Simplifying Complex Logical Systems through Box Operations

Box Logic

Box logic, though seemingly simple on the surface, is a sophisticated language used to create complex logical systems. In this segment we will investigate some of the latest innovations in logic technology by learning about box logic. Today our goal was to look at box logic.
There are three constants: a large box, a medium box, and a small box. They're all oriented one way or another. It's easy to form compound sentences from our consonants—if we want to negate a sentence, we just invert it (Not medium box; medium box).

To form this kind of junction from our constants, we stack them up (small box on top of medium box) or place multiple boxes or stacks of boxes on the table (colossal for).
Now let's try to apply the rules we've learned to solve some problems. For example, let's consider the following conditional: If it's Monday, Mary loves Pat or Quincy.

If Mary loves Pat, then Mary loves Quincy. Let's see how we can express this statement in our new language.
In order to convert a conditional statement of the form "If p, then q" into an equivalent statement of the form "not p or q," we need to understand that "p" implies "q" as equivalent to "not p or q." In other words, if it is true that "p" is true, then it must also be true that "q" is true.

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Similarly, if it is false that "p" is true, then it must also be false that "q" is true. In this case, if Mary loves Pat, then she must love Quincy as well. So we write: if it is Monday (a true statement), then Mary loves Pat or Quincy (a false statement). However, if it is not Monday (a false statement), then Mary does not love either Pat or Quincy (also a false statement).
The solution to the problem, which requires only one step, is shown below. First, we represent each piece of information in the problem with a box. Then, we apply resolution on literals inside boxes. Here, we have two boxes with medium boxes down; we can combine these two and eliminate their complementary literals to form answers. Since it's a duplicate, we can throw that one away. That leaves us with just one box on the stack: the conclusion. If it's Monday, Mary loves Quincy.
The language of logic is not concerned with particular sets of symbols, and it is possible to deduce logical truths by manipulating those symbols. Any instantiation of the symbols is equally good, and any operations on them are equally good provided that they respect that basic abstraction.