Possible world in the context of Many-worlds interpretation

Reality is the state of everything that exists, not how they might be imagined. Different cultures and academic disciplines conceptualize it in various ways.

Philosophical questions about the nature of reality, existence, or being are considered under the rubric of ontology, a major branch of metaphysics in the Western intellectual tradition. Ontological questions also feature in diverse branches of philosophy, including the philosophy of science, religion, mathematics, and logic. These include questions about whether only physical objects are real (e.g., physicalism), whether reality is fundamentally immaterial (e.g., idealism), whether hypothetical unobservable entities posited by scientific theories exist (e.g., scientific realism), whether God exists, whether numbers and other abstract objects exist, and whether possible worlds exist. Skeptics question whether any of those claims are true, and suggest more extreme postulates.

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.

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.

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.

Importance is a property of entities that matter or make a difference. For example, World War II was an important event and Albert Einstein was an important person because of how they affected the world. There are disagreements in the academic literature about what type of difference is required. According to the causal impact view, something is important if it has a big causal impact on the world. This view is rejected by various theorists, who insist that an additional aspect is required: that the impact in question makes a value difference. This is often understood in terms of how the important thing affects the well-being of people. So in this view, World War II was important, not just because it brought about many wide-ranging changes but because these changes had severe negative impacts on the well-being of the people involved. The difference in question is usually understood counterfactually as the contrast between how the world is and how the world would have been without the existence of the important entity. It is often argued that importance claims are context- or domain-dependent. This means that they either explicitly or implicitly assume a certain domain in relation to which something matters. For example, studying for an exam is important in the context of academic success but not in the context of world history. Importance comes in degrees: to be important usually means to matter more within the domain in question than most of the other entities within this domain.

The term "importance" is often used in overlapping ways with various related terms, such as "meaningfulness", "value", and "caring". Theorists frequently try to elucidate these terms by comparing them to show what they have in common and how they differ. A meaningful life is usually also important in some sense. But meaningfulness has additional requirements: life should be guided by the agent's intention and directed at realizing some form of higher purpose. In some contexts, to say that something is important means the same as saying that it is valuable. More generally, however, importance refers not to value itself but to a value difference. This difference may also be negative: some events are important because they have very bad consequences. Importance is often treated as an objective feature in contrast to the subjective attitude of caring about something or ascribing importance to it. Ideally, the two overlap: people subjectively care about objectively important things. Nonetheless, the two may come apart when people care about unimportant things or fail to care about important things. Some theorists distinguish between instrumental importance relative to a specific goal in contrast to a form of importance based on intrinsic or final value. A closely related distinction is between importance relative to someone and absolute or unrestricted importance.

The phrase "the best of all possible worlds" (French: Le meilleur des mondes possibles; German: Die beste aller möglichen Welten) was coined by the German polymath and Enlightenment philosopher Gottfried Leibniz in his 1710 work Essais de Théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal (Essays of Theodicy on the Goodness of God, the Freedom of Man and the Origin of Evil), more commonly known simply as the Theodicy. The claim that the actual world is the best of all possible worlds is the central argument in Leibniz's theodicy, or his attempt to solve the problem of evil.

In philosophy, specifically in the area of metaphysics, counterpart theory is an alternative to standard (Kripkean) possible-worlds semantics for interpreting quantified modal logic. Counterpart theory still presupposes possible worlds, but differs in certain important respects from the Kripkean view. The form of the theory most commonly cited was developed by David Lewis, first in a paper and later in his book On the Plurality of Worlds.

Formal semantics is the scientific study of linguistic meaning through formal tools from logic and mathematics. It is an interdisciplinary field, sometimes regarded as a subfield of both linguistics and philosophy of language. Formal semanticists rely on diverse methods to analyze natural language. Many examine the meaning of a sentence by studying the circumstances in which it would be true. They describe these circumstances using abstract mathematical models to represent entities and their features. The principle of compositionality helps them link the meaning of expressions to abstract objects in these models. This principle asserts that the meaning of a compound expression is determined by the meanings of its parts.

Propositional and predicate logic are formal systems used to analyze the semantic structure of sentences. They introduce concepts like singular terms, predicates, quantifiers, and logical connectives to represent the logical form of natural language expressions. Type theory is another approach utilized to describe sentences as nested functions with precisely defined input and output types. Various theoretical frameworks build on these systems. Possible world semantics and situation semantics evaluate truth across different hypothetical scenarios. Dynamic semantics analyzes the meaning of a sentence as the information contribution it makes.

