* Material conditional, in propositional calculus, or logical calculus in mathematics.

( this is the basis of calculus ) – one can therefore have an infinite series of logical arguments

We fix some axiomatization of the predicate calculus: logical axioms and rules of inference.

Depending on the particular formalism adopted for the calculus, it may be seen as a simple application of a " functional substitution " rule of inference, as in Gödel's paper, or it may be proved by considering the formal proof of, replacing in it all occurrences of Q by some other formula with the same free variables, and noting that all logical axioms in the formal proof remain logical axioms after the substitution, and all rules of inference still apply in the same way.

A recent study argues that the force of Berkeley's criticisms has been overestimated ; that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof ; and that Leibniz's system for differential calculus was free of logical contradictions.

Traditionally, in Aristotle's classical logical calculus, in evaluating any proposition there are only two possible truth values, " true " and " false.

In general terms, a calculus is a formal system that consists of a set of syntactic expressions ( well-formed formulæ or wffs ), a distinguished subset of these expressions ( axioms ), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions.

English translation of title: " Completeness of the logical calculus ".

English translation of title: " The completeness of the axioms of the calculus of logical functions ".

In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "|" ( see vertical bar, not to be confused with "||" which is often used to represent disjunction ), " Dpq ", or "↑", denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as " not both ".

* Sir Isaac Newton's Method of Fluxions ( 1671 ), describing his method of differential calculus, is first published ( posthumously ) and Thomas Bayes publishes a defense of its logical foundations ( anonymously ).

Traditionally, in Aristotle's logical calculus, there were only two possible values ( i. e., " true " and " false ") for any proposition.

In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler.

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 the " natural " way of reasoning.

He wrote on the hypothetical syllogism and on the propositional calculus, which were both part of the Stoic logical tradition.

# An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst ( published anonymously in 1736 ), in which he defended the logical foundation of Isaac Newton's calculus (" fluxions ") against the criticism of George Berkeley, author of The Analyst

* The 1997 Rolf Schock Prize in logic and philosophy from the Royal Swedish Academy of Sciences for his conceptually oriented logical works, especially the creation of domain theory, which has made it possible to extend Tarski's semantical paradigm to programming languages as well as to construct models of Curry's combinatory logic and Church's calculus of lambda conversion ; and

Gentzen ( 1934 ) further introduced the idea of the sequent calculus, a calculus advanced in a similar spirit that better expressed the duality of the logical connectives, and went on to make fundamental advances in the formalisation of intuitionistic logic, and provide the first combinatorial proof of the consistency of Peano arithmetic.

In terms of a syntactical constraint for a propositional calculus, it is necessary, but not sufficient, that premises and conclusion share atomic formulae ( formulae that do not contain any logical connectives ).

* Boolean algebra, a logical calculus of truth values or set membership

Construed literally, it is absurd, and yet it is successful ; identities derived via the umbral calculus can also be derived by more complicated methods that can be taken literally without logical difficulty.

A non-intensional statement is also known as an extensional statement, since substitution of co-extensive expressions into it always preserves logical value.

These tools can give a visual summary of a topic that preserves its logical structure, with lines used to show how different parts link together.

Note that these arguments may indeed have validity, but they require some independent justification of the connection between their terms: otherwise the argument ( as a logical tool ) remains fallacious.

Theoretical statistics concerns both the logical arguments underlying justification of approaches to statistical inference, as well encompassing mathematical statistics.

Empiricism ( the evidence of the senses ), authoritative testimony ( the appeal to criteria and authority ), and logical deduction are often involved in justification.

Then the formula Cold ( r )→ High ( t ) is true for any t and therefore any t gives a correct control given r. A rigorous logical justification of fuzzy control is given in Hájek's book ( see Chapter 7 ) where fuzzy control is represented as a theory of Hájek's basic logic.

Although the position of antinomianism and the doctrine of Sola Fide, where justification is through faith alone, are related, antinomianism takes the notion of the relative weight of law to its logical extreme, opposing law as a groundwork.

An appeal to probability is a justification based on probability, sometimes regarded as a logical fallacy, when an unwarranted assumption that something will happen, because it can happen, or when the odds of an occurrence are unrealistically played down in lieu of appropriate precaution.

Each scale has a logical justification to explain the use of each such differing numerical name and value within that scale.

Coherentism only allows for justification based on logical relations between all the beliefs within a system of beliefs.

