Possible world in the context of "Modal logic"

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.

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.