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See searches and references containing LOGICAL CONSEQUENCE!LOGICAL CONSEQUENCE
Relationship where one statement follows from another
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that
Logical_consequence
Type of logical system
statements which are true in all models are provable. Although the logical consequence relation is only semidecidable, much progress has been made in automated
First-order_logic
Statement that is true regardless of the truth or falsity of its constituent propositions
thought of as providing accounts of the nature of logical truth, as well as logical consequence. Logical truths are generally considered to be necessarily
Logical_truth
Polish–American mathematician (1901–1983)
pointing out that his definition of logical consequence depends upon a division of terms into the logical and the extra-logical and he expresses some skepticism
Alfred_Tarski
Statement that is taken to be true
then }}\Sigma \vdash \phi } that is, for any statement that is a logical consequence of Σ {\displaystyle \Sigma } there actually exists a deduction of
Axiom
Method of deriving conclusions
"Logical Consequence". Internet Encyclopedia of Philosophy. Retrieved 28 March 2025. McKeon, Matthew W. (2010). The Concept of Logical Consequence: An
Rule_of_inference
Form of reasoning
deductive argument is usually referred to as "logical consequence". According to Alfred Tarski, logical consequence has 3 essential features: it is necessary
Deductive_reasoning
Logical principles
cognitive psychology. Logic is concerned with the relationships of logical consequence between propositions or sentences. The laws are not universally accepted
Law_of_thought
Subfield of mathematics
compactness theorem, demonstrating the finitary nature of first-order logical consequence. These results helped establish first-order logic as the dominant
Mathematical_logic
Mathematical model for deduction or proof systems
with the deductive nature of the system. The logical consequence (or entailment) of the system by its logical foundation is what distinguishes a formal system
Formal_system
If and only if relation
In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or bidirectional implication or biimplication
Logical_biconditional
Logical connective
logic. Many textbooks reserve the term logical consequence (or logical implication) for the semantic consequence relation with the symbol ⊨ {\displaystyle
Material_conditional
Study of the semantics, or interpretations, of formal and natural languages
validity, and logical consequence. While logical syntax concerns the formal rules for constructing well-formed expressions, logical semantics establishes
Semantics_(logic)
Logical connective AND
\wedge } ) is the truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as ∧ {\displaystyle
Logical_conjunction
Fundamental theorem in mathematical logic
be expressed more generally in terms of logical consequence. We say that a sentence s is a syntactic consequence of a theory T, denoted T ⊢ s {\displaystyle
Gödel's_completeness_theorem
Logical operation
{\displaystyle P\rightarrow \bot } (where → {\displaystyle \rightarrow } is logical consequence and ⊥ {\displaystyle \bot } is absolute falsehood). Conversely, one
Negation
Symbol connecting formulas in logic
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is an operator that combines or modifies
Logical_connective
Whether a decision problem has an effective method to derive the answer
logic are not. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for
Decidability_(logic)
Argument whose conclusion must be true if its premises are
argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a necessary consequence of its premises
Validity_(logic)
Impossible task in computing
general problem of deciding whether a given first-order sentence is a logical consequence of a given finite set of sentences, but validity in first-order theories
Entscheidungsproblem
Programming paradigm based on formal logic
the logical semantics, any result of a computation of a concurrent logic program is a logical consequence of the program, even though not all logical consequences
Logic_programming
Non-contradiction of a theory
for every formula φ in its language, at least one of φ or ¬φ is a logical consequence of the theory. Presburger arithmetic is an axiom system for the natural
Consistency
Establishment of a theorem using inference from the axioms
semantics (i.e. what they mean). A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive
Formal_proof
Symbol representing a property or relation in logic
In logic, a predicate is a non-logical symbol that represents a property or a relation, though, formally, does not need to represent anything at all.