A popular approach in some circles is the approach of Reformed epistemologists, such as Alvin Plantinga and Nicholas Wolterstorff, who assert that belief in God might be foundational ( or, properly ' basic ') and warranted without the need for logical or evidential justification, like belief in other minds or the external world, rather than inferentially derived from other beliefs ; it can, however, be subject to defeaters, rationally requiring that one give up the belief.

As retinal images are conceded to be an integral function of the brain it seems logical to suppose that the nerves, between the inner brain and the eyes, carry the direct drive for cooperation from the various brain centers -- rather than to theorize on the transmission of an image which is already in required location.

But Sapir had since become influenced by a current of logical positivism, such as that of Bertrand Russel and the early Ludwig Wittgenstein, particularly through Ogden and Richards ' The Meaning of Meaning, from which he adopted the a view that natural language potentially obscures, rather than facilitates, the mind to perceive and describe the world as it really is.

For compatibility and ease of implementation, the 1571's double-sided format of one logical disk side with 70 tracks was created by putting together the lower 35 physical tracks on each of the physical sides of the disk rather than using two times 40 tracks, even though there were no more quality problems with the mechanisms of the 1571 drives.

Otherwise, the description of the boundaries would be rather more logical, if schenisghe meant the city of Szczecin.

Lesbian feminism, which was most influential from the mid 1970s to the mid 1980s, encouraged women to direct their energies toward other women rather than men, and advocated lesbianism as the logical result of feminism.

A British documentary on the Krupp family and firm included footage of German-speakers of the 1930s who would have had speaking contact with the family, which attests the long, thus or rather than what would be the logical German spelling pronunciation, or.

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.

This does not imply that poetry is illogical or lacks narration, but rather that poetry is an attempt to render the beautiful or sublime without the burden of engaging the logical or narrative thought process.

In other words, Ramsey held that epistemic probabilities simply are degrees of rational belief, rather than being logical relations that merely constrain degrees of rational belief.

He argued that modern philosophies legitimized their truth-claims not ( as they themselves claimed ) on logical or empirical grounds, but rather on the grounds of accepted stories ( or " metanarratives ") about knowledge and the world — comparing these with Wittgenstein's concept of language-games.

Kuhn's model of scientific change differs here, and in many places, from that of the logical positivists in that it puts an enhanced emphasis on the individual humans involved as scientists, rather than abstracting science into a purely logical or philosophical venture.

The consistency of a relational database is enforced, not by rules built into the applications that use it, but rather by constraints, declared as part of the logical schema and enforced by the DBMS for all applications.

In the 1980s, deconstruction and its emphasis on the fundamental ambiguity of language rather than its crystalline logical structure — became popular.

* They say that sedevacantists indulge in the logical fallacy of post hoc ergo propter hoc when they attribute problems that the Church has experienced in the Western world since the reforms of the Second Vatican Council to the reforms themselves rather than to the general decrease in religiosity in the West.

Conventional usage of the terms data processing and information systems restricts their use to refer to the algorithmic derivations, logical deductions, and statistical calculations that recur perennially in general business environments, rather than in the more expansive sense of all conversions of real-world measurements into real-world information in, say, an organic biological system or even a scientific or engineering system.

Although the metric system has changed and developed since its inception, its basic concepts have hardly changed-it was designed for transnational use ; it consisted of a basic set of units of measurement, now known as base units ; derived units were built up from the base units using logical rather than empirical relationships while multiples and submultiples of both base and derived units were decimal based and identified by a standard set of prefixes.

Apps Hungarian notation strives to encode the logical data type rather than the physical data type ; in this way, it gives a hint as to what the variable's purpose is, or what it represents.

Opponents of evolution theory have often maintained that social Darwinism is a logical entailment of a belief in evolutionary theory, while biologists and historians maintain that it is rather a perversion of Charles Darwin's ideas.

The standard approach to ontological commitment has been that, once a theory has been regimented and / or " paraphrased " into an agreed " canonical " version, which may indeed be in formal logical notation rather than the original language of the theory, ontological commitments can be read off straightforwardly from the presence of certain ontologically committing expressions ( e. g. bound variables of existential quantification ).

Scripture does not compel a mere intellectual assent to its doctrine, resting on logical argumentation, but rather it creates the living agreement of faith.

Although the concepts of a law or principle in nature is borderline to philosophy, and presents the depth to which mathematics can describe nature, scientific laws are considered from a scientific perspective and follow the scientific method ; they " serve their purpose " rather than " questioning reality " ( philosophical ) or " statements of logical absolution " ( mathematical ).

The physical layer defines the means of transmitting raw bits rather than logical data packets over a physical link connecting network nodes.