Doubts About Logical Implication and Cause

Thesis: “logical implication” and “causation” are terms that correspond to ideas that probably don’t describe reality. The instances in which “logical implication” does not describe reality are those in which they mirror “causation” in terms of the images of thoughts that correspond with them. I will first take a look at logical implication, and then the notion of causation based off the contentions of Hume, and then describe possible strategies to describe reality that are free of both the ideas of logical implication and cause. (the notions of logical implication and cause are not themselves subject to logical implication and cause). The bivalence functor may be argued to be talking about synonyms at times. De Morgan’s law may be an instance of logical grammar determining instances, and may be related to when multiple descriptions result in the same outcome. This may be another class. These things should be said to be what is not being described in the essay. Any abstraction using logical implication is itself something that exists in reality and does not, it seems, have a relationship of cause.

     It is said in the received tradition of western logic that there are many statements that imply other ones, and that the cavalcade of one statement to the next constitutes, at least to an appreciable degree, what logical reasoning is. This something that can be seem throughout the advent of western philosophy, perhaps findings its most exact version in Leibniz, under the title of “principle of sufficient reason”. This principle can be summed up as follows: “nothing is without a reason or there is no effect without a cause” or, in latin, “nihil est sine ratione”. Typically, this sort of statement can be represented as something of the form “A→B”. Depending on who you ask, some will say that “A” is the cause, or “→” is the cause or causal event between things A and B, or perhaps some mixture of those two positions. In logic, what is considered “cause” or, perhaps, “logical causation”, can be represented as either one-way implication “-->” or two way implication “<-->”, which is called “bivalence”. Bivalence translates to “if and only if”, so that a statement like “A<-->B” can be translated as “A is the case if and only if B is the case”. One-way logical implication can be understood as an idiom that can be instantiated into of the form “if . . . then” such that “A-->B” can be understood “If A then B”. Note that “If A then B” doesn’t necessarily mean “If B then A”.   

  Let’s look at an example of how this works in terms of real world claims. If someone believed that every time a certain wavelength frequency produced that subjective sensation of the middle C followed, and that, conversely, all subjectively middle C notes only happened under the condition of the activated frequency of 440 Hz, then they could accurately portray their belief in terms of bivalence, and say “c notes happen if and only if there is an activation of the frequency 440 Hz”. Let’s imagine that there is a second person who is a musician with perfect pitch who sometimes hears middle C in the form of a subjectively remembered tunes, and questions whether there is some proximate activation of 440 Hz that occurs simultaneously, in other words, that the subjective middle C note can happen without 44O Hz, yet they still maintain the belief that in all cases that 440 Hz is stimulated that the subjective impression of C middle note is created, then this second person would aptly portray their belief of the relationship of C and 440 Hz as “If 44O HZ is stimulated then the middle C note is subjective experienced” but would not necessarily join this statement with the reverse “ If the middle C note is subjective experienced then  44O HZ is stimulated”. Now it is possible, if there is some material reality outside of us, that a 44O HZ can be stimulated without producing the subjective sensation of the middle C note is created, such that 44O HZ and middle C note sometimes converge sufficiently yet there some cases where 440 HZ occurs by itself, and it could also be the case that along with this states of affairs there are some independently occurring cases of the subjectively experienced C note, however I am bringing up these points of view and the according language suitable to describe them in order to illustrate what logical phrasing gets at, and so considering these more complicated possibilities misses the point and purpose of this explanation. 

