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I have an expression that includes a variable that might and might not be present. So I would like to denote that variable with something that logically expressess maybe I was unable to find one. Is there other creative way to express maybe using logic operators?

GENIVI-LEARNER
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  • That depends on the expression. If the variable is added in, it can just take value $0$ when it is not there. If it is multiplied, it can be $1$. You can have a random variable that says whether to consider a term which takes value $0$ if the term is not included and $1$ if the term is included, then multiply it by the term. – Ross Millikan Mar 30 '20 at 14:01
  • How about a non-zero probability? – Dan Christensen Mar 30 '20 at 14:05
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    You could possibly (pun intended) use the symbol $\Diamond$ for "it is possible that" from modal logic (see also here). – Dave L. Renfro Mar 30 '20 at 14:19
  • @RossMillikan how would you encode that in a logic expression? Offcouse I can have a binary condition which could implicitly convey that the variable do take one of the two values but I was looking for more simplistic solution which I think Dave has already pointed out. – GENIVI-LEARNER Mar 30 '20 at 15:30
  • +1 I think that should work well so can I use the following expression to say "A and maybe B" as $$ A & \Diamond B$$ – GENIVI-LEARNER Mar 30 '20 at 15:32

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A logical expression is determined completely by the characters in the expression. It's not possible to have a single logical expression in which a variable is maybe present. It's as impossible as asking a set to maybe contain an element.

The situation you seem to be in is that you have two different expressions that refers to the same quantity, just like the expressions "evening star" and "morning star" refers to the same physical object.

One can conceive of a function, which assigns the quantity you're considering the logical expression that fit with the circumstances (this is roughly what happens when you use random variables). You could also state in writing that the two expressions refer to the same quantity, and which expression you will use in what circumstance.

However, it is not possible to use a logical operator to make a logical expression sometimes contain and not contain a variable.

Daniel Ahlsén
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  • what about the modal logic which @DaveL.Renfro mentioned? $\Diamond$ – GENIVI-LEARNER Mar 30 '20 at 15:33
  • The possibility operator is interpreted to mean that the proposition which the expression in the scope of the operator refers to is possibly true. It does not apply to the expressions themselves.

    Again, we must distinguish between the logical expression and the proposition to which it refers.

     – Daniel Ahlsén Apr 03 '20 at 08:24