By Tom Veatch
Logic and Her DaughtersEvolution, Cognition, Emotion.
Looking back, years later, my friend, Mohammad Abd-Rabbo, the Palestinian linguistics professor, said this was the most boring and useless class in his entire career. I give him some credit because Tautology, my subject, can be understood as meaninglessness itself, and also because Symbolic Logic can draw one into incomprehensible thickets of technical noise with a gradually-approximating-zero signal to noise ratio. On the other hand, Aristotle's syllogistic, itself, the basis of Greek, Roman, Arabic, and medieval logic, is both mere tautology and a thicket of technical noise enabling, with difficulty, formal and correct, non-trivial and bulletproof, reasoning. As I show here.
But if you will be patient, I hope you will soon find traction, and an accessible easy path to true insight, from which you start to see vistas opening. Many vistas, for science, and for yourself.
I'm arguing for a somewhat unfashionable, but radically ambitious view of logic. I will shoehorn this view, using the slogan "H" for H Notation, onto various topics like evolution, psychology, and human evolution, hoping to favor simplicity, understandability, and true insight, in all these fields. So put your seat belt on.
The Queen of All Knowledge
My first Penn phonetics professor, Leigh Lisker, told me, The less you say, the less you lie: the less you are wrong. At the time I thought he was telling me to shut up, but later realized it was more than and different from that.
Tautology is just repeating yourself. As in 1 = 1. Simple. Nothing to it. Yes, on the one hand, tautology says nothing. On the other hand, look: it cannot be denied!
It is like the water way of the Tao, it is has no more substance than an imagined cloud, and yet it is more powerful than rock, or steel, unbreakable. It is trustworthy, for the patient. But these enthusiasms are for you to discover, not for me to say, or sell.
I believe: Logic, math and 1=1 are not functions of time; they are true before and after now, before and after any now, before the now of the first beginning at the big bang and after now of the final end in the universal maximum-entropy dissolution. Before and after the origin of life, and irrespective of any mind that knows it or doesn't know it. Describe some process today, with truth, irrespective whether an instance of it occurred yesterday, today, or tomorrow your description IS, WAS, WILL BE true; define something today, your definition applies outside time, to things your eyes do see or will see but also to things in the past you can never see again. Tautology is automatically true; and truth is outside of time.
As an example, consider logical possibility in the case of repeated evolution into an unoccupied niche, as in the Galapagos finches, or post-dinosaur-apocalypse predation, etc. You have a species B that evolves to fill a niche which was previously or elsewhere filled by another species A. Now, did the evolutionary niche exist before B evolved to fill it? Obviously Yes, because A had filled it before or elsewhere, so of course it existed, as a possibility. Well then did that niche exist before A filled the niche? Well there is nothing different between A filling it and B filling it, so if it existed before B filled it it must also have existed before A filled it. Therefore evolutionary niches exist, let's say in possibility, or in logic, BEFORE they are occupied.
Does not the same argument apply to life and physics and everything? Was life possible before life happened? Yes it was, otherwise it couldn't have. Do the laws of physics, where they are universal and true, that is, which are merely definitional, describing tautological logic amongst terms, do these laws not apply outside of time itself, and therefore also inside of time and therefore before, during, and after the Big Bang and everywhere in the universe? Yes, that's what universal means.
Abstraction in natureThe relationship between language, or let us reduce it to, abstraction, and reality is the fundamental mystery here. But that's not so complicated. Two things in the real world that are similar share some sameness about them. When multi-use, or shall we say, effective mechanisms (since without their ability to be used consistently or repeatedly or predictably even within slightly-variable circumstances we would hardly call them very effective), when they operate, they obviously operate over the multiple things or circumstances in the real world for each of which they are effective. But this is nothing but abstraction: the class of things picked out by the mechanism over which it can operate is thereby no less than an abstract category of things, an agglomeration of similars into a category each member of which can be treated the same, which is treated the same, by that mechanism. If the mechanism is nuclear or electrical or gravitational attraction or repulsion or quark interaction before the big bang, its sameness of input/output relations makes it an abstraction, a thing that applies in a general way.
Okay then, yes, if you have a general mechanism in nature, it IS itself an abstraction, by definition because of its generality. Despite diversity, nature does happen to be full of similars and samenesses, of consistents and repeatables, and those are the parts and aspects that organisms, and we, are interested in and need to know or be able to control, being as we all are indeed organisms with, usefully, a will to power, that is, a will to the control of general mechanisms. So if we humans, as an evolved, social, teamwork type of species, have evolved to control and use an inventory or library of abstractions in a lexicon of signals that we can communicate to each other, that makes a lot of sense. Contemplate the opposite: for us to be unable to detect sameness, in general, would be quite the detriment to our capabilities, or to be unable to communicate them, that would limit our capacities for social achievement as well.
By these links, abstraction connects to reality on the one hand and lexicon on the other hand. I haven't, indeed, derived syntax from first principles, no, but animal cognition, and the cognitive capability to reason tautologically, to apply logic, I think we have enough here to connect them: abstractions, minds, and nature itself, in a tight circle of mutual dependency, if not perfect equivalence.