    Let’s take a look at a few example of common tautologies that use both logical implication and bivalence. It seems helpful to start from the relatively simple to relative complex tautologies. The simplest tautology is “B<-->B” which seems to state the obvious, that if something exists, then something exists, as in, “if a jacaranda exist, then a jacaranda  exist”. It is not entirely evident what exactly the difference is between statements of this form, and statements that are merely repetitive or demonstrate idempotency as Boole dealt with in the algebraic formalism he made for his laws of thought. In other words, it is unclear why “B exists and B exists” is different from the statement “B exists, therefore B exists”, and therefore perhaps they are, largely or wholly, synonymous. For instance, by a certain application of the of idempotency saying “B exists” is synonymous “B exists and B exists”, which is to say, in formal parlance, “B ∧ B =B”. Perhaps, then, “B exists therefore B exist” is synonymous along with “B exists” and “B exists and B exists”.  Perhaps, though, another logician would say that by me asserting that “B exists therefore B exists” is to be understood as synonymous to “B exists and B exists” that I am fundamentally misunderstanding what “B exists therefore B exists” communicates, perhaps because the “therefore” (corresponding with formal bivalence) in communicating something more than the existence of B, materially twice repeated, and/or is to be understood differently than the Boolean operator “∧” (formal for “and”). If a logician were to respond with something roughly to this effect, I would then ask what they mean by “communicate”; for instance, are they saying that “communicate” means “corresponds with something that exists”, where exists can be understood as corresponding abstractly, or in the real world, or in “both realms”, as it were? Inasmuch as this logician would appeal to an abstract realm of logical and/or mathematical truths, I would be somewhat reluctant to feel that I needed to argue about communicating in this sense; I think the idea of a “platonic heaven of mathematical propositions”, though statistically modish with some mathematicians, is something that largely dismissed by a modern understanding of reality; if this appeal to popular opinion isn’t sufficient, then I would imagine that the onus is on a logician who believes in this abstract reality to show that correspondence occurs to this abstract realm, which would be a precondition to us producing arguments about which particular correspondences occur and which statements are synonymous. Personally, I think that what we call the abstract realm is probably just what occurs subjectively in the mind, bound to a particular a particular section in space if the standard model of science is correct, and that the theory of everything can simply be phrased as talking about the particulars of what exists, perhaps without there being a significant distinction between abstract aspects of that reality and material aspects of that reality, which features many minds that at certain time entertain certain abstract ideas. The way of viewing a model of reality provides some clarity on the question of an abstract world, which is a point that I bring up in order to follow the queue of the question of tautologies and synonym.

     When it comes to the question of whether form of tautologies like “B exists therefore B exists” correspond with material or concrete or, at the very least, “non-abstract reality”, the question then becomes what “therefore” adds to a portrait of reality, that is not already covered by a statement like “B exists” or “B exists and B exists”. When looked at this way, the question becomes decidedly Leibnizian because there is a conflation, seemingly, between logical implication and causation, since claims about cause occurring between two or more things will plausibly be couched in the language of one-way implication and bivalence (as I gave examples of when talking about C notes and instances of the activation of 440 HZ). Leibniz, the logicians’ logician, thought that if you were able to figure out all the implications that exist, that you could calculate how things cause one another in order to figure out all truths, whether scientific, theological, or political, an idea that he called “characteristica universalis”, an idea that is admittedly picturesque. He never completed by idea, but thought that it could be an Esperanto-like language using something like Chinese characters, that could systematically compute out most or all states of affairs.

     It is not evident, however, that the formalism of logical necessity reflect a sense of implication that occurs in reality, as it seems Leibniz thought was the case. In order to take this point further, it seems useful to invoke an aspect of Hume’s analysis of cause. Hume said that any time you see two things converge that all you can strictly say is that two things have happened at the same time, not necessarily that causation has occurred between the two. I want to push this point a little further and say that if we are not able to determine whether cause exists or not based on observation, that that may very well be an indication that the idea of cause does not make sense or is not a “clear” or “well-formed hypothesis” (where the set of all well-formed statements has as a subset of it the set of statements that correspond with reality to the degree that correspondence is achieved). This is similar to a point that logical positivism make, namely, in order for a hypothesis to be sensible or well-formed it has to reduce to meaningful experience, though the Quine-Duhem thesis and the fact that it seems possible to come up a consistent hypothesis that doesn’t correspond with anything immediately observable and/or exists in some other part of the universe make the positivist point best ornamented with qualification.

    This flows into the larger question, though, of what is meant by something being “clear” and “well-formed” or “unclear” and “ill-formed”. This idea itself seems “unclear” and ambiguous to me, and my reasoning for this is as follows: all concepts in minds seem to be part of reality, and reality (to whatever degree it has parts or not), seems to be consistent. If reality is consistent then that which is in reality in consistent, by dint of syllogism. This would seem to imply that all thoughts have a certain clarity, if only because the fact that they are part of reality means that they are capable of being isomorphic or forming correspondence to reality; this point is something like if reality is square-like and its square-like property distributes to all its parts, then all its parts are square-like and capable of fitting directly onto any other square-like thing in reality, and there is nothing that can be circle-like; perhaps this metaphor is not helpful. Perhaps the fact that I am lead in this direction means that the descriptions of "clarity" or "unclear" or "confused" need to be understood in a different way, perhaps by details reports of phenomenological experience. 

