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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
Inference in propositional logic
Biconditional elimination is the name of two valid rules of inference of propositional logic. It allows for one to infer a conditional from a biconditional
Biconditional_elimination
Inference in propositional logic
"I'm breathing if and only if I'm alive". Biconditional introduction is the converse of biconditional elimination. The rule can be stated formally as: P
Biconditional_introduction
Branch of logic
representing the truth functions of conjunction, disjunction, implication, biconditional, and negation. Some sources include other connectives, as in the table
Propositional_logic
Propositional logic theorem
combined into a single biconditional formula: ¬ ¬ P ↔ P . {\displaystyle \neg \neg P\leftrightarrow P.} Since biconditionality is an equivalence relation
Double_negation
\varphi }}} φ ↔ ψ {\displaystyle \varphi \leftrightarrow \psi } Biconditional elimination φ ↔ ψ {\displaystyle \varphi \leftrightarrow \psi } φ _ {\displaystyle
List_of_rules_of_inference
Kind of proof calculus
immediately by its elimination can be turned into an equivalent derivation without this detour. It is a check on the strength of elimination rules: they must
Natural_deduction
Overview of and topical guide to logic
Subalternation Tautology Theorem Rule of inference (list) Biconditional elimination Biconditional introduction Case analysis Commutativity of conjunction
Outline_of_logic
Rule of logical inference
(from Latin 'mode that by affirming affirms'), conditional elimination, implication elimination, or affirming the antecedent, is a deductive argument form
Modus_ponens
True when either but not both inputs are true
logical inequality is a logical operator whose negation is the logical biconditional. With two inputs, XOR is true if and only if the inputs differ (one
Exclusive_or
Rule of inference in predicate logic
instantiation (UI; also called universal specification or universal elimination,[citation needed] and sometimes confused with dictum de omni)[citation
Universal_instantiation
Pair of logical equivalences
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
De_Morgan's_laws
Rule of inference in predicate logic
predicate logic, existential instantiation (also called existential elimination) is a rule of inference which says that, given a formula of the form
Existential_instantiation
Logical rule of inference
propositional logic, disjunctive syllogism (also known as disjunction elimination and or elimination, or abbreviated ∨E), is a valid rule of inference. If it is
Disjunctive_syllogism
Rule of inference of propositional logic
In propositional logic, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination) is the valid argument form and rule
Disjunction_elimination
Rule of logical inference
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Modus_tollens
Commonly used rules of replacement in propositional logic
either of two commonly used rules of replacement. The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical
Tautology_(rule_of_inference)
Property of a mathematical operation
disambiguation. An example where this does not work is the logical biconditional ↔. It is associative; thus, A ↔ (B ↔ C) is equivalent to (A ↔ B) ↔ C
Associative_property
Line-by-line system for natural deduction proofs
not not P [assumption, want P] 6 | | P [negation elimination: 5] | 7 | P iff not not P [biconditional introduction: 1 - 4, 5 - 6] The null assumption,
Fitch_notation
Mathematical logic concept
equivalent to a given conditional statement, though not sufficient for a biconditional. Similarly, take the statement "All quadrilaterals have four sides,"
Contraposition
Inference rule in logic
In propositional logic, conjunction elimination (also called and elimination, ∧ elimination, or simplification) is a valid immediate inference, argument
Conjunction_elimination
Property involving two mathematical operations
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Distributive_property
Rule of replacement in propositional logic
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Exportation_(logic)
Claimed as largest named number
θ ∧ ( ¬ ξ ) ) ) {\displaystyle (\neg (\theta \land (\neg \xi )))} . Biconditional: ( θ ⇔ ξ ) {\displaystyle (\theta \Leftrightarrow \xi )} as ( ¬ ( (
Rayo's_number
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Modus_non_excipiens
Rule of inference in predicate logic
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Existential_generalization
Rule of replacement in propositional logic
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Material implication (rule of inference)
Material_implication_(rule_of_inference)
Logical rule of inference
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Negation_introduction
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Absorption_(logic)
Formal proof
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Conditional_proof
