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Formal proof
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to
Conditional_proof
Topics referred to by the same term
Y Conditional probability, the probability of an event A given that another event B Conditional proof, in logic: a proof that asserts a conditional, and
Conditional
Type of consensus algorithm
of proof of capacity was Burstcoin (now Signum). The Proof of Capacity (PoC) consensus algorithm is used in some cryptocurrencies. Conditional Proof of
Proof_of_space
Syllogism with conditional premise(s)
hypothetical syllogism is a valid argument form, a deductive syllogism with a conditional statement for one or both of its premises. Ancient references point to
Hypothetical_syllogism
Proposition in mathematics that is unproven
proof, some have even proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called conditional proofs:
Conjecture
Mathematical logic concept
of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive
Contraposition
science, the method of conditional probabilities is a systematic method for converting non-constructive probabilistic existence proofs into efficient deterministic
Method of conditional probabilities
Method_of_conditional_probabilities
Mathematical paradox
The standard method for proving conditional sentences (sentences of the form "if A, then B") is called "conditional proof". In this method, in order to
Curry's_paradox
Conjecture in number theory
conjecture has been stated) and conjectures for which it gives a conditional proof. The consequences include: Roth's theorem on Diophantine approximation
Abc_conjecture
Metatheorem in mathematical logic
logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that
Deduction_theorem
Expected value of a random variable given that certain conditions are known to occur
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated
Conditional_expectation
Logical connective
The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol → {\displaystyle
Material_conditional
American mathematician (born 1972)
Combined with theorems of Gross–Zagier and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has
Christopher_Skinner
Logical operation
and elimination are just special cases of implication introduction (conditional proof) and elimination (modus ponens). In this case one must also add as
Negation
Conjecture in number theory
{\displaystyle {\frac {20}{19}}C} . In 1967, Christopher Hooley published a conditional proof for the conjecture, assuming certain cases of the generalized Riemann
Artin's conjecture on primitive roots
Artin's_conjecture_on_primitive_roots
Measure of relative information in probability theory
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable Y {\displaystyle
Conditional_entropy
Theorem in abstract algebra
circle of ideas was connected to a purity conjecture; Laumon gave a conditional proof based on such a conjecture, for unitary groups. Laumon and Ngô (2008)
Fundamental lemma (Langlands program)
Fundamental_lemma_(Langlands_program)
Summatory function of the Möbius function
this conjecture is given by Nathan Ng. In particular, Ng gives a conditional proof that the function e − y / 2 M ( e y ) {\displaystyle e^{-y/2}M(e^{y})}
Mertens_function
Estimatation in number theory
the three forms have yet been proven or disproven. Cramér gave a conditional proof of the much weaker statement that p n + 1 − p n = O ( p n log p
Cramér's_conjecture
Probability theory concept
hypothesis. It is the opposite of conditional dependence. Conditional independence is usually formulated in terms of conditional probability, as a special case
Conditional_independence
Theorem in algebraic number theory relating p-adic L-functions and ideal class groups
Combined with theorems of Gross-Zagier and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has
Main conjecture of Iwasawa theory
Main_conjecture_of_Iwasawa_theory
Reasoning for mathematical statements
statements then results from the transitivity of the material conditional. A probabilistic proof is one in which an example is shown to exist, with certainty
Mathematical_proof
Rule of inference of propositional logic
The validity of this argument structure can be shown by using both conditional proof (CP) and reductio ad absurdum (RAA) in the following way: Hurley,
Destructive_dilemma
Branch of logic