Predicate_(logic)
Mathematical theory of data types
Gregory Bateson introduced a theory of logical types into the social sciences; his notions of double bind and logical levels are based on Russell's theory
Type_theory
Doctrine of multiplicity in contrast with monism
instrumentalism). Pluralism about logical consequence says that because different logical systems have different logical consequence relations, there is therefore
Pluralism_(philosophy)
Logical connective OR
logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as
Logical_disjunction
Term in logic and deductive reasoning
that can be proven in the system is logically valid with respect to the logical semantics of the system. These two properties are different but closely
Soundness
In mathematics, a statement that has been proven
theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms
Theorem
In logic, a statement which is always true
interpretation of its component terms, with only the logical constants having a fixed meaning. It is a logical truth. For example, a formula that states "the
Tautology_(logic)
Concept in logic
logically equivalent if they have the same truth value in every model. The logical equivalence of p {\displaystyle p} and q {\displaystyle q} is sometimes
Logical_equivalence
Characteristic of some logical systems
if it can express the subject matter for which it is intended. A set of logical connectives associated with a formal system is functionally complete if
Completeness_(logic)
Set of sentences in a formal language
are taken as axioms. In a deductive theory, any sentence that is a logical consequence of one or more of the axioms is also a sentence of that theory. More
Theory_(mathematical_logic)
Mathematical table used in logic
functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In
Truth_table
Problem in computer science
on the Programming System under consideration. Logical Limitations to Machine Ethics, with Consequences to Lethal Autonomous Weapons - paper discussed
Halting_problem
Paradox in set theory
axioms of set theory while maintaining a standard logical language, while Russell modified the logical language itself. The language of ZFC, with the help
Russell's_paradox
Mathematical logical symbol of 3 dots
In logical argument and mathematical proof, the therefore sign, ∴, is generally used before a logical consequence, such as the conclusion of a syllogism
Therefore_sign
Steps in reasoning
Inferences are steps in logical reasoning, moving from premises to logical consequences. Inference is traditionally divided into deduction and induction
Inference
Attempt to persuade or to determine the truth of a conclusion
deductive argument asserts that the truth of the conclusion is a logical consequence of the premises: if the premises are true, the conclusion must be
Argument
Rules used for constructing, or transforming the symbols and words of a language
formal language need not be symbols of anything. For instance there are logical constants which do not refer to any idea, but rather serve as a form of
Syntax_(logic)
Value indicating the relation of a proposition to truth
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical
Truth_value
Type of logical argument that applies deductive reasoning
Greek: συλλογισμός, syllogismos, 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based
Syllogism
Study of correct reasoning
Metaphysics Research Lab, Stanford University. McKeon, Matthew. "Logical Consequence". Internet Encyclopedia of Philosophy. Archived from the original
Logic
Assignment of meaning to the symbols of a formal language
interpretation that satisfies ψ then φ is said to be a logical consequence of ψ). Some of the logical symbols of a language (other than quantifiers) are truth-functional
Interpretation_(logic)
Branch of logic
currently studied in universities, is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered
Propositional_logic
Concept in propositional logic
In propositional logic, tautological consequence is a strict form of logical consequence in which the tautologousness of a proposition is preserved from
Tautological_consequence
Symbol in mathematical logic
metalogic, the study of formal languages; the turnstile represents syntactic consequence (or "derivability"). This is to say, that it shows that one string can
Turnstile_(symbol)
Hypothetical megastructure around a star
Infrared Radiation". Dyson speculated that such structures would be the logical consequence of the escalating energy needs of a technological civilization and
Dyson_sphere
Study of the scope and nature of logic
