[FOM] Modal logics of contingency

[Åke Persson]

Joao Marcos wrote:

> Consider the unary modal operators L, for necessity, and M, for
> possibility, and define the following operators:
>
> C(A) = M(A) & M(~A)
> D(A) = L(A) v L(~A)
>
> So, C is the modal operator for contingency, and D is its dual.  Among
> other things, it is immediate to check, for instance, that C(A) is
> equivalent to C(~A) and to ~D(A), and that D(A) is equivalent to D(~A)
> and to ~C(A).  It is equally obvious that D respects the G-rule, but
> does not validate the K-axiom.
>
> This much to fix some intended meaning for the above operators.  Now,
> start from a modal logic which has only C (and D) as primitive
> operators, alongside with the classical ones.  Call any such a logic a
> *modal logic of contingency*.  Question:
>
> -o- Were such logics already investigated?  References?
>
> Obviously, given the above properties, there are no formulas depending
> only on C-formulas and D-formulas which allow us to define L and M in
> terms of them.  Question:
>
> -o- Is there any modal logic of contingency in which L or M can be
> introduced by some other kind of definition?
>
> If we restrict our attention to canonical modal frames, a direct proof
> of com-pleteness for such modal logics of contingency may present some
> gearbox diffi-culties.  (In particular, notice from the above that they
> will not in general constitute examples of normal modal logics.)
> Question:
>
> -o- Does anyone have an idea of how this proof could be done?  Or had we
> better change the underlying semantical structures?
>
> Comments and references welcome!


In my article "[FOM] negation vs intuitionism" posted 2003-03-23 I defined
necessity and possibility using quantity relations and then applied them
to a context that has a third alternative 'unknown'. In such a context
reductio ad absurdum proofs takes place. The primitives L and M was defined
as:

    (10)    Lp    =df    (Rp == 1)        ; 'p is necessary 100% true'
    (11)    ~Lp    =df    (Rp =/= 1)      ; 'p is not necessary 100% true'
    (12)    Mp    =df    (Rp =/= 0)       ; 'p is not necessary 0% true'
    (13)    ~Mp    =df    (Rp == 0)       ; 'p is necessary 0% true'

 and the relation between L and M was be derived as:

    (14)    Lp    ==    ~M(~p v ?p)
    (15)    Mp    ==    ~L(~p v ?p)

My summary was:

    "Intuitionism and constructivism has rejected reductio ad
    absurdum as an unsure method of proof. Above is shown that
    from ~M~p we can not conclude  Lp. We need to show ~M
    (~p v ?p), i.e. both ~M~p ("it's not possible that not-p") and
    ~M?p ("it's not possible that p is unknown") to conclude Lp.
    Intuitionists are right in their critisism but hits the wrong property
    of logic. Double negation reduction is a most basically
    foundation of all logic and will still remain untouched. There are
    still no problems to reduce ~~p to p, to toggle beween a
    meaning and its contrary meaning for every negation. Also for
    every negation of truth we still can toggle between the contraries
    Tp and Fp. As reductio ad absurdum proof takes place in
    modal contexts we need to handle them by modal logic but for
    that we need a modal logic strict constructive derived, like the
    logic refered above."

By the same formalism, contingency easily can be defined and its properties
be studied. As a contingent proposition is both possible and not necessary,
and a non-contingent is not possible or necessary, the modal operators for
contingency and non-contingency could best be defined in this logic as:

    (i)    Cp    =df    (~Lp & Mp)        ; 'p is contingent'
    (ii)   Dp    =df    (Lp v ~Mp)         ; 'p is non-contingent'

In this refered quantity relation logic de Morgans's laws and double
negation elimination are theorem. Following properties of C and D can now be
derived as (in a context with 'unknown'):

    (iii)  Cp    =df    (~Lp & Mp)            ; (i)
    == ~(Lp v ~Mp)                              ; de Morgan
    == ~Dp                                           ; (ii)

    (iv)  Cp    =df    (~Lp & Mp)               ; (i)
    == (~~M(~p v ?p) & Mp)                   ; (14)
    == (Mp & M(~p v ?p))                       ; ~~p == p

    (v)  C~p == (M~p & M(~~p v ?~p))       ; (iv), ~p/p
    == (M(p v ?~p) & M~p)                        ; ~~p == p
    == (M(p v ?p) & M~p)                          ; ?~p == ?p

Cp is necessary equal to ~Dp but not to C~p in this context. However, in a
context without 'unknown', ?p = ?~p = 0 and the relations (iv) and (v)
are reduced to:

    (iv')    Cp = (Mp & M~p)
    (v')     Cp = (Mp & M~p)

This result corresponds now fully to your definition of C(A), that is equal
and equaivalent to both C(~A) and ~D(A). However, to use M(A) & M(~A) to
define contingency requires that LEM hold, which the use of ~L(A) & M(A) not
does. Only in simpler contexts they are equivalent.

Finally a question: What can a logic of 'p is contingent' and 'p is
non-contingent' be used for?

Åke Persson