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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
Logical connective AND
) is the truth-functional operator of conjunction or logical conjunction. The logical connective of this operator is typically represented as ∧ {\displaystyle
Logical_conjunction
Logical connective OR
disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as ∨ {\displaystyle
Logical_disjunction
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
List of symbols used to express logical relations
suggested set of logical symbols Logic gate § Symbols Logical connective Mathematical operators and symbols in Unicode Non-logical symbol Polish notation
List_of_logic_symbols
Statement that is taken to be true
primitive connectives can be alternatively constructed. These axiom schemata are also used in the predicate calculus, but additional logical axioms are
Axiom
Logical operation
of a negation is called a negand or negatum. Negation is a unary logical connective. It may furthermore be applied not only to propositions, but also
Negation
Branch of logic
Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication
Propositional_logic
Type of logical system
"Plato". Due to the ability to speak about non-logical individuals along with the original logical connectives, first-order logic includes propositional logic
First-order_logic
Logical connective
a sentential connective within a formal language. It should not be confused with the relation of logical consequence (also called logical implication or
Material_conditional
If and only if relation
implication or biimplication or bientailment or exclusive nor, is the logical connective used to conjoin two statements P {\displaystyle P} and Q {\displaystyle
Logical_biconditional
Statement that is true regardless of the truth or falsity of its constituent propositions
is true because of the logical terms it contains which are logical connectives (e.g. "or", "and", and "nor"). Not all logical truths are tautologies of
Logical_truth
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
True when either but not both inputs are true
disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical biconditional. With two inputs
Exclusive_or
Study of correct reasoning
example, the expression " p ∧ q {\displaystyle p\land q} " uses the logical connective ∧ {\displaystyle \land } (and). It could be used to express a sentence
Logic
Binary operation that is true if and only if both operands are false
\leftrightarrow P\downarrow (Q\downarrow R)} . The logical NOR, taken by itself, is a functionally complete set of connectives. This can be proved by first showing
Logical_NOR
Topics referred to by the same term
linguistics, a word or phrase like "therefore" or "in other words". Logical connective Connective (botany), in the stamen of flowers, the sterile tissue that
Connective
Assignment of meaning to the symbols of a formal language
an interpretation does not have anything to say about logical symbols, e.g. logical connectives " a n d {\displaystyle \mathrm {and} } ", " o r {\displaystyle
Interpretation_(logic)
Logical connective
two statements are equal. It is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where
If_and_only_if
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
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
Term in logic
sentence contains no logical connectives, variables, or quantifiers. A sentence consisting of one or more sentences and a logical connective is a compound (or
Atomic_sentence
Principle governing approaches to logical semantics
satisfy. The logician Gerhard Gentzen proposed that the meanings of logical connectives does not need to be defined by a world outside of logic, but could
Logical_harmony
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)
Symbols requiring interpretation
to logical constants which are required to have the same interpretation under every model, such as logical connectives and quantifiers. A non-logical symbol
Non-logical_symbol
Subfield of mathematics
'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid
Mathematical_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
Mathematical table used in logic
First-order logic Functional completeness Karnaugh maps Logic gate Logical connective Logical graph Mathematical table Method of analytic tableaux Propositional
Truth_table
Algebraic manipulation of "true" and "false"
either true or false, the meanings of these logical connectives often have the meaning of their logical counterparts. However, with descriptions of behavior
Boolean_algebra
Overview of and topical guide to logic
Predicate variable Literal Metavariable Logical constants Logical connective Quantifier Identity Brackets Logical connective Converse implication Converse nonimplication
Outline_of_logic
Digital logic gate
gate whose function is the logical complement of the exclusive OR (XOR) gate. It is equivalent to the logical connective ( ↔ {\displaystyle \leftrightarrow
XNOR_gate
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
truth values of its constituent parts must satisfy the relevant logical connectives that occur in it (most commonly: [and], [or], [not], [only if], [if
