# Are there any established logical symbols for merely possible and contingently true?

In modal logic we have:

P → ◇P - If something is true, then it is true at some possible world.

◻P → P - If something is necessarily true, then it is true.

However, the reversed conditionals don't hold universally which yields two interesting classes of statements:

Mere possibilities - ◇P & ~P

Contingent Truths - P & ~◻P

Update: As Bumble pointed out what I call a mere possibility is logically equivalent to a contingent falsehood (~P & ~◻~P).

I understand that we can express these two classes of statements logically with only the standard connectives and modal operators (as shown above). However, they seem conceptually important enough to deserve a specialized notation.

So my question:

What existing notations are there for this? Ideally, a unary operator notation `_P` for each of ◇P & ~P and P & ~◻P.

• There are always expressions of probability, typically as a fraction or a percentage. Commented Aug 9, 2022 at 20:04
• @MarkAndrews This isn't really an issue of probability. I think probability is orthogonal to the modal categories I describe. How would do you imagine the mapping between probabilities and mere possibility/contingency? Commented Aug 9, 2022 at 20:24
• We don't really need a separate unary operator to indicate such combinations. The usual notation is expressive enough to distinguish necessary from contingent. Note that your "mere possibilities" are just contingent falsehoods. Commented Aug 9, 2022 at 20:35
• There is no established symbol, but some authors introduce their own. Girle in Modal Logics and Philosophy uses ∇ for (derived) contingency operator, del Cerro and Herzig in Logics of Contingency use L+ (necessity), L− (impossibility), C+ (contingent truth), and C− (contingent falsehood) as primitives. Commented Aug 9, 2022 at 23:11
• @Conifold you've answered my question. If you put it in the answer field I'll give it a checkmark. Commented Aug 9, 2022 at 23:18

There is no established symbol, but some authors introduce their own. Girle in Modal Logics and Philosophy, p.9 uses ∇ for (derived) contingency operator, i.e. ∇p:= ◊p ∧ ◊¬p, he also has the dual Δp:= ◻p ∨ ◻¬p (p is non-contingent). There is some history to it, Routley's Conventionalist and contingency-oriented modal logics used ∇ back in 1971.

Note that ∇p is not p ∧ ¬◻p, i.e. not "p is contingently true", actuality of p is not asserted. Fan in Bimodal logics with contingency and accident, who also employs ∇, does introduce the requisite symbol ●p:= p ∧ ¬◻p spelled as "it is an accident that p" (an allusion to Aristotle's accidental properties). He then remarks:

"The meanings of contingency and accident are so close that people mix the two notions from time to time in everyday discourse and academic research. For instance, Leibniz used the term ‘contingency’ to mean what is essentially meant by ‘accident’ (e.g. [1, 13]). For another example, in Chinese, the same character has been used to express both notions."

◊ and ◻ are then dropped and a language with ∇, ● as primitives is studied. The ● notation goes back to Marcos's Logics of Essence and Accident, explicitly dedicated to Aristotelian essentialism, where ∘ is used for "essential", and ●, ∘ replace ◊,◻ as primitives. Del Cerro and Herzig in Logics of Contingency also use L+ (necessity), L− (impossibility), C+ (contingent truth), and C− (contingent falsehood) as primitives from the start.

Actuality operator is denoted ○ in Hardegree's Modal Logic and @ in Stephanou's paper. There is a problem with plainly adding it to standard modal logic like K, it trivializes modality since ◊p → p is derivable.

• You say that people confuse contingency and accident; do you mean by this that people confuse contingent with (contingent and true) or does accident mean something else? Commented Aug 10, 2022 at 4:23
• @DavidGudeman I think Marcos and Fan mean the former, "contingent" is confused with "contingent and actual", which is what they call "accident". This is dual to Kripke's distinction between necessary and essential (if actual then necessary). Commented Aug 10, 2022 at 4:56

I suspect that there's a good chance that somewhere, someone has written about modal logic such that their system takes contingency as a primitive, and uses a "new" symbol for this. That, or maybe they repurpose the box and diamond, there. But this is just speculative. I've been reading more of/about a wide range of Central/South American logicians, and Spanish (IDK about Portuguese) tends to use certain punctuation marks more often than, or above those in, English. (At least, the rotated question mark shows up, bracketing entire sentences, like an unusual parenthesis over those sentences.) So maybe there's a place to start looking?

Besides all that, I do recall, though not by the author's name, a system where there was an actuality operator, and a circle was used for it. The author was not quite famous, even "for a logician/philosopher," but at least well-known enough (I think) that his notation might have found a few subscribers (among them: me).

As Bumble observes, so far your account of mere possibility is reducible to contingent falsity. It is not totally obvious that the same state of affairs obtains in the light of a definition of mere possibility as "possibly X and ~actually X." For now we can refer to necessarily merely possible objects: "possibly X and necessarily not actually X." This ends up being impossibility, and the same point can be made in a logic with no actuality operator, but only existential quantification and the flat assertion of a sentence sans a box or diamond (or whatever); still, the point seems more interesting when made in terms of an actuality operator (if only because of how the point reinforces the (apparently possible, but not indubitable) fact that actual possibility and possible actuality are relatively equivalent).