# How to define ‘impossible’ using propositional modal logic?

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## What is modal proposition in logic?

Any proposition at least one of whose constituent concepts is a modal concept is a modal proposition. All other propositions are nonmodal. Any modal proposition can be represented in our conceptual notation by a wff containing one or more modal operators, e.g., “•”, “0”, etc.

## What is modal logic with example?

Even in modal logic, one may wish to restrict the range of possible worlds which are relevant in determining whether ◻A is true at a given world. For example, I might say that it is necessary for me to pay my bills, even though I know full well that there is a possible world where I fail to pay them.

## What Is syntax of modal logic?

The symbols of modal logic consistute of an infinite countable set P of proposi- tional variables, logical connectives, parenthesization, and the modal operator D. The choice of logical connectives depends on the development of proposi- tional logic one wants to follow; below I choose negation and implication.

## What is the modal possible?

Modal verbs express if something is certain, probable or possible. If something is possible in the future, we use ‘could’, ‘might’ or ‘may’ before the main verb to talk about it. If something is possible now or was possible in the past we add the word ‘have’ after the modal verb. This still expresses possibility.

## How do you read modal logic?

The box means what just means it is necessary that or necessarily the diamond means it is possible that or just possibly.

## What are the types of modal logic?

Modal logic can be viewed broadly as the logic of different sorts of modalities, or modes of truth: alethic (“necessarily”), epistemic (“it is known that”), deontic (“it ought to be the case that”), or temporal (“it is always the case that”) among others.

## Is modal logic first order?

First-order modal logics are modal logics in which the underlying propositional logic is replaced by a first-order predicate logic. They pose some of the most difficult mathematical challenges.

## Is modal logic useful?

Simply put, modal logics are useful any time that you want to reason about truths that are, well, modal. The example you gave contrasts first order logic and modal logic, but a more common starting point is to build modal logics upon propositional logics.

## Is modal logic math?

Publisher Summary. Formal modal logic is mostly mathematical in its methods, regardless of area of application. This chapter presents a wide variety of mathematical techniques developed over decades of studying the intricate details of modal logic.

## Where is propositional logic used?

It has many practical applications in computer science like design of computing machines, artificial intelligence, definition of data structures for programming languages etc. Propositional Logic is concerned with statements to which the truth values, “true” and “false”, can be assigned.

## What is possibility and necessity?

Possibility and necessity are related. Something is possible if its failing to occur is not necessary; if something is necessary, its failure to occur is not possible. Divers (2002), 3-4, provides a nice summary: “Possibility rules out impossibility and requires (exclusively) contingency or necessity.

## How do you know if something is logically possible?

Logical possibility is usually considered the broadest sort of possibility; a proposition is said to be logically possible if there is no logical contradiction involved in its being true.

## What does Epistemically possible mean?

An epistemic possibility is something that may be true, given the relevant epistemic constraints (for example, “Given what we know about the weather, it might rain tomorrow”), while an epistemic necessity is something that must be true given the relevant epistemic constraints (for example, “I don’t see Julie’s car in …