Practical Logic

Most recently, we built up a working knowledge of predicate logic and how to quantify predicates into propositions. Thus, from statements such as x > 10, we can conjure statements like 100 > 10. The result is we can compute the logic truth value as True or False. That's all well and good for high academic performance in a discrete structures or discrete mathematics course. The operationally important question is, how practical is propositional logic.

I suggest four components are necessary to apply propositional logic in a practical manner. We need objects we can instantiate as variables. An easy way to conceptualize variables is to imagine subject phrases, nouns, and forth. An important note is the set of variables must be finite. We also need at least one connective. Here, we can conceptualize connectives as those words we refer to as conjunctions in English. These two are the essential propositional statement building blocks. The other two components are the tools to arrange the essential building blocks. We refer to the tool components as axioms and rules of inference.

As far as I can tell, these four components are naturally present in statements. Further, statements are practically manifest in declarations, claims, inquiries, opinion, fact, and so forth. Thus, it appears as if we can apply propositional logic to everything. Of course, absolutes carry unknown future exceptions and perhaps in a future essay we can explore potential areas where an exception may exist to the applicability of propositional logic. Until then, let's see just what we mean by apply.

Practicality

What I personally find simultaneously interesting and useful about propositional logic is it practically applies to multiple areas of life. Furthermore, these areas are intrinsically linked through a meta-context which is itself subject to propositional logic. Let me explain.

On one hand, we can process information and generate a set of outputs used to evaluate the information. Yes, this is computational albeit in a metadata context. That is to say, the outputs such as true or false, fallacious or consistent, logical or illogical, are properties of the information rather than the information proper. In this we have a practical ability to evaluate the information we encounter.

I'd like to point out one think, however. The information itself can attest to its own truth, its own consistency, and its own logic state. Importantly, this practical action requires the entirety of the information. We cannot process partial information and expect to get truthful, consistent, logical results. By the way, don't lose track of this metadata in our examination of practicality.

Now, let we can process information and generate a set of outputs used to evaluate the information be our example.

On the other hand, we can atomize complex problems using propositional logic. Our example statement isn't necessarily complex but it works for our purposes. We can break the information into sub-parts, which individually may represent sub-information, and compartmentalize the bits. Then, we can feed the bits back into the processing machinery and repeat. Thus, we can process information can become we, process, information or p, q, and r. Scale that up to multi-statement, convoluted word problems. The same process applies and iteratively can produce such simple variables. This is a practical tool-chain for decomposing problems, wouldn't you agree?

Meta-context

Lastly, we have the meta-context. Here, we take the metadata from the information processor and encapsulate the atomized sub-information with it. This seems like we'd simply be reconstituting we can process information from p, q, and r but that's inaccurate. Instead, I'm talking about a linked layer of true, consistent, logical superimposed over the variables in a manner which produces a richness or perhaps better stated as understanding. Moreover, the meta-context is itself a context encapsulating this structure. Hence, the entire clause can be false whereas individual sub-information can be true.

I find this interesting and practical because it serves as a reliable, robust tool to illuminate the inherent complexity in our information, in our life. What do you think?

Going Forward

I think we've come close to exhausting what we can do with propositional logic as an isolated topic. However, the combination of theory and practical knowledge ought to position us to move into the next subject.