(conjunction), "or" (disjunction), "not" (negation), "if" (material conditional), and "if and only if" (biconditional). Examples of such compound sentences
Propositional_logic
four or five distinct schools concerning the correct understanding of conditional ("if...then...") statements. Sextus Empiricus described one school as
Connexive_logic
Family of logics for natural-language and counterfactual conditionals
tying acceptability to conditional probability, and belief revision frameworks based on the Ramsey test. Corresponding proof-theoretic systems range
Conditional_logic
Style of formal logical argumentation
every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is
Sequent_calculus
Set of conjectures in algebraic geometry
conjectures remain open problems, so that their application gives only conditional proofs of results. In quite a few cases, including that of the Weil conjectures
Standard conjectures on algebraic cycles
Standard_conjectures_on_algebraic_cycles
Mathematical conjecture about elliptic curves
papers. Further results are conditional on improved forms of the Arthur–Selberg trace formula. Harris has a conditional proof of a result for the product
Sato–Tate_conjecture
Approach to handling inferred information
two types of justification for a node. They are: Support list [SL] Conditional proof (CP) Many kinds of truth maintenance systems exist. Two major types
Reason_maintenance
given that another event has already occurred. conditional proof A method in logic for proving a conditional statement by assuming the antecedent and showing
Glossary_of_logic
B} , as well, and, if conditional events are admitted, the event ( A ∪ B ) → A {\displaystyle (A\cup B)\rightarrow A} . The proof derives a contradiction
Lewis's_triviality_result
Topics referred to by the same term
top-level domain Software testing test (Unix), a Unix command for evaluating conditional expressions TEST (x86 instruction), an x86 assembly language instruction
Test
Overview of and topical guide to logic
the antecedent Theorem Axiom Axiomatic system Axiomatization Conditional proof Invalid proof Degree of truth Truth Truth condition Truth function Double
Outline_of_logic
Concept in Christian theology
immortality and proof of conditional immortality. Fudge, Edward William; Cousins, Peter (1994). The Fire That Consumes: The Biblical Case for Conditional Immortality
Christian_conditionalism
Conditional statement which is true because the antecedent cannot be satisfied
vacuous truth is a conditional or universal statement (specifically a universal statement that can be converted to a conditional statement) that is true
Vacuous_truth
Measure of algorithmic complexity
formal proofs in S by some procedure def nth_proof(n: int) which takes as input n and outputs some proof. This function enumerates all proofs. Some of
Kolmogorov_complexity
Type of modal logic
P → Q ) {\displaystyle K_{a}(P\rightarrow Q)} with N, and using a conditional proof with the axiom K, we can then derive K a P → K a Q {\displaystyle
Epistemic_modal_logic
Rule of logical inference
premise is a conditional ("if-then") claim, such as P implies Q. The second premise is an assertion that Q, the consequent of the conditional claim, is not
Modus_tollens
Truth-based approach to semantics
based on inference, such as proof-theoretic semantics, provides a better foundation for this model than truth-conditional semantics does. Some authors
Truth-conditional_semantics
Kind of proof calculus
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to
Natural_deduction
Way of arriving to a mathematical proof
without making any further assumptions. In order to directly prove a conditional statement of the form "If p, then q", it suffices to consider the situations
Direct_proof
Inequality between integrals in Lp spaces
side of the conditional Hölder inequality, 0 times ∞ as well as ∞ times 0 means 0. Multiplying a > 0 with ∞ gives ∞. Proof of the conditional Hölder inequality:
Hölder's_inequality
Generalization of conditional expectation
In mathematics, non-commutative conditional expectation is a generalization of the notion of conditional expectation in classical probability. The space
Non-commutative conditional expectation
Non-commutative_conditional_expectation
Version of classical propositional calculus that uses only one connective
propositional calculus that uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if .