     Since the question of whether of clarity and lack of clarity is itself not clear or doesn’t reduce to a decision procedure that helps sort out the phenomenological experiences in order to locate what models are those which are clear, it seems hard for me to strictly prove the point that I am trying to make in this paper now. To say it differently, it seems to me that I can prove this point about logical implication and cause being not well-formed, if and only if, I make a logical or valid point on this topic and logical implication and cause are both clarified and “well-formed” is clarified. To state this more clearly, the words “logical implication” and “cause” and need to be “well-formed” terms corresponding with “well-formed concepts”. “This perhaps forms of a sort of problem of event horizon of reasoning, though, because what this would require is for there to be a well-formed definition of “well-formed”, and this seems like it has the assume the property that it is seeking to establish, and therefore exhibits circular reasoning. One can invoke Neurath's boat--that the process of understanding reality is something like using some planks to stay afloat on in a boat while other planks are readjusted--and simply say that this paper is attempting to go towards some paradigm of clarity and understanding. I would want to go further into nuance on this matter, however it seems like to do so would be to go beyond the scope of this paper.

      Back to the primary queue. Perhaps someone would insist that they do have a clear thought of logical implication or causation when they think about it, whether it ends up corresponding with reality or not. This makes the point perhaps a little bit difficult because its reduces to phenomenology (though, in a sense, all arguments reduce to phenomenology in terms of things “clicking” for one or many participants in a discussion). I would say to this that I am not convinced that just because someone says something is clear in their mind that that does not mean that it is the case. For instance, to a person who sees a book and pencil on a desk that there are three distinct objects in front of them and think that their belief about reality qualifies, but a logically minded philosopher might respond to this and say that the book, pencil, and desk are arbitrary ways we impose notions of objecthood and/or boundaries upon reality, and a physicist might respond that this seemingly clear perception is unclear or fuzzy because in fact the book, pencil, and desk are made of microphysical states--like atoms, fermions, bosons and/or small constituents of matter--and that the microphysical states evince a sort of continuity that makes the seemingly parceled and parsed aspects of the objects as illusory. This is just one example of things that are thought of as clear that are contested; it seems like there may be many examples of this. The fact that people can have thoughts that they think are obviously clear that in fact are incorrect makes me think that issues like logical implication and cause may be similar, however it is possible that it is clear, at least to some people. It seems worth noting that it is not something that I experience as clear, and perhaps in order for someone to make the claim that the concept is clear that it would have to be case that it is clear for all thinkers or states of consciousness that are attempting to understand the term or terms that is synonymous or similar in meaning.  

      At first glance it may seem like “B-->B” is different from De Morgan's law, because one uses the same symbol twice whereas the other one uses a different langauge packaging to capture the same state of affairs, however it is not evident to me that there is a different significant to this argument. My reasoning for this is as follows: in “B-->B” there are either printed on paper or apparent on a computer screen two separate “B” marks, and to say the expression of De Morgan’s law, “~(P/\Q)<-->(~P)\/(~Q)” is also to have two separate set of marks that, namely, “~(P/\Q)” “(~P)\/(~Q)”. In both the case that of “B” and “B” that are understood as corresponding as a reproduction to ““B-->B” and the case of  of “~(P/\Q)” and  “(~P)\/(~Q)” that are understood as corresponding as a reproduction of ““~(P/\Q)<-->(~P)\/(~Q)”, the two terms that are broken apart into smaller chucks are understood to be to synonymous to one another. This is the case even though in the case of  ““B-->B”, the symbols and “B” and “B” are said to be “the same”--which is to say that they seem to be similar color patterns occupying relatively comparable locations given an understanding, of, say, Euclidean geometry in the form of 3-dimensional coordinate triples, of the form “(x, y, z)”--whereas the expressions “~(P/\Q)” and  “(~P)\/(~Q)” seem different that are understood as corresponding to a reproduction to ““B-->B” and the case of  of “~(P/\Q)” and  “(~P)\/(~Q)” that are understood as corresponding as a reproduction of ““~(P/\Q)<-->(~P)\/(~Q)”, the two terms that are broken apart into smaller chucks are understood to be synonymous to one another in both cases. This fact seems to be what is material to matter at hand. 

