Modal logic in the context of "Truth function"

Early Islamic philosophy or classical Islamic philosophy is a period of intense philosophical development beginning in the 2nd century AH of the Islamic calendar (early 9th century CE) and lasting until the 6th century AH (late 12th century CE). The period is known as the Islamic Golden Age, and the achievements of this period had a crucial influence in the development of modern philosophy and science. For Renaissance Europe, "Muslim maritime, agricultural, and technological innovations, as well as much East Asian technology via the Muslim world, made their way to western Europe in one of the largest technology transfers in world history." This period starts with al-Kindi in the 9th century and ends with Averroes (Ibn Rushd) at the end of 12th century. The death of Averroes effectively marks the end of a particular discipline of Islamic philosophy usually called the Peripatetic Arabic School, and philosophical activity declined significantly in Western Islamic countries, namely in Islamic Spain and North Africa, though it persisted for much longer in the Eastern countries, in particular Persia and India where several schools of philosophy continued to flourish: Avicennism, Illuminationist philosophy, Mystical philosophy, and Transcendent theosophy.

Intellectual innovations, achievements, and advancements of this period included, within jurisprudence, the development of ijtihad, a method or methodological approach to legal reasoning, interpretation, and argument based on independent inquiry and analogical deduction; within science and the philosophy of science, the development of empirical research methods emphasizing controlled experimentation, observational evidence, and reproducibility, as well as early formulations of empiricist epistemologies; commentaries and developments in Aristotelian logic, as well as innovations in non-Aristotelian temporal modal logic and inductive logic; and developments in research practice and methodology, including, within medicine, the first documented peer review process and within jurisprudence and theology, a strict science of citation, the isnad or "backing".

Modal metaphysics is a branch of philosophy that investigates the metaphysics underlying statements about what is possible or necessary. These include propositions such as "It is possible that I become a dentist" or "Necessarily, 2 + 2 = 4." Unlike ordinary factual statements, modal statements concern not just what is actual but what could or must be the case. Modal metaphysics seeks to understand what makes such statements true or false—what grounds their truth.

One influential framework for understanding modal claims comes from the development of modal logic, especially in the work of Saul Kripke. Kripke introduced the use of possible worlds as a formal tool: abstract representations of how things could have been. On this view, a statement like "Possibly, p" is true if there exists at least one possible world where p is true; "Necessarily, p" is true if p holds in every possible world.

In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am always hungry", "I will eventually be hungry", or "I will be hungry until I eat something"). It is sometimes also used to refer to tense logic, a modal logic-based system of temporal logic introduced by Arthur Prior in the late 1950s, with important contributions by Hans Kamp. It has been further developed by computer scientists, notably Amir Pnueli, and logicians.

Temporal logic has found an important application in formal verification, where it is used to state requirements of hardware or software systems. For instance, one may wish to say that whenever a request is made, access to a resource is eventually granted, but it is never granted to two requestors simultaneously. Such a statement can conveniently be expressed in a temporal logic.

An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.

The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of an object language. For example, an interpretation function could take the predicate symbol ${\displaystyle T}$ and assign it the extension ${\displaystyle \{(\mathrm {a} )\}}$. All our interpretation does is assign the extension ${\displaystyle \{(\mathrm {a} )\}}$ to the non-logical symbol ${\displaystyle T}$, and does not make a claim about whether ${\displaystyle T}$ is to stand for tall and ${\displaystyle \mathrm {a} }$ for Abraham Lincoln. On the other hand, an interpretation does not have anything to say about logical symbols, e.g. logical connectives "${\displaystyle \mathrm {and} }$", "${\displaystyle \mathrm {or} }$" and "${\displaystyle \mathrm {not} }$". Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.

Subjunctive possibility (also called alethic possibility) is a form of modality studied in modal logic. Subjunctive possibilities are the sorts of possibilities considered when conceiving counterfactual situations; subjunctive modalities are modalities that bear on whether a statement might have been or could be true—such as might, could, must, possibly, necessarily, contingently, essentially, accidentally, and so on. Subjunctive possibilities include logical possibility, metaphysical possibility, nomological possibility, and temporal possibility.

In logic, contingency is the feature of a statement making it neither necessary nor impossible. Contingency is a fundamental concept of modal logic. Modal logic concerns the manner, or mode, in which statements are true. Contingency is one of three basic modes alongside necessity and impossibility. In modal logic, a contingent statement stands in the modal realm between what is necessary and what is impossible, never crossing into the territory of either status. Contingent and necessary statements form the complete set of possible statements. While this definition is widely accepted, the precise distinction (or lack thereof) between what is contingent and what is necessary has been challenged since antiquity.

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.

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.

Logic translation is the process of representing a text in the formal language of a logical system. If the original text is formulated in ordinary language then the term natural language formalization is often used. An example is the translation of the English sentence "some men are bald" into first-order logic as ${\displaystyle \exists x(M(x)\land B(x))}$. The purpose is to reveal the logical structure of arguments. This makes it possible to use the precise rules of formal logic to assess whether these arguments are correct. It can also guide reasoning by arriving at new conclusions.

Many of the difficulties of the process are caused by vague or ambiguous expressions in natural language. For example, the English word "is" can mean that something exists, that it is identical to something else, or that it has a certain property. This contrasts with the precise nature of formal logic, which avoids such ambiguities. Natural language formalization is relevant to various fields in the sciences and humanities. It may play a key role for logic in general since it is needed to establish a link between many forms of reasoning and abstract logical systems. The use of informal logic is an alternative to formalization since it analyzes the cogency of ordinary language arguments in their original form. Natural language formalization is distinguished from logic translations that convert formulas from one logical system into another, for example, from modal logic to first-order logic. This form of logic translation is specifically relevant for logic programming and metalogic.