An experiment in propositional calculus

Q: Are truths simply objective reasons whose truth-values may or may not be verifiable?


This question seems to possess a native paradox, but that simply arises from a logical error in the semantics: we can’t address unverifiable statements as “truths”. Instead, they are logically contingent statements.

Even so: As Wittgenstein says in the preface of his Tractatus Logico-Philosophicus, “In order to draw a limit of thinking, we should have to think both sides of this limit.” Similarly, in order to establish the objectivity of a statement, its subjectivity must be conclusively denied as well as its independence of subjective considerations verified.

The attainment of these conditions can be explored through Sir Ayer’s verification principle, the tenets of which were established in his 1926 opus, Language, Truth and Logic. However, it must be noted that Ayer denied, reasonably, that unempirical hypotheses may be formed on the basis of empirical engagements with reality. By extension, there exists an inherent denial of any transcendent reality, which in turn eliminates the presence of any objective truths.

At the same time, however, there exist objective literal truths, which are closer to being tautologies than truths themselves simply because they are a repetition of meaning whose propositional variables are actually fixed and whose truth-value is also fixed.

During an argument, negation and affirmation are used to establish the value of a propositional formula. The formula could be any statement whose propositional variables can assume different values. For instance, the statement S has an unverified propositional value.

S: Smoking is disagreeable; drinking is agreeable.

To some, S will make sense while, to some others, S won’t make any sense at all. In order to establish the truth-value of S, we explore the existence of a logical system that is consistent with the value of S being both true and false. This is unlikely because it contradicts our logical framework itself. Then, the next step is to understand the structure of a logical system in which S is either true or false and such that the value of one propositional variable impacts the value of the second propositional variable directly.

In other words, we make S a formula with two variables, X and Y, and find out how the values of X and Y are consistent/inconsistent with each other while they exist in the framework of the same set of logical principles.

S: X • Y

If we now hypothesize that X cannot retain its value while Y’s value is held fixed, then we pursue the negation of this hypothesis in order to establish that S is true. If we affirm the hypothesis, then we will prove that S is false. In the course of either of these arguments, we repeatedly hypothesize and evaluate the truth-value of each, and proceed until we have with a hypothesis that corroborates or denies the parent hypothesis and so renders the statement as either true or false.

However, if a rhetorical tautology cannot be assumed to constitute a reason (because it is a repetition of meaning), and if Wittgenstein’s proposition that tautologies are statements deducible logically and therefore meaningless is true, then the tenets of propositional logic are neither tautologies nor analytic truths.

Moreover, no literal significance can be assigned to logically valid statements according to Sir Ayer! In this context, the existence of any literal significance of logically valid statements depends not on their analytic proposition but their synthetic proposition – as affirmed by Sir Ayer. (Here, according to George Berkeley: “esse est percipi”!)