List_of_fallacies
the larger system. subordinate connective A logical connective that occurs within the scope of another logical connective. substitution The act of replacing
Glossary_of_logic
Concept in mathematical logic
In logic, a functionally complete set of logical connectives or Boolean operators is one that can be used to express all possible truth tables by combining
Functional_completeness
Branch of mathematical logic
advanced in a similar spirit that better expressed the duality of the logical connectives, and went on to make fundamental advances in the formalisation of
Proof_theory
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)
Topics referred to by the same term
implication), a logical connective and binary truth function typically interpreted as "If p, then q" Material implication (rule of inference), a logical rule of
Implication
Any logic with four truth values
his 4-valued system. Belnap addressed the challenge of extending logical connectives to A4. Since it is the power set on {T, F}, the elements of A4 are
Four-valued_logic
Precisely specified semantic version of a statement
language and involving logical connectives, which are joined by juxtaposition to other sentences, which in turn may have logical structure. Medieval logicians
Logical_form
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)
Axiomatic system of formal deduction in logic
calculus and absorbed into the logical axioms of another. A Hilbert calculus need not take every familiar connective as primitive. For classical propositional
Hilbert_system
Logical operation
Boolean functions and propositional calculus, the Sheffer stroke denotes a logical operation that is equivalent to the negation of the conjunction operation
Sheffer_stroke
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
Function in logic
exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every
Truth_function
Set of elements common to all of some sets
identities and relations – Equalities for combinations of sets Logical conjunction – Logical connective AND MinHash – Data mining technique Naive set theory –
Intersection_(set_theory)
In logic, a statement which is always true
propositions. A formula consists of propositional variables connected by logical connectives, built up in such a way that the truth of the overall formula can
Tautology_(logic)
Type of infinite structure
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
O-minimal_theory
Possessing negative truth value
is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of
False_(logic)
Method of deriving conclusions
from premises. They are integral parts of formal logic, serving as the logical structure of valid arguments. If an argument with true premises follows
Rule_of_inference
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
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
Syntactically correct logical formula
quantifier-free formula. An atomic formula is a formula that contains no logical connectives nor quantifiers, or equivalently a formula that has no strict subformulas
Well-formed_formula
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)
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
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
Concept in mathematical logic
are P. Philosophy portal Aristotle Contraposition Inverse (logic) Logical connective Obversion Term logic Transposition (logic) Robert Audi, ed. (1999)
Converse_(logic)
3-volume treatise on mathematics, 1910–1913
the behaviour of the symbols "⊢" (assertion of truth), "∾" (logical not), and "V" (logical inclusive OR). Truth-values: PM embeds the notions of "truth"
Principia_Mathematica
Rule of mathematical logic
the logical discipline of proof theory, a structural rule is an inference rule of a sequent calculus that does not refer to any logical connective but
Structural_rule
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
Set of all things that may be the input of a mathematical function
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Domain_of_a_function
Impossible task in computing
statement is universally valid if and only if it can be deduced using logical rules and axioms, so the Entscheidungsproblem can also be viewed as asking
Entscheidungsproblem
Basic framework of mathematics
Foundations of mathematics are the logical and mathematical frameworks that allow the development of mathematics without generating self-contradictory
Foundations_of_mathematics
Range of application for a quantifier or connective in a logical formula
dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope. The scope of a logical connective
Scope_(logic)
Greek Stoic philosopher (c.279–c.206 BC)
propositions by the use of logical connectives. Chrysippus enumerated five kinds of molecular propositions according to the connective used: Thus several types
Chrysippus
Approach to the semantics of logic that locates meaning in inferential role
the semantics of logic in which the meaning of propositions and logical connectives is explained by the role they play within a system of inference.
