Names and Lemmon numbers for modal logic rules


Stanislav Hall once said

Using the term as a nonlinear science is like referring to the bulk of zoology as a study of non-elephant animals.

There is only one way to be linear, but there are many ways to be non-linear.

A similar observation applies to non-classical logic. There are many ways not to be classic.

Modal logic axioms

Modal logic Expands classical logic by adding modal □ and □ operators. The interpretation of these operators depends on the application, but a common interpretation is “necessarily” and “may”.

There are an awkward number of possible axioms to choose from to create a system of modal logic, and often these have mysterious names like “K”, “G” and “5”. Rarely is any reason given for names, and after digging into it a bit, I think I know why: there is not much reason. Or rather, the reasons are historical and not anonymous.

For example, Axiom T is so named because Kurt grew up publishing an article calling some set of “System T” axioms. Of course this raises the question of why Growing up called his system “T”.

Axiom 5 gets its name from the classification of CI Lewis logical systems. But Lewis’ numbering of systems S1 to S5 was inherently arbitrary, except for larger numbers corresponding to more axioms.

Unless you are interested in history for its own sake, you probably should not ask why rules are called what they are.

Lemon numbering

Although there is not much logic behind the names of the logic axioms of the model, there is a convenient way to catalog many of them. E.J. Lemon [1] Note that many axioms for modal logic can be written in a form

Below is a list of some common modal logical axioms, along with their names and lemon numbers.

 begin center  begin tabular lll name and requirement and lemon \  hline B &  (p  to  boxempty  Diamond p ) & (0, 0, 1, 1) \ C &  ( ( boxempty p  wedge  boxempty q)  to  boxempty (p  wedge q) ) & NA \ D &  ( boxempty p  to  Diamond p ) & (0, 1, 0, 1) \ G &  ( Diamond  boxempty p  to  boxempty  Diamond p ) & (1, 1, 1, 1) \ K &  ( boxempty (p  to q)  to ( boxempty p  to  boxempty q) ) & NA \ T &  ( boxempty p  to p ) & (0, 1, 0, 0) \ 4 &  ( boxempty !  boxempty p  to  boxempty p ) & (0, 2, 1, 0) \ 5 &  ( Diamond p  to  boxempty  Diamond p ) & (1, 0, 1, 1) \  end Tables  end center

[1] EJ Lemmon, An Introduction to Modal Logic, American Philosophical Quarterly, Monograph 11. Oxford, 1977.




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