Classical logic in the context of "True or false"

The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol ${\displaystyle \to }$ is interpreted as material implication, a formula ${\displaystyle P\to Q}$ is true unless ${\displaystyle P}$ is true and ${\displaystyle Q}$ is false.

Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language.

Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, and public intellectual. He influenced mathematics, logic, set theory, and various areas of analytic philosophy.

He was one of the early 20th century's prominent logicians and a founder of analytic philosophy, along with his predecessor Gottlob Frege, his friend and colleague G. E. Moore, and his student and protégé Ludwig Wittgenstein. Russell with Moore led the British "revolt against idealism". Together with his former teacher Alfred North Whitehead, Russell wrote Principia Mathematica, a milestone in the development of classical logic and a major attempt to reduce the whole of mathematics to logic (see logicism). Russell's article "On Denoting" has been considered a "paradigm of philosophy".

In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (true or false). Truth values are used in computing as well as various types of logic.

Pluralism is a term used in philosophy, referring to a worldview of multiplicity, often used in opposition to monism (the view that all is one) or dualism (the view that all is two). The term has different meanings in metaphysics, ontology, epistemology and logic. In metaphysics, it is the view that there are in fact many different substances in nature that constitute reality. In ontology, pluralism refers to different ways, kinds, or modes of being. For example, a topic in ontological pluralism is the comparison of the modes of existence of things like 'humans' and 'cars' with things like 'numbers' and some other concepts as they are used in science.

In epistemology, pluralism is the position that there is not one consistent means of approaching truths about the world, but rather many. Often this is associated with pragmatism, or conceptual, contextual, or cultural relativism. In the philosophy of science it may refer to the acceptance of co-existing scientific paradigms which though accurately describing their relevant domains are nonetheless incommensurable. In logic, pluralism is the relatively novel view that there is no one correct logic, or alternatively, that there is more than one correct logic. Such as using classical logic in most cases, but using paraconsistent logic to deal with certain paradoxes.

Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic, which includes additional topics like how to define logic or a discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic.

An important issue for philosophical logic is the question of how to classify the great variety of non-classical logical systems, many of which are of rather recent origin. One form of classification often found in the literature is to distinguish between extended logics and deviant logics. Logic itself can be defined as the study of valid inference. Classical logic is the dominant form of logic and articulates rules of inference in accordance with logical intuitions shared by many, like the law of excluded middle, the double negation elimination, and the bivalence of truth.

In classical logic, disjunctive syllogism (historically known as modus tollendo ponens (MTP), Latin for "mode that affirms by denying") is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.

An example in English:

In classical logic, a 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 the works of Theophrastus and Eudemus for the first investigation of this kind of syllogisms.

In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as ${\displaystyle \lor }$ and read aloud as "or". For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula ${\displaystyle S\lor W}$, assuming that ${\displaystyle S}$ abbreviates "it is sunny" and ${\displaystyle W}$ abbreviates "it is warm".

In classical logic, disjunction is given a truth functional semantics according to which a formula ${\displaystyle \phi \lor \psi }$ is true unless both ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an inclusive interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument, Heisenberg's uncertainty principle, as well as the numerous mismatches between classical disjunction and its nearest equivalents in natural languages.

In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition ${\displaystyle P}$ to another proposition "not ${\displaystyle P}$", written ${\displaystyle \neg P}$, ${\displaystyle {\mathord {\sim }}P}$, ${\displaystyle P^{\prime }}$ or ${\displaystyle {\overline {P}}}$. It is interpreted intuitively as being true when ${\displaystyle P}$ is false, and false when ${\displaystyle P}$ is true. For example, if ${\displaystyle P}$ is "The dog runs", then "not ${\displaystyle P}$" is "The dog does not run". An operand of a negation is called a negand or negatum.

Negation is a unary logical connective. It may furthermore be applied not only to propositions, but also to notions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition ${\displaystyle P}$ is the proposition whose proofs are the refutations of ${\displaystyle P}$.