Proof-theoretic_semantics
Logic formula
greater than three" or propositional variables such as p and q, using connectives or logical operators such as NOT, AND, OR, or IMPLIES; for example: (p AND
Propositional_formula
Standard system of axiomatic set theory
choose a different set of connectives or quantifiers. For example, the logical connective NAND alone can encode the other connectives, a property known as
Zermelo–Fraenkel_set_theory
Any one of the distinct objects that make up a set in set theory
this case, the domain of Px, which is the set containing all dependent logical values x that satisfy the stated conditions for membership in y, is called
Element_of_a_set
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
Value indicating the relation of a proposition to truth
Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of truth tables. Logical biconditional becomes the
Truth_value
Attempt to persuade or to determine the truth of a conclusion
premises—sentences, statements, or propositions—directed towards arriving at a logical conclusion. The purpose of an argument is to give reasons for one's thinking
Argument
Measure of algorithmic complexity
3821. doi:10.1145/321526.321530. S2CID 12584692. Kolmogorov, A. (1968). "Logical basis for information theory and probability theory". IEEE Transactions
Kolmogorov_complexity
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)
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
Subdiscipline of proof theory
explicitly realized by the left and right introduction rules for each logical connective. For example, the rules for conjunction ( ∧ {\displaystyle \land }
Structural_proof_theory
Example of a Semigroup
the right zero semigroup of order two. ({0,1}, ∧) (where "∧" is the logical connective "and"), or equivalently the set {0,1} under multiplication: the only
Semigroup_with_two_elements
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
Problem in computer science
"Generic algorithms for halting problem and optimal machines revisited". Logical Methods in Computer Science. 12 (2) 1633: 1. arXiv:1505.00731. doi:10
Halting_problem
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
Logical connective
{\displaystyle P} is true and Q {\displaystyle Q} is false. It may be written using logical notation as P ↛ Q {\displaystyle P\nrightarrow Q} , P ⊅ Q {\displaystyle
Material_nonimplication
Defining elements of a set in terms of other elements in the set
a logical connective. ¬ (p ∧ q) is a wff, because (p ∧ q) is a wff. (¬ p ∧ ¬ q) is a wff, because ¬ p and ¬ q are wffs and ∧ is a logical connective. Logic
Recursive_definition
Mathematical operation with two operands
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Binary_operation
Target set of a mathematical function
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Codomain
Form of logic that allows quantification over predicates
semantics, the interpretations of the first-order quantifiers and the logical connectives are the same as in first-order logic. Only the ranges of quantifiers
Second-order_logic
In mathematics, a statement that has been proven
of a 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
Theorem
Theorem in set theory
opposition Venn diagram Propositional Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued
Kőnig's_theorem_(set_theory)
Whether a decision problem has an effective method to derive the answer
decidable if there exists an effective method for deriving the correct answer. Logical systems are decidable if membership in their set of logically valid formulas
Decidability_(logic)
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
Area of mathematical logic
mathematical structures and logic for logical theories; and model theory = algebraic geometry − fields. where logical formulas are to definable sets what
Model_theory
Circa the end of the 19th century, several paradoxes made questionable the logical foundation of mathematics, and consequently the validity of the whole of
Philosophy_of_mathematics
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
Variable that can either be true or false
¬X is a formula. Given two formulas X and Y, and a binary connective b (such as the logical conjunction ∧), the expression (X b Y) is a formula. (Note
Propositional_variable
Fundamental theorem in mathematical logic
formal deduction. The theorem can be expressed more generally in terms of logical consequence. We say that a sentence s is a syntactic consequence of a theory
Gödel's_completeness_theorem
Logical connective
In logic, converse nonimplication is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of
Converse_nonimplication
Non-contradiction of a theory
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T {\displaystyle T} is consistent if there is no
Consistency
Proof calculus based on introduction and elimination rules
characteristic feature of natural-deduction systems is that the logical connectives are normally governed by paired introduction and elimination rules
Natural_deduction
Theorem in mathematical logic
Birkhäuser. p. 20. ISBN 978-3-7643-7259-0. Dov M. Gabbay, ed. (1994). What is a logical system?. Clarendon Press. p. 380. ISBN 978-0-19-853859-2. Jouko Väänänen
Lindström's_theorem
Mathematical use of "for all"
universal (and existential) quantifier moves unchanged across the logical connectives ∧, ∨, →, and ↚, as long as the other operand is not affected; that
Universal_quantification
Proposition in mathematical logic
doi:10.1017/S1755020311000359. S2CID 33807508. Shelah, Saharon (2003). "Logical dreams". Bulletin of the American Mathematical Society. New Series. 40
Continuum_hypothesis
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