The first three treatises form the core of the logical theory stricto sensu: the grammar of the language of logic and the correct rules of reasoning.

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

A formula of propositional logic is said to be satisfiable if logical values can be assigned to its variables in a way that makes the formula true.

Two typical components of a CPU are the arithmetic logic unit ( ALU ), which performs arithmetic and logical operations, and the control unit ( CU ), which extracts instructions from memory and decodes and executes them, calling on the ALU when necessary.

Under the Curry – Howard correspondence, the existence of currying and uncurrying is equivalent to the logical theorem, as tuples ( product type ) corresponds to conjunction in logic, and function type corresponds to implication.

Both the logical complexity ( needing very large logic design and logic verification teams and simulation farms with perhaps thousands of computers ) and the high operating frequencies ( needing large circuit design teams and access to the state-of-the-art fabrication process ) account for the high cost of design for this type of chip.

In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true.

The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued, e. g., fuzzy 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.

* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.

In this stratum we elaborate a " pure grammar " or a logical syntax, and he would call its rules " laws to prevent non-sense ", which would be similar to what logic calls today " formation rules ".

Husserl also talked about what he called " logic of truth " which consists of the formal laws of possible truth and its modalities, and precedes the third logical third stratum.

The work of both authors was heavily influenced by Kurt Gödel's earlier work on his incompleteness theorem, especially by the method of assigning numbers ( a Gödel numbering ) to logical formulas in order to reduce logic to arithmetic.

Logical empiricism ( aka logical positivism or neopositivism ) was an early 20th century attempt to synthesize the essential ideas of British empiricism ( e. g. a strong emphasis on sensory experience as the basis for knowledge ) with certain insights from mathematical logic that had been developed by Gottlob Frege and Ludwig Wittgenstein.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

In logic and related fields such as mathematics and philosophy, if and only if ( shortened iff ) is a biconditional logical connective between statements.

In logic formulae, logical symbols are used instead of these phrases ; see the discussion of notation.

It puts one in the position of asserting or implying that truth or standards of logical consistency are relative to a particular thinker or group and that under some other standard, the position is correct despite its failure to stand up to logic.

The early Wittgenstein was concerned with the logical relationship between propositions and the world, and believed that by providing an account of the logic underlying this relationship he had solved all philosophical problems.

* Lambda is the set of logical axioms in the axiomatic method of logical deduction in first-order logic.

The language ’ s grammar is based on predicate logic, which is why it was named Loglan, an abbreviation for " logical language ".

In the 20th century, following the development of formal logic, the ampersand became a commonly used logical notation for the binary operator or sentential connective AND.

In logic and mathematics, the logical biconditional ( sometimes known as the material biconditional ) is the logical connective of two statements asserting " p if and only if q ", where q is a hypothesis ( or antecedent ) and p is a conclusion ( or consequent ).

In logic, false is a truth value or a nullary logical connective.

This is the definition of negation in some systems, such as intuitionistic logic, and can be proven in propositional calculi where negation is a fundamental connective.

This reflects the fact that the usual language of first-order logic does not include the " is not implied by " connective that would be the De Morgan dual of implication.

Proof-theoretic semantics is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical connective plays within the system of inference.

Russell also refers to Aristotle's ethics as " repulsive ", and calls his logic " as definitely antiquated as Ptolemaic astronomy ".

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

Closely linked to these cohomology theories, he originated topos theory as a generalisation of topology ( relevant also in categorical logic ).

Schaefer ’ s concept of " vocality " offers neither a compromise nor a synthesis of the views which see the poem as on the one hand Germanic, pagan, and oral and on the other Latin-derived, Christian, and literate, but, as stated by Monika Otter: "... a ' tertium quid ', a modality that participates in both oral and literate culture yet also has a logic and aesthetic of its own.

Software ( or firmware ) is also used in video games and for the configurable parts of the logic systems of automobiles, televisions, and other consumer electronics.

In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.

This activity is performed through the verbal impersonation of the characters by the players, while also employing a variety of social and other useful cognitive skills, such as logic, basic mathematics and imagination.

In propositional logic, disjunctive syllogism ( also known as disjunction elimination and or elimination, or abbreviated ∨ E ), is a valid rule of inference.

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

In the same essay he also said that the " logic of the system " puts developers into " dysfunctional roles ", with bad code the result.

Later, in the first volume of his Logical Investigations, the Prolegomena of Pure Logic, Husserl, while attacking the psychologistic point of view in logic and mathematics, also appears to reject much of his early work, although the forms of psychologism analysed and refuted in the Prolegomena did not apply directly to his Philosophy of Arithmetic.

Rudolf Carnap was also influenced by Husserl, not only concerning Husserl's notion of essential insight that Carnap used in his Der Raum, but also his notion of " formation rules " and " transformation rules " is founded on Husserl's philosophy of logic.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

He taught logic to Demosthenes, and he is also said to have taught Apollonius Cronus, the teacher of Diodorus Cronus, and the historian Euphantus.

Research is also very active in France, where researchers focus on the automation of reasoning and logic engines.

The expert system that uses that logic is also called a zeroth-order expert system.

Many expert systems are also penalized by the logic used.

Alphabets can also be infinite ; e. g. first-order logic is often expressed using an alphabet which, besides symbols such as ∧, ¬, ∀ and parentheses, contains infinitely many elements x < sub > 0 </ sub >, x < sub > 1 </ sub >, x < sub > 2 </ sub >, … that play the role of variables.

In some applications, especially in logic, the alphabet is also known as the vocabulary and words are known as formulas or sentences ; this breaks the letter / word metaphor and replaces it by a word / sentence metaphor.