Saul Aaron Kripke (/ˈkrɪpki/; November 13, 1940 – September 15, 2022) was an American analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emeritus professor at Princeton University. From the 1960s until his death, he was a central figure in a number of fields related to mathematical and modal logic, philosophy of language and mathematics, metaphysics, epistemology, and recursion theory.

Kripke made influential and original contributions to logic, especially modal logic. His principal contribution is a semantics for modal logic involving possible worlds, now called Kripke semantics. He received the 2001 Schock Prize in Logic and Philosophy.

Counterfactual conditionals (also contrafactual, subjunctive or X-marked) are conditional sentences which discuss what would have been true under different circumstances, e.g. "If Peter believed in ghosts, he would be afraid to be here." Counterfactuals are contrasted with indicatives, which are generally restricted to discussing open possibilities. Counterfactuals are characterized grammatically by their use of fake tense morphology, which some languages use in combination with other kinds of morphology including aspect and mood.

Counterfactuals are one of the most studied phenomena in philosophical logic, formal semantics, and philosophy of language. They were first discussed as a problem for the material conditional analysis of conditionals, which treats them all as trivially true. Starting in the 1960s, philosophers and linguists developed the now-classic possible world approach, in which a counterfactual's truth hinges on its consequent holding at certain possible worlds where its antecedent holds. More recent formal analyses have treated them using tools such as causal models and dynamic semantics. Other research has addressed their metaphysical, psychological, and grammatical underpinnings, while applying some of the resultant insights to fields including history, marketing, and epidemiology.

Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Joyal. It was first conceived for modal logics, and later adapted to intuitionistic logic and other non-classical systems. The development of Kripke semantics was a breakthrough in the theory of non-classical logics, because the model theory of such logics was almost non-existent before Kripke (algebraic semantics existed, but were considered 'syntax in disguise').

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 ${\displaystyle w}$.

David Kellogg Lewis (September 28, 1941 – October 14, 2001) was an American philosopher. Lewis taught briefly at UCLA and then at Princeton University from 1970 until his death. He is closely associated with Australia, whose philosophical community he visited almost annually for more than 30 years.

Lewis made significant contributions in the philosophy of mind, philosophy of probability, epistemology, philosophical logic, aesthetics, philosophy of mathematics, philosophy of time, and philosophy of science. In most of these fields he is considered among the most important figures of recent decades. Lewis is most famous for his work in metaphysics, philosophy of language and semantics, in which his books On the Plurality of Worlds (1986) and Counterfactuals (1973) are considered classics. His works on the logic and semantics of counterfactual conditionals are broadly used by philosophers and linguists along with a competing account from Robert Stalnaker; together the Stalnaker–Lewis theory of counterfactuals has become perhaps the most pervasive and influential account of its type in the philosophical and linguistic literature. His metaphysics incorporated seminal contributions to quantified modal logic, the development of counterpart theory, counterfactual causation, and the position called "Humean supervenience". Most comprehensively in On the Plurality of Worlds, Lewis defended modal realism: the view that possible worlds exist as concrete entities, and that our world is one among many equally real possible ones.

A logical possibility is a logical proposition that cannot be disproved, using the axioms and rules of a given system of logic. The logical possibility of a proposition will depend upon the system of logic being considered, rather than on the violation of any single rule. Some systems of logic restrict inferences from inconsistent propositions or even allow for true contradictions. Other logical systems have more than two truth-values instead of a binary of such values. Some assume the system in question is classical propositional logic. Similarly, the criterion for logical possibility is often based on whether or not a proposition is contradictory and as such, is often thought of as the broadest type of possibility.

In modal logic, a logical proposition is possible if it is true in some possible world. The universe of "possible worlds" depends upon the axioms and rules of the logical system in which one is working, but given some logical system, any logically consistent collection of statements is a possible world. The modal diamond operator ${\displaystyle \lozenge }$ is used to express possibility: ${\displaystyle \lozenge P}$ denotes "proposition ${\displaystyle P}$ is possible".

Situation semantics is a framework in formal semantics and situation theory in which the meanings of linguistic expressions are evaluated with respect to situations—partial, concrete parts or aspects of the world—rather than complete possible worlds. It was developed in the late 1970s and early 1980s by Jon Barwise and John Perry as an alternative to extensional model theory and possible-worlds semantics, with a particular focus on perception reports, attitude reports and other context-dependent constructions in natural language.

Situation semantics is underpinned by situation theory, a general mathematical theory of information developed by Barwise, Perry, Keith Devlin and others, which introduces formal objects such as infons (units of information), constraints and types to model how information is carried and flows between situations.