Modal logic

Modal logic (from the Latin. Modus - method, measure) - logic , in which in addition to standard logical connectives, variables and / or predicates there are modalities (modal operators).

Content

Comparison with formal logic

Formal logic can be simplified to the chain of true knowledge → process → conclusions.

Where does one get true knowledge for formal logics if only single true knowledge is universal? ..

Logic should respond to real life situations, and few universal truths .

Modal logic in a broad sense operates:

knowledge
assumptions (what we do not know)
questions (partly in the logic of knowledge )
tasks (what to do to gain knowledge) ^{[ clarify ]}

That is, it is a more realistic / practical extension of the propositional logic and the first order logic .

Claim Examples

For example, modal logic is able to operate with statements such as “Moscow has always been the capital of Russia” or “St. Petersburg, once in the past, was the capital of Russia”, which is impossible or extremely difficult to express in a non-modal language. In addition to temporal and spatial modalities, there are others, for example, “it is known that” (the logic of knowledge) or “it is possible to prove that” ( logic of provability ).

Usually, to denote a modal operator, $\Box$ ${\ displaystyle \ Box}$ and dual to it $\diamondsuit$ ${\ displaystyle \ diamondsuit}$ :

\diamondsuit A=\neg \Box \neg A.

{\ displaystyle \ diamondsuit A = \ neg \ Box \ neg A.}

This reflects the fact that saying “Moscow was once the capital of Russia” is the same as saying “it is not true that Moscow has never been the capital of Russia”.

Modalities

Modalities are different; the most common are temporal (“sometime in the future”, “always in the past”, “always”, etc.) and spatial (“here”, “somewhere”, “close”, etc.).

Aletic (from ancient Greek ἀλήθεια is true) modal concepts:
- Brain teaser:
  - L - is necessary
  - M - maybe
  - C - by chance.
- Actual:
  - $\Box$ ${\ displaystyle \ Box}$ - necessary
  - $\Diamond$ ${\ displaystyle \ Diamond}$ - possibly,
  - $\triangle$ ${\ displaystyle \ triangle}$ - by chance.

Axiological ( ancient-Greek. Ἀξίᾱ - value) modal concepts:
- OK,
- neutral
- badly.

Axiological logic was developed by the philosopher A. A. Ivin .

Temporary:
- past,
- the present,
- future.

Spatial:
- there,
- here,
- nowhere.

Knowledge Logic

Operates with the concepts of "knows", "believes".

Deontic Logic

Operates with concepts: obligation , permission , norm .

“You have to do it” (“Your duty is to do it”) or “You can do it”

These concepts were attempted to be introduced quite a long time ago, but only George von Wrygt had significant results in Deontic Logic, Mind, New Series, Vol. 60, No. 237. (Jan., 1951), pp. 1-15. ^[one]

Article 2007 on the implementation of deontic logic. A Formal Language for Electronic Contracts ^[2] using µ-calculus and mu-cke implementation from A. Biere ^[3]

Semantics

In mathematical logic and computer science, the most common is Kripke semantics , there are also algebraic semantics , topological semantics, and a number of others.

Syntax

A modal formula is defined recursively as a word in the alphabet consisting of a countable set of propositional variables. $P L$ ${\ displaystyle PL}$ classic bundles $\to ,\bot$ ${\ displaystyle \ to, \ bot}$ brackets $($ ${\ displaystyle (}$ , $)$ ${\ displaystyle)}$ and modal operator $\Box$ ${\ displaystyle \ Box}$ . Namely, the formula is

$p$ ${\ displaystyle p}$ for anyone $p\in PL$ ${\ displaystyle p \ in PL}$ .
$\bot$ ${\ displaystyle \ bot}$ .
$(A\to B)$ ${\ displaystyle (A \ to B)}$ , if a $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ - formulas.
$(\Box A)$ ${\ displaystyle (\ Box A)}$ , if a $A$ ${\ displaystyle A}$ - formula.

Normal modal logic is a set of modal formulas containing all classical tautologies , the axiom of normality.

\Box (p\to q)\to (\Box p\to \Box q)

{\ displaystyle \ Box (p \ to q) \ to (\ Box p \ to \ Box q)}

and closed relative to the rules of Modus ponens ${\frac {A,A\to B}{B}}$ ${\ displaystyle {\ frac {A, A \ to B} {B}}}$ , substitutions ${\frac {A(p)}{A(B)}}$ ${\ displaystyle {\ frac {A (p)} {A (B)}}}$ and the introduction of modality ${\frac {A}{\Box A}}$ ${\ displaystyle {\ frac {A} {\ Box A}}}$ .

The minimum normal modal logic is denoted by $K$ ${\ displaystyle K}$ .

Remarks

the theory of twins provides a translation of the language of quantified modal logic into a first-order theory (but not vice versa) without any intentional operators of the type “possible” and “necessary” ^[4]

Notes

↑ http://links.jstor.org/sici?sici=0026-4423%28195101%292%3A60%3A237%3C1%3ADL%3E2.0.CO%3B2-C
↑ DOI : 10.1007 / 978-3-540-72952-5_11
↑ A. Biere. mu-cke - efficient mu-calculus model checking. In O. Grumberg, editor, International Conference on Computer-Aided Verification (CAV'97), number 1254 in Lecture Notes in Computer Science, pages 468–471. © Springer-Verlag, 1997
↑ Karpenko Alexander Stepanovich in Questions of Philosophy 2016 № 12

Literature

Chagrov A., Zakharyaschev M. Modal Logic. — Oxford University Press, 1997. (in English)
Blackburn P., de Rijke M., Venema Y. Modal Logic.— CambridgeUniversity Press, 2002.
Kondakov N. I. Logical dictionary-directory. - M: Science, 1976. - 720s.
R. Faith, Modal Logic. - The Main Editorial Board for Physics and Mathematics of the Science Publishing House, M., 1974.
D. Shkatov, Modal Logic and Modal Fragments of Classical Logic.— Institute of Philosophy, RAS, 2008. ISBN 978-5-9540-0128-0 (see book description: in Ozone )

Links

http://plato.stanford.edu/entries/logic-modal/
https://cgi.csc.liv.ac.uk/~frank/MLHandbook Handbook of Modal Logic by Patrick Blackburn , Johan van Benthem , Frank Wolter

[1] ttp://links.jstor.org/sici?sici=0026-4423%28195101%292%3A60%3A237%3C1%3ADL%3E2.0.CO%3B2-C

[2] DOI : 10.1007 / 978-3-540-72952-5_11

[3] A. Biere. mu-cke - efficient mu-calculus model checking. In O. Grumberg, editor, International Conference on Computer-Aided Verification (CAV'97), number 1254 in Lecture Notes in Computer Science, pages 468–471. © Springer-Verlag, 1997

[4] Karpenko Alexander Stepanovich in Questions of Philosophy 2016 № 12