This is something I am wondering about, because all the formulations I've seen of the logic seem fairly difficult to grasp, e.g. lists of abstract axioms that have a few missing versus classical logic. Yet, I've had the following idea floating around and wonder if it could work.
For me, ordinary classical logic is "more intuitive" because I can put it to a concrete evaluation algorithm, in the sense that if you know whether the elementary facts (atomic variables) of a given situation are true or false, you can easily evaluate the truth of a compound statement about them by using truth tables, which also have simple, intuitive interpretations, e.g. that "and" is only true when both its inputs are true, which makes sense from casual usage in English. Yet, as is well-known, for intuitionistic logic, not only is this not possible, but any "truth-functional", i.e. truth table-based approach must utilize a system with infinitely many truth values. Hence, what I am wondering about is - is there some non-truth functional "evaluation algorithm" for intuitionistic logic?
To illustrate what I mean. For the infinite truth values approach, one such system that works is to use the open subsets of $(0, 1)$ (I believe), where a full subset (i.e. $(0, 1)$ itself) is maximum truth, an empty subset is maximum falsehood, and the "middle ground" is every other subset, where many subsets are incomparable as truth values - we can only say one is "more truthful" than another if the former includes the latter as a subset, thus forming in effect a sort of fibrated structure to the truth values. This is, admittedly, a bit more "intuitive", but what would it mean for a realistic proposition to have, say, a truth value of $(0, 0.5)$ or $(0, 0.1) \cup (0.2, 0.9)$?
Now I suppose that, if one interprets the underlying epistemic basis of intuitionist logic as starting from that "truth means verifiability", i.e. to say that a proposition $P$ is "true" means we can prove it to be true, and to say that a proposition $P$ is "false" means we can prove it to be false, these may represent, perhaps, states of incomplete proof, i.e. they correspond to a situation in which we have some, but not all, evidence a given proposition is true, and perhaps we can understand the subsets of the interval as "coloring in" different areas where we have and don't have evidence with a suitable understanding of what the different numbers in the interval mean. After all, topologies or, better, open sets, have been described as generally specifying "semi-decidable properties", i.e. properties for which an assertion of their truth can be verified.
In addition, we perhaps could use more complicated topological spaces, too, as they will also form Heyting algebras in the same fashion, which then gives a ready example of what I'm talking about. Suppose an atomic proposition is "The car is red". Now we have a "car-shaped" topological space (i.e. a surface in $\mathbb{R}^3$ shaped like at least the hull of the car), which then induces a Heyting algebra. We've not seen the whole car, but only its back end, and seen that yes, it is red. Thus we should valuate that atomic formula to the open subset consisting only of the back end of the car. Would this make any sense? In this regard, it would become a richer version of fuzzy logic, where we can only give a percentage (0% to 100%) in which something is true.
But if it is not sensible, and also for its own sake, I will also say what I mean by a non-truth functional approach, which was the original thrust of this question. First, it would seem natural and intuitive to, given that the "excluded middle" is rejected, first expand the set of truth values to include an additional "unknown" or "unverifiable" value (U), in addition to "true" and "false". Now we have three, but not two, truth values. It is clear this is insufficient for a truth-functional interpretation. Hence, what I wonder then is if we can come up with some algorithm - which need not be efficient - that:
- Takes as input an intuitionistic propositional formula,
- Takes as input an assignment of T/F/U values to each atomic variable,
- Evaluates the formula to the correct T/F/U value. Most importantly, that tautological formulas always evaluate to T and contradictory ones always to F.
Is such an algorithm possible? If not, why not? If so, what does it look like?