often seen as the study of correct reasoning, valid inference, or logical consequence. It is a formal science that investigates how conclusions follow
Philosophy_of_logic
Topics referred to by the same term
Look up consequence in Wiktionary, the free dictionary. Consequence may refer to: Logical consequence, also known as a consequence relation, or entailment
Consequence
Logical operator in propositional calculus
Logical equality is a logical operator that compares two truth values, or more generally, two formulas, such that it gives the value True if both arguments
Logical_equality
Limitative results in mathematical logic
Jr. (1996). Logical dilemmas: The life and work of Kurt Gödel. Taylor & Francis. ISBN 978-1-56881-025-6. Dawson, John W. Jr. (1997). Logical dilemmas: The
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Mathematical use of "for all"
mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or
Universal_quantification
consequence relation is a non-monotonic consequence relation satisfying certain properties listed below. A rational consequence relation is a logical
Rational_consequence_relation
Mathematical use of "for all" and "there exists"
∈ D P ( x ) {\displaystyle \forall x\in D\;P(x)} is equivalent to the logical conjunction P ( a 1 ) ∧ . . . ∧ P ( a n ) {\displaystyle P(a_{1})\land
Quantifier_(logic)
1921 philosophical work by Ludwig Wittgenstein
tautology is thus central to Wittgenstein's Tractarian account of logical consequence, which is strictly deductive. 5.13 When the truth of one proposition
Tractatus Logico-Philosophicus
Tractatus_Logico-Philosophicus
Type of logic diagram
the subject. Every categorical proposition can be reduced to one of four logical forms, named A, E, I, and O based on the Latin affirmo (I affirm), for
Square_of_opposition
Collection of mathematical objects
specific logical framework. For the branch of mathematics that studies sets, see Set theory; for an informal presentation of the corresponding logical framework
Set_(mathematics)
Term in mathematical logic
combinations of the sentences being true or false are consistent. Since 2000, logical independence has become understood as having crucial significance in the
Independence (mathematical logic)
Independence_(mathematical_logic)
Approach to the semantics of logic that locates meaning in inferential role
that takes the proper subject matter of logic to be the relation of logical consequence rather than truth, and that treats meaning as a function of inferential
Proof-theoretic_semantics
Computation model defining an abstract machine
first-order logic] is solved when we know a procedure that allows for any given logical expression to decide by finitely many operations its validity or satisfiability
Turing_machine
Number of arguments required by a function
plus, the increment and decrement operators in C-style languages (not in logical languages), and the successor, factorial, reciprocal, floor, ceiling, fractional
Arity
truths can be reduced to logical truths, and all objects forming the subject matter of those branches of mathematics are logical objects. In other words
Mathematical_object
Philosophical concept
They are a logical consequence of lower-level facts about the world, similar to how a clock's ability to tell time is a logical consequence of its clockwork
Hard_problem_of_consciousness
Algebraic manipulation of "true" and "false"
the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted
Boolean_algebra
Mathematical-logic system based on functions
substitution, as used in β-reduction Harrop formula – A kind of constructive logical formula such that proofs are lambda terms Interaction nets Kleene–Rosser
Lambda_calculus
Mathematical symbol
double turnstile. It is often read as "entails", "models", "is a semantic consequence of" or "is stronger than". It is closely related to the turnstile symbol
Double_turnstile
Soviet general, pilot and cosmonaut (born 1942)
and also discovered the Dzhanibekov Effect on that same mission, a logical consequence of the Tennis racket theorem. Dzhanibekov was born Vladimir Aleksandrovich
Vladimir_Dzhanibekov
Mathematical use of "there exists"
existence of an object with a given property. It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable
Existential_quantification
Branch of mathematics that studies sets
Arithmetic. In his work, Frege tries to ground all mathematics in terms of logical axioms using Cantor's cardinality. For example, the sentence "the number
Set_theory
Infinite cardinal number
Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory Theorems (list), paradoxes Gödel's