     I think it is helpful to view argument from a “God’s eye perspective”, which I think is useful whether one actually believes in God or not. God, perhaps, would have something like a grand unified theory (GUT) or theory of everything (TOE), at least in terms of syntax or outline, and I think we can meaningfully ask certain questions about this what this theory would like even if we don’t know all or nearly all of the details in it. It seems to be that the traditional elements of symbolic of logic, notably logical implication and bivalence, wouldn’t be needed; repetition or redundancy wouldn’t be needed to boot. Also, it seems like reality could have correspondence to it in terms of statements that are merely affirmative without the use of negation, which typically is symbolized by a tilde symbol (~). I wanted to mention these points, however probably each of those figurations is best discussed in a separate paper; logical implication and cause are, of course, the intended scope of this paper. To invoke a simple idea, if there were three things that existed, A, B, and C, then there would simply need to be three concepts or propositional states in order for some manner of correspondence to statements, which we can call A’, B’, and C’, that correspond with A, B, C, respectively, without the need for language that features repetition).

Perhaps someone is inclined to appeal to the notion of heritability in order to explain logical implication; I think that the notion of heritability of truthhood as we have inherited it is ambiguous (or possibly clear but not true of reality). Traditionally, the standard formats of syllogisms, which can include constructions like "if A is B and B is C, then A is C" are said to demonstrate "inheritance" of certain properties, and if we recast this syllogistic construction to include ratings of truth and falsity as is done in sentential logic, where truthhood and falsity are properties ascribed to statements, then truthhood could be inherited or, as it were, "spread out over” subsequent elements just as A is inherited by C in the above example. However, it is not evident to me that dividing things into parts, as the terms A, B, and C can be employed to do, does not reflect an arbitrary gerrymandering the mind somehow superimposes onto reality. If they are an instance of arbitrary gerrymandering, then perhaps heritability doesn't occur, due to the fact that reality, at its outset, doesn't need to inherit anything from itself. 

     We can see this in a syllogism that is said to apply to the real world. Three three-part statement “Socrates is a man, men are mortal, therefore Socrates is moral” is an example of one form of syllogism that is often brought up as exemplar. I think it is important to note that, as it were, Socrates-ness, humanity, and morality as things that in real life come interlocked in such a way

There are many breeds of syllogism, which can use ingredients like the qualifiers “all”, “some” or “none” to form three part deductions. A syllogism that is often touted as exemplar of syllogism is the statement.

     This is my fundamental contention and forms the heart of my paper. If two things can be understood as significantly different--or different enough to be considered “different”--then they are the sort things that don’t imply one another. In other words, the idea that something is the way that it is, seems to be better understood as as quality that doesn’t involve implication or cause in any way. It seems better to think of these qualities or states as “free states”, which is to say, “states without any cause”. This seems to be a consequence of the point of “law from no law”, a point brought up by physicist John Wheeler, the person who coined the termed “black hole”; the point can be understood as that the “laws” themselves (whatever that means) come into existence at the beginning of the universe but not prior to it, just as space, time, and the way that energy instantiates is thought to have also been brought into existence of the universe. This point is actually consistent with thull conclusion, a conclusion which Chris Langan--man, incidentally, with the highest IQ in America makes-- which is that it is reasonable to believe that God or creativity exists. In Chris Langan’s paradigm, however, the idea of logical implication or cause is still employed in some; it seems to me however that it is possible and logical to go one step further, and say that all logical implication and cause is likely something that doesn’t make sense. Although I do have doubts about this conclusion that I am making, it also seems to me currently unclear what is meant by logical implication and cause, at least given how I am have seened them explained to me so far.

     If what I am saying about logical implication and causation holds water, then I think the triality of axiom/rule of inference/theorem (an abstract parallel, perhaps, to the paradigm of initial state/scientific law/subsequent state) also reflects more about human thinking than reality itself. I don't think reality has to be described as having either rules of inference or scientific laws. I would be closer to advocating for a subject-predict language, of the form E(AB) where E "is distributed over" A and B (the quoted may be an example of an inexact phrase), however it is not evident to that reality contains either predicates or parts, and so an algebra of that form I suggest tempered with hesitance.