Soundness is among the most fundamental properties of mathematical logic.

Soundness properties come in two main varieties: weak and strong soundness, of which the former is a restricted form of the latter.

* Soundness: if the statement is false, no prover, even if it doesn't follow the protocol, can convince the honest verifier that it is true, except with some small probability.

* Soundness: if the string is not in the language, no prover, however malicious, will be able to convince the verifier to accept the string with probability exceeding 1 / 3.

* ( Soundness ) Every provable second-order sentence is universally valid, i. e., true in all domains under standard semantics.

# Soundness: if the statement is false, no cheating prover can convince the honest verifier that it is true, except with some small probability.

* Soundness, a logical term meaning that an argument is valid and its premises are true

Soundness guarantees that all possible behaviours are preserved while completeness guarantees that no behaviour is added by the encoding.

* October 1992 – The Housing and Community Development Act of 1992 codifies within its language the Federal Housing Enterprises Financial Safety and Soundness Act of 1992 that creates the Office of Federal Housing Enterprise Oversight, and mandates HUD to set goals for lower income and underserved housing areas for the GSEs Fannie Mae and Freddie Mac

* Soundness: If x ∉ L then for every π, V < sup > π </ sup >( x ) accepts with probability at most s ( n ).

Validity and Soundness in the Internet Encyclopedia of Philosophy.

In Quality of Management they scored second place, third in Long Term Investments, fourth in Financial Soundness, and ninth in Global Competitiveness.

It was established by the Federal Housing Enterprises Financial Safety and Soundness Act of 1992.

Certain features we have touched upon: philosophy as a logical, deductive system from which a social science methodology can be built up ; ;

Even Plato had difficulties with logic ; although he had a reasonable conception of a deductive system, he could never actually construct one and relied instead on his dialectic.

Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.

A deductive system consists of a set of logical axioms, a set of non-logical axioms, and a set of rules of inference.

A desirable property of a deductive system is that it be complete.

Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system.

There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms.

Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.

To formally state, and then prove, the completeness theorem, it is necessary to also define a deductive system.

A deductive system is called complete if every logically valid formula is the conclusion of some formal deduction, and the completeness theorem for a particular deductive system is the theorem that it is complete in this sense.

Thus, in a sense, there is a different completeness theorem for each deductive system.

A converse to completeness is soundness, the fact that only logically valid formulas are provable in the deductive system.

If some specific deductive system of first-order logic is sound and complete, then is it " perfect " ( a formula is provable iff it is a semantic consequence of the axioms ), thus equivalent to any other deductive system with the same quality ( any proof in one system can be converted into the other ).

Gödel's completeness theorem says that a deductive system of first-order predicate calculus is " complete " in the sense that no additional inference rules are required to prove all the logically valid formulas.

A converse to completeness is soundness, the fact that only logically valid formulas are provable in the deductive system.

It is possible to produce sound deductive systems for higher-order logics, but no such system can be complete.

On the face of it, it is because he employs deductive techniques alien to official police routine.

However, as to whether inductive or deductive reasoning is more valuable still remains a matter of debate, with the general conclusion being that neither is prominent.

A brilliant London-based " consulting detective " residing at 221B Baker Street, Holmes is famous for his intellectual prowess and is renowned for his skillful use of astute observation, deductive reasoning, and forensic skills to solve difficult cases.

The logician is free to see the extension of this deductive, theoretical sphere of pure logic.

Popper argued that this would require the inference of a general rule from a number of individual cases, which is inadmissible in deductive logic.

However, if one finds one single black swan, deductive logic admits the conclusion that the statement that all swans are white is false.

Common to all deductive systems is the notion of a formal deduction.

Strong soundness of a deductive system is the property that any sentence P of the language upon which the deductive system is based that is derivable from a set Γ of sentences of that language is also a logical consequence of that set, in the sense that any model that makes all members of Γ true will also make P true.

In a deductive theory, any sentence which is a logical consequence of one or more of the axioms is also a sentence of that theory.

* as basically a transformation of a deductive categorical syllogism ( but in 1903 he offered a variation on modus ponens instead, and by 1911 he was unconvinced that any one form covers all hypothetical inference ).

* " Class " as an analytical feature of any Category or Categorical term, in the language of deductive reasoning

The term ' syntactic ' has a slightly wider scope than ' proof-theoretic ', since it may be applied to properties of formal languages without any deductive systems, as well as to formal systems.

Though they require some filling in, enthymemes are intended to have the form of valid deductive syllogisms, so a complete enthymeme has the same premise-premise-conclusion structure as any syllogism, and is intended to guarantee the truth of its conclusion based on the truth of its premises.

The structural criterion requires that one who argues for or against a position should use an argument that meets the fundamental structural requirements of a well-formed argument, using premises that are compatible with one another, that do not contradict the conclusion, that do not assume the truth of the conclusion, and that are not involved in any faulty deductive inference.

: during the period of elaboration of any deductive theory we choose the ideas to be represented by the undefined symbols and the facts to be stated by the unproved propositions ; but, when we begin to formulate the theory, we can imagine that the undefined symbols are completely devoid of meaning and that the unproved propositions ( instead of stating facts, that is, relations between the ideas represented by the undefined symbols ) are simply conditions imposed upon undefined symbols.

* A. Padoa ( 1900 ) " Logical introduction to any deductive theory " in Jean van Heijenoort, 1967.