Aleph_number
Formal systems of logic that significantly differ from standard logical systems
departures is to make it possible to construct different models of logical consequence and logical truth. Philosophical logic is understood to encompass and focus
Non-classical_logic
Mathematical set containing no elements
at least two ways: Standard first-order logic implies, merely from the logical axioms, that something exists, and in the language of set theory, that
Empty_set
Theorem for proving more complex theorems
Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory Theorems (list), paradoxes Gödel's
Lemma_(mathematics)
Set of the elements not in a given subset
(X\times Y)\setminus R.} Here, R {\displaystyle R} is often viewed as a logical matrix with rows representing the elements of X , {\displaystyle X,} and
Complement_(set_theory)
Index of articles associated with the same name
predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. Specifically, we say that a set of clauses of the form Q
Stratification_(mathematics)
Set whose elements all belong to another set
is an element of B. The validity of this technique can be seen as a consequence of universal generalization: the technique shows ( c ∈ A ) ⇒ ( c ∈ B
Subset
Form of logic that allows quantification over predicates
and a variety of other powerful logical theories could be formulated axiomatically without appeal to any more logical apparatus than first-order quantification
Second-order_logic
Symbol with a fixed meaning in logic
types of logical constants are logical connectives and quantifiers. The equality predicate (usually written '=') is also treated as a logical constant
Logical_constant
Axioms for the natural numbers
the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular,
Peano_axioms
Reasoning for mathematical statements
starting with an assumption, and with each subsequent formula a logical consequence of the preceding ones. This definition makes the concept of proof
Mathematical_proof
Logical incompatibility between two or more propositions
(or perceived as due) to presuppositions which are contradictory in the logical sense. Proof by contradiction is used in mathematics to construct proofs
Contradiction
In mathematical logic, a well-formed formula with no free variables
values, the truth value of such a formula may vary. Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy
Sentence_(mathematical_logic)
Mathematical term; concerning axioms used to derive theorems
the logical deduction of other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical
Axiomatic_system
Set of elements common to all of some sets
{\displaystyle A\cap A=A} . All these properties follow from analogous facts about logical conjunction. Intersection distributes over union and union distributes
Intersection_(set_theory)
Final event in a sequence
A result (also called upshot) is the outcome or consequence of a sequence of actions or events. Possible results include gain, injury, value, and victory
Result
Function that preserves distinctness
Formal semantics Foundations of mathematics Information theory Lemma Logical consequence Model Theorem Theory Type theory Theorems (list), paradoxes Gödel's
Injective_function
Symbols requiring interpretation
In mathematical logic, especially model theory, non-logical symbols are elements of a formal language whose interpretation may change depending on the
Non-logical_symbol
Set of elements in any of some sets
{\displaystyle A\cup A=A} . All these properties follow from analogous facts about logical disjunction. Intersection distributes over union A ∩ ( B ∪ C ) = ( A ∩
Union_(set_theory)
Logic theorem
different conceptions of the law of non-contradiction. One can interpret a logical law ontologically, e. g. to say nothing in reality is contradictory; one
Law_of_noncontradiction
Collection of sets in mathematics that can be defined based on a property of its members
a metalanguage, the classes can be described as equivalence classes of logical formulas: If A {\displaystyle {\mathcal {A}}} is a structure interpreting
Class_(set_theory)
Book on the philosophy of mathematics
everything that is a logical consequence of N + M must also be a logical consequence of just N. However, the concept of logical consequence is ambiguous. It
Science_Without_Numbers
Reasoning about equations with free variables
contrary to function theory. A given relation may be represented by a logical matrix; then the converse relation is represented by the transpose matrix
Algebraic_logic
System of formal deduction in logic
