There are many systems of logic. Examples include propositional logic, description logic, first-order logic, fuzzy logic, modal logic. (Logic - Wikipedia)
Maybe somebody should develop a package that defines the meaning of operators like <, <=, ==, ===, >=, >, ⇔, ⇒, `→’, etc. based on the type of logic the user want to apply.
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Or new Julia’s feature could be used:
Custom infix operators can now be defined by appending Unicode combining marks, primes, and sub/superscripts to other operators. For example, +̂ₐ″ is parsed as an infix operator with the same precedence as + (source: https://github.com/JuliaLang/julia/blob/master/NEWS.md)
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That seems like an intriguing possibility, however, I feel that typing those operators would be a royal pain! 
(like many of the Unicode operators that can be used in Julia)