Tautology (logic) in the context of "Law of thought"

Positivism is a philosophical school that holds that all genuine knowledge is either true by definition or positive – meaning a posteriori facts derived by reason and logic from sensory experience. Other ways of knowing, such as intuition, introspection, or religious faith, are rejected or considered meaningless.

Although the positivist approach has been a recurrent theme in the history of Western thought, modern positivism was first articulated in the early 19th century by Auguste Comte. His school of sociological positivism holds that society, like the physical world, operates according to scientific laws. After Comte, positivist schools arose in logic, psychology, economics, historiography, and other fields of thought. Generally, positivists attempted to introduce scientific methods to their respective fields. Since the turn of the 20th century, positivism, although still popular, has declined under criticism within the social sciences by antipositivists and critical theorists, among others, for its alleged scientism, reductionism, overgeneralizations, and methodological limitations. Positivism also exerted an unusual influence on Kardecism.

Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement which is not only true, but one which is true under all interpretations of its logical components (other than its logical constants). Thus, logical truths such as "if p, then p" can be considered tautologies. Logical truths are thought to be the simplest case of statements which are analytically true (or in other words, true by definition). All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.

Logical truths are generally considered to be necessarily true. This is to say that they are such that no situation could arise in which they could fail to be true. The view that logical statements are necessarily true is sometimes treated as equivalent to saying that logical truths are true in all possible worlds. However, the question of which statements are necessarily true remains the subject of continued debate.

Modal logic is a kind of logic used to represent statements about necessity and possibility. In philosophy and related fieldsit is used as a tool for understanding concepts such as knowledge, obligation, and causation. For instance, in epistemic modal logic, the formula ${\displaystyle \Box P}$ can be used to represent the statement that ${\displaystyle P}$ is known. In deontic modal logic, that same formula can represent that ${\displaystyle P}$ is a moral obligation. Modal logic considers the inferences that modal statements give rise to. For instance, most epistemic modal logics treat the formula ${\displaystyle \Box P\rightarrow P}$ as a tautology, representing the principle that only true statements can count as knowledge. However, this formula is not a tautology in deontic modal logic, since what ought to be true can be false.

Modal logics are formal systems that include unary operators such as ${\displaystyle \Diamond }$ and ${\displaystyle \Box }$, representing possibility and necessity respectively. For instance the modal formula ${\displaystyle \Diamond P}$ can be read as "possibly ${\displaystyle P}$" while ${\displaystyle \Box P}$ can be read as "necessarily ${\displaystyle P}$". In the standard relational semantics for modal logic, formulas are assigned truth values relative to a possible world. A formula's truth value at one possible world can depend on the truth values of other formulas at other accessible possible worlds. In particular, ${\displaystyle \Diamond P}$ is true at a world if ${\displaystyle P}$ is true at some accessible possible world, while ${\displaystyle \Box P}$ is true at a world if ${\displaystyle P}$ is true at every accessible possible world. A variety of proof systems exist which are sound and complete with respect to the semantics one gets by restricting the accessibility relation. For instance, the deontic modal logic D is sound and complete if one requires the accessibility relation to be serial.

A priori ('from the earlier') and a posteriori ('from the later') are Latin phrases used in philosophy & linguistics to distinguish types of knowledge, justification, or argument by their reliance on experience. A priori knowledge is independent of any experience. Examples include mathematics, tautologies and deduction from pure reason. A posteriori knowledge depends on empirical evidence. Examples include most fields of science and aspects of personal knowledge.

The terms originate from the analytic methods found in Organon, a collection of works by Aristotle. Prior analytics (a priori) is about deductive logic, which comes from definitions and first principles. Posterior analytics (a posteriori) is about inductive logic, which comes from observational evidence.

Verificationism, also known as the verification principle or the verifiability criterion of meaning, is a doctrine in philosophy and the philosophy of language which holds that a declarative sentence is cognitively meaningful only if it is either analytic or tautological (true or false in virtue of its logical form and definitions) or at least in principle verifiable by experience. On this view, many traditional statements of metaphysics, theology, and some of ethics and aesthetics are said to lack truth value or factual content, even though they may still function as expressions of emotions or attitudes rather than as genuine assertions. Verificationism was typically formulated as an empiricist criterion of cognitive significance: a proposed test for distinguishing meaningful, truth-apt sentences from "nonsense".