While all sources that refer to an "axiomatic" logical proof system characterize it simply as a logical proof system with axioms, sources that use variants
Hilbert_system
Axiom of set theory
which R varies over all formulas or over all formulas of a particular logical form. Zermelo 1904. Jech 1977, p. 351. Jech 1977, p. 348 ff; Mac Lane 1986
Axiom_of_choice
Subfield of automated reasoning and mathematical logic
ISBN 9780080917283. Loveland, Donald W. (2016) [1978]. Automated Theorem Proving: A Logical Basis. Fundamental Studies in Computer Science. Vol. 6. Elsevier. ISBN 9781483296777
Automated_theorem_proving
View that there are statements that are both true and false
is that dialetheism cannot describe logical consequences, once we believe in the relevance of logical consequences, because of its inability to describe
Dialetheism
Property of many systems of logic
Monotonicity of entailment is a property of many logical systems such that if a sentence follows deductively from a given set of sentences then it also
Monotonicity_of_entailment
Standard system of axiomatic set theory
been extensively studied. Landmark results in this area established the logical independence of the axiom of choice from the remaining Zermelo-Fraenkel
Zermelo–Fraenkel_set_theory
Study of computable functions and Turing degrees
and incompleteness theorems. Gödel's proofs show that the set of logical consequences of an effective first-order theory is a computably enumerable set
Computability_theory
LOGICAL CONSEQUENCE
LOGICAL CONSEQUENCE
Boy/Male
Indian, Sanskrit
Endowed with Mind; Logical
Girl/Female
Hindu
Boy/Male
Indian, Sanskrit
Logician
Girl/Female
African, Arabic, French, Indian, Muslim, Swahili, Tamil
Intelligent; Logical; Intelligent One who Reasons; Wise
Girl/Female
Tamil
Give light to others
Girl/Female
Indian, Modern, Sanskrit
Magical
Girl/Female
Hindu, Indian
Give Light to Others
Boy/Male
Indian
Intelligent, Logical
Girl/Female
Indian, Tamil
King Rama's Wife
Girl/Female
Indian
Successful; Logical Thinkers
Boy/Male
Gujarati, Hindu, Indian, Sanskrit
Logical Science
Boy/Male
German, Swedish
Elf; Magical Army; Warrior
Boy/Male
Hindu, Indian
Logical
Boy/Male
Hindu, Indian
A Magical Sword
Girl/Female
Danish, Hindu, Indian, Japanese
Ray of Light; Logical
Boy/Male
Tamil
Intelligent, Logical
Girl/Female
Australian, French, Swedish
Elf; Magical Counsel
Boy/Male
Tamil
Love and kindness, Analytical, Logical
Girl/Female
Native American
Magical dancer.
Boy/Male
Hindu
Love and kindness, Analytical, Logical
LOGICAL CONSEQUENCE
LOGICAL CONSEQUENCE
Boy/Male
Arabic
Helper
Male
English
Short form of English Rodger, RODGE means "famous spear."
Boy/Male
Hindu
Voice, Audible
Female
English
Anglicized form of Irish Gaelic Fionnghuala, FINUALA means "white shoulder."
Girl/Female
American, Australian, British, English, Latin
Honey; Blend of Melissa and Linda; Sweet
Surname or Lastname
English
English : patronymic from the personal name Stephen (see Steven).
Girl/Female
Latin
Gray eyes.
Boy/Male
Hindu
Lord Vishnu
Boy/Male
Indian, Punjabi, Sikh
One Having All Treasure; Contented
Boy/Male
Indian
Flower
LOGICAL CONSEQUENCE
LOGICAL CONSEQUENCE
LOGICAL CONSEQUENCE
LOGICAL CONSEQUENCE
LOGICAL CONSEQUENCE
adv.
In a logical manner; as, to argue logically.
a.
Having the form of, or resembling, a geometrical cone; round and tapering to a point, or gradually lessening in circumference; as, a conic or conical figure; a conical vessel.
a.
Of or pertaining to logic; used in logic; as, logical subtilties.
a.
Exciting mirth; droll; laughable; as, a comical story.
a.
Excessively logical; adhering too closely to the forms or rules of logic.
a.
Having or observing logical sequence; logically consistent and rigorous; consecutive in development or transition of thought.
a.
Skilled in logic; versed in the art of thinking and reasoning; as, he is a logical thinker.
a.
Half logical; partly logical; said of fallacies.
a.
Ignorant or negligent of the rules of logic or correct reasoning; as, an illogical disputant; contrary of the rules of logic or sound reasoning; as, an illogical inference.
n.
A logical deduction.
a.
According to the rules of logic; as, a logical argument or inference; the reasoning is logical.
v. t.
Consistent; logical.
n.
Of or pertaining to a place; limited; logical application; as, a topical remedy; a topical claim or privilege.
n.
A treatise on logic; as, Mill's Logic.
a.
Having a mixture of seriousness and sport; serious and comical.
a.
Logical.
pl.
of Lorica
n.
A person skilled in logic.
a.
Of or pertaining to the nodes; from a node to the same node again; as, the nodical revolutions of the moon.
n.
See Logic.