Implicational propositional calculus
Implicational_propositional_calculus
Proposition in probability theory
{G}}_{1}]=\operatorname {E} [X\mid {\mathcal {G}}_{1}]\quad {\text{(a.s.)}}.} Proof. Since a conditional expectation is a Radon–Nikodym derivative, verifying the following
Law_of_total_expectation
Certain type of mistaken proof
simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical
Mathematical_fallacy
Notation system for natural deductive logic
into conditionals where the antecedent is a conjunction. These sequents are often listed above the proof, as Modus Tollens is above. The above proof is
Suppes–Lemmon_notation
Austrian mathematician (1898–1962)
fixed and p varies. These are unproven; in 1967, Hooley published a conditional proof for the second conjecture, assuming certain cases of the generalized
Emil_Artin
Combined with theorems of Gross-Zagier and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that E has
Eric_Urban
In mathematics, a statement that has been proven
are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises. In light of the interpretation of proof as
Theorem
Theorem in probability theory
expresses the variance of a random variable Y in terms of its conditional variances and conditional means given another random variable X. Informally, it states
Law_of_total_variance
Relationship where one statement follows from another
consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical consequence
Logical_consequence
Concept in probability theory
regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. The resulting conditional probability
Regular conditional probability
Regular_conditional_probability
Mathematical logic concept
Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of
Gentzen's_consistency_proof
Averages of repeated trials converge to the expected value
\sin(X)e^{X}X^{-1}} has no expected value according to Lebesgue integration, but using conditional convergence and interpreting the integral as a Dirichlet integral, which
Law_of_large_numbers
Approach to logic
"Bocardo" and "Ferison". In the following table, a syllogism with "conditional" validity is one whose validity depends on existential import (i.e.,
Term_logic
Rule of replacement in propositional logic
logic, material implication is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is
Material implication (rule of inference)
Material_implication_(rule_of_inference)
Lemma in measure theory
the proof is very similar to the one for the standard version of Fatou's lemma above, however the monotone convergence theorem for conditional expectations
Fatou's_lemma
Rule of replacement in propositional logic
allows conditional statements having conjunctive antecedents to be replaced by statements having conditional consequents and vice versa in logical proofs. It
Exportation_(logic)
Formula in probability theory
theory, the law of total covariance, covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are random variables on
Law_of_total_covariance
Provisional version pending further research
hypotheses in order to investigate its consequences and formulate conditional proofs. Materials scientists Hosono et al. (1996) developed a working hypothesis
Working_hypothesis
Arminian religious doctrine
The conditional preservation of the saints, or conditional perseverance of the saints, or commonly conditional security, is the Arminian Christian belief
Conditional preservation of the saints
Conditional_preservation_of_the_saints
Rule of logical inference
deductive proofs that includes the "rule of definition" and the "rule of substitution". Modus ponens allows one to eliminate a conditional statement from
Modus_ponens
Signed odd unit fractions sum to π/4
denominator is the nearest multiple of 4 to the numerator. The product is conditionally convergent; its terms must be taken in order of increasing p. List of
Leibniz_formula_for_π
Formal statement in logic
In logic, a strict conditional (symbol: ◻ {\displaystyle \Box } , or ⥽) is a conditional governed by a modal operator, that is, a logical connective of
Strict_conditional
Logical paradox
modern logicians call "logical conditionals". Uncle Joe concludes his proof reductio ad absurdum, meaning in English "proof by contradiction". What Carroll
Barbershop_paradox
Concept in computer science
barriers to entry and resist sybil attacks include proof of authority, proof of space, proof of burn, or proof of elapsed time. Contrasting with the above permissionless
Consensus_(computer_science)
Form of reasoning
premise a conditional statement ( P → Q {\displaystyle P\rightarrow Q} ) and as second premise the antecedent ( P {\displaystyle P} ) of the conditional statement
Deductive_reasoning
Terms to describe a conditional relationship between two statements
concepts used to denote a conditional or implicational relationship between two statements. For example, in the conditional statement: "If P then Q",
Necessity_and_sufficiency
Proving or disproving the correctness of certain intended algorithms
verification of these systems is done by ensuring the existence of a formal proof of a mathematical model of the system. Examples of mathematical objects
Formal_verification
Concept in cryptography theory
Conditional disclosure of secrets (CDS) is a primitive, studied in information-theoretic cryptography, that allows distributed, non-communicating parties
Conditional disclosure of secrets
Conditional_disclosure_of_secrets
Mathematical conjecture
others. Some of these results were conditional upon the generalized Riemann hypothesis (GRH) being true. In fact, the proof of the full conjecture under GRH
