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Beyond the Presumption of Universal Applicability

Instead of letting it reside only in my head and occasionally slipping into casual conversations with others, I have concluded that I should put forth a written description of my logic system. Though it has some similarities with other logic systems, it still varies significantly from any I have been able to find out about, and addresses some pitfalls that I believe have been overlooked by other logicians.

Now, obviously, there is too much to my system of logic to possibly do justice to it in one blog post, so I will instead do it as a series of blog posts that will go on and on for as long as I have more material to add. Today I merely present my first installment in this series.

Though there are other pitfalls in pre-existing logic systems, the one I find most troublesome is what I call the Presumption of Universal Applicability. This is the insistance that, in a truth table, every statement must have an applicable truth-value, no matter what that applicable truth-value is.

In crisp logic, this assumption means that every statement must be either true or false. Even logicians such as Kleene, who acknowledge that we don’t always know if something is true or false don’t really dispute the ultimate notion of everything has to be true or false, even if beyond our scope of in formation. Granted, Kleene claimed to use “unknown” as a third truth-value — but really, if you dissect and analyze it, “unknown” really means nothing more than “This statement is either true or false – but I don’t know which one it is”. (Either that, or he’d have no way, without cheating, to prove that “A ∨ (!A)” is true when “A” is “unknown”.)

Granted, the concept of “unknown” is very important in applying logic to real-live situations – least one succumb to a fallacy that I call “Imposition of the Default”. However, it is not a truth-value in it’s own right. It is not it’s own spot on the truth-table, but merely an acknowledgement that we are not sure where on the truth-table the case-in-point resides.

In fuzzy logic, the Presumption of Universal Applicability means that every statement’s truth-value has only one dimension – it’s degree of truth (or membership) – denying the need of a second dimension to measure the significance, or it’s degree of applicability. Of course, for the duration of this post, I won’t go into how to get past the Presumption of Universal Applicability for fuzzy logic – beyond saying that it needs at some point to be done (which I just have finished doing). Instead, this post will focus on how to move past that notion in the area of crisp logic.

In crisp logic, the way to get past the Presumption of Universal Applicability is to realize that “true” and “false” are not the only possible values for a statement. Rather, one must realize that there is a third possible value, “nonapplicable”. If a statement is “nonapplicable” that doesn’t mean that we just don’t know if it’s true or false. It means that neither “true” nor “false” accurately describes the statement.

I strongly suspect that the reason why previous logicians have overlooked “nonapplicable” as a third possible truth-value, in favor of “unknown” (which as I have already described, isn’t really a truth-value at all, but merely an uncertainty state) may have been that though they were determined to expand the truth-table beyond the scope covered in Aristotelian logic, they were unwilling in any way to alter the portion of the truth-table that Aristotelian logic does cover.

A logic system that contains “nonapplicable” as the third possible truth value will not alter the portion of the truth-table where “and” and “or” statements are concerned. It will do this, however, where “if” statements are concerned.

Take, for example, the following table …

T F
T T F
F ? ?

In the row where the premise variable of this “if” statement is true, I have reaffirmed Aristotle’s assertion that the whole statement should have the same value as the assertion variable. However, in the lower row, where the premise variable is “falls”, I have filled the spaces with question-marks to indicate that this is an area of dispute between myself and Aristotle. Aristotle says that when the value of the premise variable is “false”, the value of the whole statement is always “true”. I disagree four a number of reasons. For one thing, this would refuse the usefulness of “a → b” to nothing more than a shorthand for “(!a) ∨ b”. But furthermore, common sense would dictate that the word “if” means that a statement is only concerned with cases where the premise is true. Therefore, if the premise statement is “false”, then that doesn’t make this a case where the whole statement is “true”. Rather, it means that this case is not one of the cases which the statement as a whole is concerned with. Hence, the correct truth-value would be “nonapplicable”.

So here, I present my trinary truth-table for “if then” statements:

T F N
T T F N
F N N N
N N N N

I could go on and expand the “and” and “or” tables, as well as discuss another operation needed in this form of logic that I call an “applicability test” – but those are for a future post.

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