For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.

* Conditional proof, in logic: a proof that asserts a conditional, and proves that the antecedent leads to the consequent.

As an example of a conditional proof in symbolic logic, suppose we want to prove A → C ( if A, then C ) from the first two premises below:

Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.

The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

Concepts such as infinite proof trees or infinite derivation trees have also been studied, e. g. infinitary logic.

In propositional logic, disjunction elimination ( sometimes named proof by cases or case analysis ), is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof.

Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, in which every proposition is either true or false, i. e., for any proposition P, the proposition " P or not P " is automatically true.

First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim – Skolem theorem and the compactness theorem.

The proof uses modal logic, which distinguishes between necessary truths and contingent truths.

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 ).

In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof, rather than with Gödel's original proof.

The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic.

He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

As is true of all axioms of logic, the law of non-contradiction is alleged to be neither verifiable nor falsifiable, on the grounds that any proof or disproof must use the law itself prior to reaching the conclusion.

Although it was based on the proof methods of logic, Planner, developed at MIT, was the first language to emerge within this proceduralist paradigm 1969.

While it is one of the most commonly used concepts in logic it must not be mistaken for a logical law ; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the " rule of definition " and the " rule of substitution " Modus ponens allows one to eliminate a conditional statement from a logical proof or argument ( the antecedents ) and thereby not carry these antecedents forward in an ever-lengthening string of symbols ; for this reason modus ponens is sometimes called the rule of detachment.

Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof to model theory of abstract truth in modern mathematics.

The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems.

The logic of creating a strong, balanced, competitive two-system railroad service in the East is so obvious that B. & O. was publicly committed to the approach outlined here.

The logic of that is impeccable, of course, except that I feel like a fool being driven up to work in a little car, by my wife, when everybody knows I have a big car and am capable of driving myself.

Since Russian was being spoken instead of Spanish, there is no violation of artistry or logic here.

With the Prior Analytics, Aristotle is credited with the earliest study of formal logic, and his conception of it was the dominant form of Western logic until 19th century advances in mathematical logic.

There is one volume of Aristotle's concerning logic not found in the Organon, namely the fourth book of Metaphysics.

In philosophy and logic, an argument is an attempt to persuade someone of something, or give evidence or reasons for accepting a particular conclusion.

In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension.

In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended " target " computer ( s ), in order for the target machines to produce output from given input ( perhaps null ).

ZFC, however, is still formalized in classical logic.

The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

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

In The Mysterious Affair at Styles, Poirot operates as a fairly conventional, clue-based detective, depending on logic, which is represented in his vocabulary by two common phrases: his use of " the little grey cells " and " order and method ".

* In the Neal Stephenson novel The Diamond Age, ubiquitous molecular nanotechology is described to make use of " rod logic " similar to that imagined by Babbage's design for the Analytical Engine.

In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

The Platonist seemed to outweigh the Aristotelian in Alan, but he felt strongly that the divine is all intelligibility and argued this notion through much Aristotelian logic combined with Pythagorean mathematics.

Jarry once wrote, expressing some of the bizarre logic of ' pataphysics, " If you let a coin fall and it falls, the next time it is just by an infinite coincidence that it will fall again the same way ; hundreds of other coins on other hands will follow this pattern in an infinitely unimaginable fashion ".

For Alexander Gottlieb Baumgarten aesthetics is the science of the sense experiences, a younger sister of logic, and beauty is thus the most perfect kind of knowledge that sense experience can have.

FTA is basically composed of logic diagrams that display the state of the system and is constructed using graphical design techniques.

He wrote that Humayun will also be remembered for creating some " immortal " characters: " Misir Ali is basically a rational psychologist committed to unraveling the mysteries around him through logic.

It is frequently noted that Aristotle's logic is unable to represent even the most elementary inferences in Euclid's geometry, but Frege's " conceptual notation " can represent inferences involving indefinitely complex mathematical statements.

Aristotle's syllogistic logic, together with the Axiomatic Method exemplified by Euclid's Elements, are universally recognized as towering scientific achievements of ancient Greece.

Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I.

In the first place, he made an analogy between laws of logic and laws of geometry: once Euclid's postulates were believed to be truths about the physical space in which we live, but modern physical theories are based around non-Euclidean geometries, with a different and fundamentally incompatible notion of straight line.

Some of Euclid's successors developed logic to such an extent that they became a separate school, known as the Dialectical school.

Becker also showed how a constructive logic that denied unrestricted excluded middle could be used to reconstruct most of Euclid's proofs.