Inference introducing a disjunction in logical proofs
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Disjunction_introduction
Rule of inference in propositional logic
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Conjunction_introduction
Rule of inference of propositional logic
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Constructive_dilemma
tendency or inclination, especially in statistical or cognitive contexts. biconditional A logical connective between statements, where both statements imply
Glossary_of_logic
Logical rule of inference
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Modus_ponendo_tollens
Rule of inference of propositional logic
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Destructive_dilemma
Method of deriving conclusions
and elimination, implication introduction and elimination, negation introduction and elimination, and biconditional introduction and elimination. As a
Rule_of_inference
Syllogism with conditional premise(s)
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Hypothetical_syllogism
Rule of inference in predicate logic
introduction / elimination (modus ponens) Biconditional introduction / elimination Conjunction introduction / elimination Disjunction introduction / elimination Disjunctive /
Universal_generalization
Form of reasoning
October 2007). "Conditional reasoning and the Wason selection task: Biconditional interpretation instead of reasoning bias". Thinking & Reasoning. 13
Deductive_reasoning
Mathematical use of "there exists"
sub-derivation with that conclusion. The reasoning behind existential elimination (∃E) is as follows: If it is given that there exists an element for which
Existential_quantification
Family of philosophical theories
where truth is predicated of sentences on the left hand side of the biconditionals such as (T) above), then deflationism is false; on the other hand, if
Deflationary_theory_of_truth
Notation system for natural deductive logic
primitive (non-highlighted) Gentzen rules. The rules 8 (Double Negation Elimination) and 9 (Reductio Ad Absurdum) are equivalent. One of them can be dropped
Suppes–Lemmon_notation
Mathematics notation with operators preceding operands
in 1924 of the article of Moses Schönfinkel, already had the idea of eliminating parentheses in logic formulas. In one of his papers Łukasiewicz stated
Polish_notation
In political philosophy, a type of proposal
neutral. Throffers are those biconditional proposals that contain both a threat and an offer, as opposed to biconditional proposals containing a threat
Throffer
1969 non-fiction book by G. Spencer-Brown
tautology, simply write "A = ". If one replaces '=' in R1 and R2 with the biconditional, the resulting rules hold in conventional logic. However, conventional
Laws_of_Form
Logic puzzle by Raymond Smullyan
is to use complicated logical connectives in your questions (either biconditionals or some equivalent construction). Boolos' question was to ask A: Does
The_Hardest_Logic_Puzzle_Ever
Algebraic manipulation of "true" and "false"
bit vector. The final goal of the next section can be understood as eliminating "concrete" from the above observation. That goal is reached via the stronger
Boolean_algebra
Propositional calculus in which there are more than two truth values
negation (¬), conjunction (∧), disjunction (∨), implication (→K), and biconditional (↔K) are given by: The difference between the two logics lies in how
Many-valued_logic
Type of diagrammatic notation for logic
instance, the well known argument modus ponens, also known as implication elimination: P → Q , P ∴ Q {\displaystyle {\frac {P\to Q,P}{\therefore Q}}} This
Randolph_diagram
its Tarski biconditional: b {\displaystyle b} is true iff b {\displaystyle b} is true. The truth predicate on the right cannot be eliminated. This example
Revision_theory
System of mathematical set theory
of Bernays' axioms (intersection, complement, domain) by replacing biconditionals with implications, which means they specify only the ordered pairs or
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Indian-American philosopher (born 1949)
theory takes truth to be a circular concept, defined by the Tarski biconditionals, 'A' is true if and only if A, and interprets it in a new way. Rather
Anil_Gupta_(philosopher)
Type of logical system
connectives: ∧ for conjunction, ∨ for disjunction, → for implication, ↔ for biconditional, ¬ for negation. Some authors use Cpq instead of → and Epq instead of
First-order_logic
Logic formula
formulas can be written in Polish notation or reverse Polish notation, eliminating the need for parentheses altogether. The inductive definition of infix
Propositional_formula
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BICONDITIONAL ELIMINATION
BICONDITIONAL ELIMINATION
BICONDITIONAL ELIMINATION
BICONDITIONAL ELIMINATION
BICONDITIONAL ELIMINATION
BICONDITIONAL ELIMINATION
BICONDITIONAL ELIMINATION
BICONDITIONAL ELIMINATION
BICONDITIONAL ELIMINATION
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