How is the logical connective $\bot$ used in formulae

Question

We are working on a question in which we have to prove that a certain set of logical connectives is functionally complete. It is said that we are allowed to use the fact that the set $\{\bot,\to\}$ is functionally complete. Our question is: how is the logical connective $\bot$ used in formulae? Is it possible to write something like $p\bot$? Or does $\bot$ always have to be used with an implication such as in $p\to\bot$.

Thanks in advance,

Hugo

Answer 1

If we use a rigorous definition of formulas (or well-formed formulas or meaningful expressions), symbols like '⊥' or '0' in Polish notation qualify as propositional constants. One possible definition of meaningful expressions goes as follows:

1. All lower case letters in alphabetical order of the Latin alphabet up to 'y' qualify as meaningful expressions: {'a', 'b', ..., 'v', 'x'} are all meaningful expressions. Note that this definition could get extended by numerical subscripts in principle.
2. '⊥' is a meaningful expression.
3. If y and z are meaningful expressions, then so is Cyz.
4. Nothing else is a meaningful expression.

This definition implies that every meaningful expression is either a lower case letter, '⊥', or starts with 'C'.

'p⊥' is not a lower case letter, nor does it start with 'C', nor is it '⊥'.

Therefore, it is not possible to correctly write 'p⊥' or something like it. But, '⊥' does not have to get used with an implication, since '⊥' is a standalone formula.

Algorithms also have gotten developed to determine whether a string of symbols is or is not a meaningful expression. Previous study of those algorithms also implies the above bold text.