So before I continue with potential solutions to the many problems I perceive surrounding existing formal logic systems (the implication operator, issues surrounding the introduction of time into a formal system and the concerns regarding replacing a binary logic system with a many valued logic system), which may ultimately bring us into the realm of modal logic (but I suspect its not that simple) I think it is best we take a step back and consider what it is we are trying to accomplish by formalizing any of this.
In other words, what is the "why" of a logical formal mathematical system and what questions we would like to try to answer once we have created such a thing. What is it we are trying to accomplish exactly. I suspect we are trying to address the following general areas …
1) Did something happen (past) or will something happen (future)?
(and is it the case that we can only answer for the present tense, or past or future?)
2) What is the certainty surrounding the derived answers (the probability) and is there any such thing as absolute certainty?
3) Is all of this somehow relative, either to the evaluator, the time or the place? Is it the case that everything is relative based on an evaluator's time and place and maybe even, who or what an evaluator is?
It would seem that predicting the future is harder than predicting the past but is this true and how do we demonstrate this. Clearly, if we could prove that something absolutely must happen given a minimum set of required inputs we could then predict the future with 100% certainty. Is it even possible to enumerate a minimum set of required inputs? Is it any easier to predict what did happen versus what will happen? How does probability enter into the equation. Can we ever say something is absolute?
To answer some of these questions let's go back to my perceived problems which may not be problems at all since I tend to be a simple uneducated layperson with nothing but questions and little formal education.
Probability can sometimes be expressed as 'necessity' and 'possibility'. When we look at modal systems we will see one can be expressed in terms of the other so I don't think this is a problem we can not overcome with already understood methods. Perhaps many valued logic is solved using a predicate of this nature.
When we further investigate the concept of time. we discover that building on A.N. Prior's work Saul Kripe seems to have proposed a solution to this issue and so have Rescher and Urquhart so one might be inclined to think this is just a matter of formalism and the logical conclusions derived from using their systems. Considering Rescher was a leading researcher into many valued logic, perhaps this solution is tied to the solution of time as well. I need to understand these systems better to be certain, but I suspect the foundational problem is not time, especially if we constrain time to the past nor do I believe it to be probability, again if we constrain it to the past.
Is there anything which is absolute in our universe and if there is, can there be any operations for combining absolutes to continue the absolute chain of certainty or is everything probabilistic or does everything become probabilistic once we use operators to combine results. I suspect things which happened in the past are 100% probabilistic, the future, not so much. Is this even true though it would seem so at face value. Perhaps more important is the concept of semantic relationship.
Revisiting the issues surrounding the implication operator we see this amounts to a semantic relationship (or lack thereof) between the antecedent and the consequent. Clearly these two must hold some valid provable relationship to each to overcome this problem, but it is fair to ask if this is simply a problem with formalism. I suspect not. I am not sure we can absolutely construct a system of semantic relationships that is consistent and complete. We tread into the boundary between mathematics and concepts when we investigate semantics and to quote Frege, "Concepts are areas with fuzzy boundaries." What are we left with when boundaries are ill defined?
It is often helpful to construct a thought experiment which can elucidate some of these issues. Let's use common sense to guide us and remove some difficult issues we are aware of to gain some insight. In other words, let's start with some low hanging fruit.
To remove the issues of perspective, knowledge and time let's take a twig. A small piece of wood. Now if I burn this piece of wood it will at some point cease to be a piece of wood and will instead become ash. This ash may ultimately blow away and we are left with nothing from something. Let us not concern ourselves at the moment with the semantic relationship between a piece of wood and ash and what ash blown away by the wind is, since I suspect this is where we are ultimately heading. Let us simply capture this event using a video recording device.
So if I film the event over time, of a piece of wood burning until it becomes ash and blows away in the wind I can certainly say this event did happen with 100% certainty. My video recording of this event is proof and so even if only I and a handful of others saw this in person we could certainly share this video recording with others. The fact that at some point in time this piece of wood did exist at some place (let's say my patio) and it no longer does, could be considered an indisputable fact of something that happened in the past (assuming we don't take into consideration fake or doctored videos, etc) at a certain location (many recording devices can also capture location) at a certain time (again, let's rely on the recording device's reporting of time) and again, let's not concern ourselves with relativity and observational issues.
This seems like the easiest thing to describe using a formal system and something where we can begin. Again, let me stress these points; excluding the consideration of relativity and the definition of observer.
So clearly, a formal logic system which could represent this event as an absolute certainty would be a good place to start. We wish to construct a logical formal mathematical system which represents this event absolutely. It will always prove this did happen. It could prove that it is "necessary" that this did happen in the past. I keep stressing the term "necessity" because we will soon become exposed to this basic concept when we review what a modal logic system is. You can look up the difference between "necessary" and "possibly" under any introduction to modal logic systems if this is still gnawing at you (which I hope it is and I hope you do) and by constraining our research to the past, for now, we can perhaps make some progress in constructing a system to model reality.
I think this simple thought experiment lays bare what we are really up against here though, and I don't know of any existing modal logic system which has completely solved this problem yet. While object oriented computer science can aid us a bit in the understanding of predicates such as 'is a' and 'has a' I believe the central predicate we will have to come to terms with is 'is to' and what we are trying to accomplish is the ability to handle the manipulation of concepts and what their 'is to' predicates are. I will refer to this as the semantic predicate or the semantic problem, and I suspect once we can solve this we will be in a much better place in beginning to construct a logical formal mathematical system which can be used to express reality. Using our above example, "ash 'is to' wood' as bla", is probably where we are heading.
But we have some work to do because the current state of modal logic is a great place to start.
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