As a self-conscious movement, verificationism was a central thesis of logical positivism (or logical empiricism), developed in the 1920s and 1930s by members of the Vienna Circle and their allies in early analytic philosophy. Drawing on earlier empiricism and positivism (especially David Hume, Auguste Comte and Ernst Mach), on pragmatism (notably C. S. Peirce and William James), and on the logical and semantic innovations of Gottlob Frege and the early Wittgenstein, these philosophers sought a "scientific" conception of philosophy in which meaningful discourse would either consist in empirical claims ultimately testable by observation or in analytic truths of logic and mathematics. The verification principle was intended to explain why many traditional metaphysical disputes seemed irresolvable, to demarcate science from pseudo-science and speculative metaphysics, and to vindicate the special status of the natural sciences by taking empirical testability as the paradigm of serious inquiry.

Logical positivism, also known as logical empiricism or neo-positivism, was a philosophical movement, in the empiricist tradition, that sought to formulate a scientific philosophy in which philosophical discourse would be, in the perception of its proponents, as authoritative and meaningful as empirical science.

Logical positivism's central thesis was the verification principle, also known as the "verifiability criterion of meaning", according to which a statement is cognitively meaningful only if it can be verified through empirical observation or if it is a tautology (true by virtue of its own meaning or its own logical form). The verifiability criterion thus rejected statements of metaphysics, theology, ethics and aesthetics as cognitively meaningless in terms of truth value or factual content. Despite its ambition to overhaul philosophy by mimicking the structure and process of empirical science, logical positivism became erroneously stereotyped as an agenda to regulate the scientific process and to place strict standards on it.

A self-refuting idea or self-defeating idea is an idea or statement whose falsehood is a logical consequence of the act or situation of holding them to be true. Many ideas are called self-refuting by their detractors, and such accusations are therefore almost always controversial, with defenders stating that the idea is being misunderstood or that the argument is invalid. For these reasons, none of the ideas below are unambiguously or incontrovertibly self-refuting. These ideas are often used as axioms, which are definitions taken to be true (tautological assumptions), and cannot be used to test themselves, for doing so would lead to only two consequences: consistency (circular reasoning) or exception (self-contradiction).

The paradoxes of material implication are a group of classically true formulae involving material conditionals whose translations into natural language are intuitively false when the conditional is translated with English words such as "implies" or "if ... then ...". They are sometimes phrased as arguments, since they are easily turned into arguments with modus ponens: if it is true that "if ${\displaystyle P}$ then ${\displaystyle Q}$" (${\displaystyle P\rightarrow Q}$), then from that together with ${\displaystyle P}$, one may argue for ${\displaystyle Q}$. Among them are the following:

A material conditional formula ${\displaystyle P\rightarrow Q}$ is true unless ${\displaystyle P}$ is true and ${\displaystyle Q}$ is false; it is synonymous with "either P is false, or Q is true, or both". This gives rise to vacuous truths such as, "if 2+2=5, then this Wikipedia article is accurate", which is true regardless of the contents of this article, because the antecedent is false. Given that such problematic consequences follow from an extremely popular and widely accepted model of reasoning, namely the material implication in classical logic, they are called paradoxes. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning.

In mathematical logic, sequent calculus is a style of formal logical argumentation in which every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than David Hilbert's earlier style of formal logic, in which every line was an unconditional tautology. More subtle distinctions may exist; for example, propositions may implicitly depend upon non-logical axioms. In that case, sequents signify conditional theorems of a first-order theory rather than conditional tautologies.

Sequent calculus is one of several extant styles of proof calculus for expressing line-by-line logical arguments.

In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be that a string is a well-formed formula, or that a proposition is true. Similarly, a judgment may assert the occurrence of a free variable in an expression of the object language, or the provability of a proposition. In general, a judgment may be any inductively definable assertion in the metatheory.

Judgments are used in formalizing deduction systems: a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequence of judgments, and their conclusion is a judgment as well (thus, hypotheses and conclusions of proofs are judgments). A characteristic feature of the variants of Hilbert-style deduction systems is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context-changing rules. Thus, if we are interested only in the derivability of tautologies, not hypothetical judgments, then we can formalize the Hilbert-style deduction system in such a way that its rules of inference contain only judgments of a rather simple form. The same cannot be done with the other two deductions systems: as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided—not even if we want to use them just for proving derivability of tautologies.