Logic

Systems of mathematical logic (and of Classic logic) are too abstract and too general for actual reality. These systems are self-contained, sound and complete, but only within the context of underlying axiomatic system (a set of assumptions taken for granted).

Actual reality outside of a window, however, is governed by the Causality Principle, and in it neither contradictions, nor falsehoods exists.

Falsehoods are products of the mind, so are contradictions, and, of course, in (the context of) actual reality False implies nothing.

The notion of implication itself, which captured in a human language as if this then that or therefore, captures (in turn) the working of the Causality Principle (or the Law Of Causality) which is, perhaps, the only law out there.

Also, a truth table for mathematical inclusive or says that both T is T but this is called AND, not OR. Or means either, not both. Both is called AND. However, there is a perfect explanation for “or both” from the set-theoretic point of view - it is just an some area on a Vienn diagram, so this logic depends on a context (as it should be) and is not universally applicable (which very few people understood). Application of mathematical logic outside of a mathematical (set theoretical) context is an error.

So, the system of Classical Logic, which nowadays expressed using Boolean algebra, is applicable only to mathematical contexts and to symbolic expressions.

The system made to be nice, symmetric and complete artificially, by human minds, to fulfill some aesthetic goals. To collapse it back down to What Is, we have to sacrifice symmetry. The truth tables has to be purged from anything that uses False as a valid premise (again, falsehoods do not exist, so they cannot be a required precondition).

Notice, that all the major tautologies will be still valid, except one \[A \Rightarrow (A \land B) \] which misses the fact that \(B\) is not necessary and unrelated.

The only “true” tautology is, of course, Modus Ponens which captures the Causality Principle itself, and written this way \[((A \Rightarrow B) \land A) \Rightarrow B\] Is an application of a function - the core of The Lambda Calculus, and rewritten the other way \[(A \land (A \Rightarrow B)) \Rightarrow B\] is a functional pipeline - the core notion for a funtion composition.

This is how to ground everything back to What Is, out of which it arose.

Those who are trying to apply proof assistants to models which captures some aspects of reality should consider purging their truth tables (and introduce runtime errors when False is a premise).

What Is (not what we think might be) is the best type-checker.