Truth value in the context of Binary digit

A belief is a subjective attitude that something is true or a state of affairs is the case. A subjective attitude is a mental state of having some stance, take, or opinion about something. In epistemology, philosophers use the term belief to refer to attitudes about the world which can be either true or false. To believe something is to take it to be true; for instance, to believe that snow is white is comparable to accepting the truth of the proposition "snow is white". However, holding a belief does not require active introspection. For example, few individuals carefully consider whether or not the sun will rise the next morning, simply assuming that it will. Moreover, beliefs need not be occurrent (e.g., a person actively thinking "snow is white"), but can instead be dispositional (e.g., a person who if asked about the color of snow would assert "snow is white").

There are various ways that contemporary philosophers have tried to describe beliefs, including as representations of ways that the world could be (Jerry Fodor), as dispositions to act as if certain things are true (Roderick Chisholm), as interpretive schemes for making sense of someone's actions (Daniel Dennett and Donald Davidson), or as mental states that fill a particular function (Hilary Putnam). Some have also attempted to offer significant revisions to our notion of belief, including eliminativists about belief who argue that there is no phenomenon in the natural world which corresponds to our folk psychological concept of belief (Paul Churchland) and formal epistemologists who aim to replace our bivalent notion of belief ("either we have a belief or we don't have a belief") with the more permissive, probabilistic notion of credence ("there is an entire spectrum of degrees of belief, not a simple dichotomy between belief and non-belief").

A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. Propositions are the objects denoted by declarative sentences; for example, "The sky is blue" expresses the proposition that the sky is blue. Unlike sentences, propositions are not linguistic expressions, so the English sentence "Snow is white" and the German "Schnee ist weiß" denote the same proposition. Propositions also serve as the objects of belief and other propositional attitudes, such as when someone believes that the sky is blue.

Formally, propositions are often modeled as functions which map a possible world to a truth value. For instance, the proposition that the sky is blue can be modeled as a function which would return the truth value ${\displaystyle T}$ if given the actual world as input, but would return ${\displaystyle F}$ if given some alternate world where the sky is green. However, a number of alternative formalizations have been proposed, notably the structured propositions view.

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.

In logic, false (Its noun form is falsity) or untrue is the state of possessing negative truth value and is a nullary logical connective. In a truth-functional system of propositional logic, it is one of two postulated truth values, along with its negation, truth. Usual notations of the false are 0 (especially in Boolean logic and computer science), O (in prefix notation, Opq), and the up tack symbol ${\displaystyle \bot }$.

Another approach is used for several formal theories (e.g., intuitionistic propositional calculus), where a propositional constant (i.e. a nullary connective), ${\displaystyle \bot }$, is introduced, the truth value of which being always false in the sense above. It can be treated as an absurd proposition, and is often called absurdity.

Propositional logic is a branch of logic. It is also called statement logic, sentential calculus, propositional calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called first-order propositional logic to contrast it with System F, but it should not be confused with first-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication, biconditional, and negation. Some sources include other connectives, as in the table below.

Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

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 ${\displaystyle P}$.

In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value. The typical example is in propositional logic, wherein a compound statement is constructed using individual statements connected by logical connectives; if the truth value of the compound statement is entirely determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and any logical connectives used are said to be truth functional.

Classical propositional logic is a truth-functional logic, in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function. On the other hand, modal logic is non-truth-functional.

An accessibility relation is a relation which plays a key role in assigning truth values to sentences in the relational semantics for modal logic. In relational semantics, a modal formula's truth value at a possible world ${\displaystyle w}$ can depend on what is true at another possible world ${\displaystyle v}$, but only if the accessibility relation ${\displaystyle R}$ relates ${\displaystyle w}$ to ${\displaystyle v}$. For instance, if ${\displaystyle P}$ holds at some world ${\displaystyle v}$ such that ${\displaystyle wRv}$, the formula ${\displaystyle \Diamond P}$ will be true at ${\displaystyle w}$. The fact ${\displaystyle wRv}$ is crucial. If ${\displaystyle R}$ did not relate ${\displaystyle w}$ to ${\displaystyle v}$, then ${\displaystyle \Diamond P}$ would be false at ${\displaystyle w}$ unless ${\displaystyle P}$ also held at some other world ${\displaystyle u}$ such that ${\displaystyle wRu}$.

Accessibility relations are motivated conceptually by the fact that natural language modal statements depend on some, but not all, alternative scenarios. For instance, the sentence "It might be raining" is not generally judged true simply because one can imagine a scenario where it is raining. Rather, its truth depends on whether such a scenario is ruled out by available information. This fact can be formalized in modal logic by choosing an accessibility relation such that ${\displaystyle wRv}$ if ${\displaystyle v}$ is compatible with the information that is available to the speaker in ${\displaystyle w}$.

In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie", the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.

Assume that "this sentence is false" is true, then we can trust its content, which states the opposite and thus causes a contradiction. Similarly, we get a contradiction when we assume the opposite.

Moral relativism or ethical relativism (often reformulated as relativist ethics or relativist morality) is used to describe several philosophical positions concerned with the differences in moral judgments across different peoples and cultures. An advocate of such ideas is often referred to as a relativist.

Descriptive moral relativism holds that people do, in fact, disagree fundamentally about what is moral, without passing any evaluative or normative judgments about this disagreement. Meta-ethical moral relativism holds that moral judgments contain an (implicit or explicit) indexical such that, to the extent they are truth-apt , their truth-value changes with context of use. Normative moral relativism holds that everyone ought to tolerate the behavior of others even when large disagreements about morality exist. Though often intertwined, these are distinct positions. Each can be held independently of the others.