André–Oort_conjecture
Field in logic and theoretical computer science
shortest P-proof of τ {\displaystyle \tau } . Many proof systems of interest are believed to be non-automatable. However, currently only conditional negative
Proof_complexity
Kind of non-classical logic
necessarily cause an "explosion." This follows from the fact that a conditional with a contradictory antecedent that does not share any propositional
Relevance_logic
Concept in information processing
Z} , implying that the conditional distribution of Z {\displaystyle Z} depends only on Y {\displaystyle Y} and is conditionally independent of X {\displaystyle
Data_processing_inequality
Lawful permanent residency in the United States
resident under the conditional clause may receive an I-551 stamp as well as a permanent resident card. The expiration date of the conditional period is two
Green_card
Theorem in probability theory
necessarily 1. A powerful generalization involving conditional probability is known as the Conditional Borel–Cantelli lemma (or Lévy's extension of the
Borel–Cantelli_lemma
Concept in information theory
measuring the conditional mutual information in bits rather than nats.) Several machine based proof checker algorithms are now available. Proof checker algorithms
Inequalities in information theory
Inequalities_in_information_theory
Theorem of convex functions
convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder
Jensen's_inequality
Number measuring the chance an event occurs
number of events. Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written
Probability
American legislative proposal on immigration
is a United States legislative proposal that would grant temporary conditional residency, with the right to work, to qualified applicants following
DREAM_Act
Condition under which an odd prime is a sum of two squares
detailed proof in two articles (between 1752 and 1755). Lagrange gave a proof in 1775 that was based on his study of quadratic forms. This proof was simplified
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Mathematical rule for inverting probabilities
after Thomas Bayes (/beɪz/), gives a mathematical rule for inverting conditional probabilities, allowing the probability of a cause to be found given
Bayes'_theorem
Concept in probability theory
(X|X\geq a)} is larger than or equal to a {\displaystyle a} because the conditional expectation only takes into account of values larger than or equal to
Markov's_inequality
Average uncertainty in variable's states
{\displaystyle \lim _{p\to 0^{+}}p\log(p)=0.} One may also define the conditional entropy of two variables X {\displaystyle X} and Y {\displaystyle Y}
Entropy_(information_theory)
Brazilian philosopher and mathematician (1929–2024)
Francisco Antônio Dória, Da Costa published two papers with conditional relative proofs of the consistency of P = NP with the usual set-theoretic axioms
Newton_da_Costa
match the approximation ratio of the existence proof above. The approach is called the method of conditional probabilities. The deterministic algorithm emulates
Randomized_rounding
Problem in computer science
definition of a computer and program, usually via a Turing machine. The proof then shows, for any program f that might determine whether programs halt
Halting_problem
Tool for proving a logical formula
In proof theory, the semantic tableau (/tæˈbloʊ, ˈtæbloʊ/; plural: tableaux), also called an analytic tableau, truth tree, or simply tree, is a decision
Method_of_analytic_tableaux
Probability puzzle
he does have a choice, and hence that the conditional probability of winning by switching (i.e., conditional given the situation the player is in when
Monty_Hall_problem
Theorem about prime numbers
the generalized Hardy–Littlewood conjecture, Green and Tao stated and conditionally proved the asymptotic formula ( S k + o ( 1 ) ) N 2 ( log N ) k {\displaystyle
Green–Tao_theorem
Korean educator (born 1978)
dependencies on improved forms of the Arthur–Selberg trace formula in the conditional proofs of generalizations of the Sato–Tate conjecture by Harris (for products
Sug_Woo_Shin
for the amount paid. There are typically four types of lien waivers: Conditional waiver on progress payment – The safest waiver for claimants, this waiver
Lien_waiver
Establishes the limits to possible data compression
are recoverable from the binary bits with probability of at least 1 − ε. Proof of Achievability. Fix some ε > 0, and let p ( x 1 , … , x n ) = Pr [ X 1
Shannon's source coding theorem
Shannon's_source_coding_theorem
Probability theory term
formalized in probability theory by conditioning. Conditional probabilities, conditional expectations, and conditional probability distributions are treated on
Conditioning_(probability)
Branch of mathematics concerning probability
both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions. Certain random
Probability_theory
semi-conditional grammar is very similar to a conditional grammar, and technically the class of semi-conditional grammars are a subset of the conditional grammars
Controlled_grammar
Concept in probability theory
_{s}(\{x\})>0} . By independence of X 1 , X 2 {\displaystyle X_{1},X_{2}} , the conditional variable X 2 | { X 1 = x } {\displaystyle X_{2}|\{X_{1}=x\}} is distributed
Maxwell's_theorem
Argument for the existence of God
the rules of the adopted logical framework. However, this validity is conditional: it does not guarantee the truth of the axioms or the reality of the
Ontological_argument
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