition

(logic) A diagram representing the relations between the four basic categorical propositions, which are positioned at the four corners of a square.


all-operator

universal quantifier
universal quantifier#English

(logic) Synonym of universal quantifier.


Lévy hierarchy

Zermelo-Fraenkel set theory
Zermelo-Fraenkel set theory

(set theory, logic) A hierarchy of formulas in the formal language of the Zermelo-Fraenkel set theory. Its first level contains only formulas with no unbounded quantifiers and is denoted by Δ₀=Σ₀=Π₀. Subsequent levels are given by finding a formula in prenex normal form which is provably equivalent over ZFC, and counting the number of changes of quantifiers.


Kleene plus

After Stephen Cole Kleene (1909-1994), American mathematician.

(logic, computing theory) The plus symbol, +, used as an operator to concatenate one or more strings from a given set, widely used in regular expressions.


paralogics

[…] obtained by adding the standard modal operators □ (necessity) and ◇ (possibility) to paralogics.

The great majority of mathematicians and philosophers regard the principle of contradiction as the over-arching principle of logic, and are therefore inclined to regard paralogics (in the improbable case they have heard of them) not just as deviations from orthodoxy but as aberrations.

2012, M. Bunge, Epistemology & Methodology III, page 58:

(logic) The study of paralogic rhetoric; the study of understanding language by successive approximation.


pseudosyllogism

(logic) A false syllogism, one that is not logically coherent.


Aristotle's dictum

(logic) The principle that whatever is affirmed or denied of a whole kind may be affirmed or denied (respectively) of any subkind of that kind.


Church-Rosser property

Church-Rosser property (plural Church-Rosser properties)

Alonzo Church and J. Barkley Rosser proved in 1936 that lambda calculus has this property.

(logic, computer science) The property of a rewriting system for which x stackrel *↔y implies x mathbin ↓y for all objects x, y.


discretive proposition

discretive proposition (plural discretive propositions)

(logic) A compound proposition in which contrasting judgments are made, connected by a word such as "but" or "though".


nonconjunction

nonconjunction (countable and uncountable, plural nonconjunctions)

(logic) The negation of the conjunction operation, expressed in ordinary language as "not both".


alternative denial

alternative denial (countable and uncountable, plural alternative denials)


hyperschema

hyperschema (plural hyperschemata or hyperschemas)

hyperschemata (plural)
hyperschemas (plural)

(logic) A generalization of the idea of a schema, having variable arity.


material nonimplication

material implication
material implication

(logic) The negation of material implication, such that for any two propositions P and Q, the material nonimplication from P to Q is true if and only if the negation of the material implication from P to Q is true.


regressive argument

regressive argument (plural regressive arguments)

regress argument
regress argument#English

(philosophy, logic) Synonym of regress argument.


regress argument

regress argument (plural regress arguments)

(philosophy, logic) An argument derived from (an infinite sequence of) previously accepted propositions.


Aumann's agreement theorem

Proved by Robert Aumann in a 1976 paper.

(logic) A theorem stating that two Bayesian agents with the same prior beliefs cannot "agree to disagree" about the probability of an event if their individual beliefs are common knowledge.