We happen to have the right kind of minds, we'll get to what that means later, but even alien minds must share logical reasoning.
Let's try an example. Two 90 degree rotations are the same, in a sense, as one 180 degree rotation. Does this depend on human cognition, or the words or sentence in which it is expressed? Does such cognition depend on or require a symbolic language to capture the insight as mathematical expressions? I say No; rather, cognition itself depends on, itself perceives, the logic itself, or some abstraction of physicality from which this logical equivalence is derived.
My purpose will be to deconstruct logic into tautology, then expand this reliable zone of perfect understanding out from there.
Formal Logic as TautologyEven so-called rules of formal logic themselves reduce to tautology.
Where THIS and THAT reference the same thing, THIS=THAT both says nothing and is absolutely true: it is tautological. Thus also Definition = Tautology.
Newton's Tautological ReasoningApply definition = tautology to Isaac Newton. We know he knew about the hourglass, the yardstick, and the scale, and that there he was, speculating about the laws of nature, about different physical relationships. Under the mythological apple tree, he considered what he did know, measureable qualities starting with 1) weight (with the scale), 2) change in location (with the yardstick) and 3) time (by the hourglass), therefore in principle he could certainly define and measure change in location divided by time i.e. 4) velocity, therefore also (since you can measure velocity, you can also measure velocity twice, and then find the difference, which is how you measure ...) change in velocity over time i.e. 5) acceleration. So, W for weight, let's say, and A for acceleration.
Now the clever bit is, he decided to relate weight (under gravity) to acceleration by a discovered, no, an invented, no, a proposed, a DEFINED scaling he called mass, by saying M≝W/A, where here '≝' sign indicates a DEFINITION, or as we write it W=M*A, where here the '=' sign rather conceals the definition in the equation. So in the context of terrestial objects, it was already a tautology to start with, the definition of the mass of a thing is its weight divided by its acceleration (under gravity). The idea is, if you take the movement out of the weight, you get a degree of non-movingness, of deadness, or unreactiveness, or uninfluenceability, a quantity of resistance to acceleration, that is its mass. So far so good, so pleasant, so firm: we believe it because you can define anything you want, and it'll always be true by definition, in its imaginary definitional world. One can imagine how he was diddling around idly with the units for different kinds of things, and making up imaginary relationships to see where they lead, a formal game of units manipulation under the rules of tautological definition-making.
But today we stand in awe of this quite especially useful definition, we call it Newton's amazing universal Second Law of Motion, that miraculously seems to apply equally throughout the universe, not just to an apocryphal apple bonking dear Isaac on the head in about 1666, but also over there, out to our planets, far out to other stars, and down inside here also even to molecules and atoms, and in time from the first beginning to the last end, or beyond. Was Newton's miracle the empirical universality of this law? Or was it just a definition, where as we know a definition is necessarily universal?
It is universal because it is a definition, which is to say, a tautology. Because of course M is DEFINED as F/A (in the context of gravity F=W; that is, an object's weight is the force it exerts being pulled down by gravity). A definition holds conceptually, that is, outside of time and the mind that conceived it: as a tautology, it asserts nothing, and saying nothing, it cannot be false, it is everywhere and always undeniable, true. It is like a perspective, come over here and look at it this way, then you'll see how things line up in this view. Every perspective is true; it's not the perspective that can be false but well or ill-observed conclusions of fact that may be drawn from viewing the world from that perspective.
So the miracle is not that F=MA is true, but that it is useful. Where gravity is disregardable, mass still is measureable by applying non-gravitational force and seeing how much is needed to push the thing around. Nowadays we think the concept of mass that Newton defined into existence is as real as, or more real than, the measurements he started with to measure and define it. Flip a cognitive switch and mass becomes the primitive characteristic out of which complexities like force under gravity are built up, rather than the other way around. Think of it this way or that, however you find useful, the truth is it became true by tautology, by logic, the queen, the king of science, before and after, outside time itself. Tautologus Rex.
I like to say I'm a lazy boy, very lazy, where a choice exists. So here's a lazy boy trick. When I'm doing math, almost all the work intellectually is writing down what I know as an expression or equation and then thinking about it, primarily by substituting equal things into the expression. You can always substitute equals for equals and the result always is an equivalent expression, equally as true as you started with, because you could in principle substitute backwards, too, and get to the same thing.
Lazy Equals, or H notation
My problem is penmanship. I suck at it, and I hate it. I have to re-write the whole equation all over and then over again when all I did was another tiny substitution of some part of it with something equal to that part of it. Maybe it's a very long equation, and maybe I have three or four parts to substitute, or maybe some parts get substituted time after time. And it makes for a lot of stupid, stupid work doing copying, when the only thing I really need to re-write in full is the last version at the end, when I want to talk about that one. So instead I just use what I call H-notation: It's a Lazy Equals sign (remember "Lazy" applied to a cow brand means a 90 degree rotation), an H.
H-notation is just my way of writing an equals sign vertically. It's not a curly brace! Curly braces suck: they are SO hard to draw attractively when large, or with economy of space on the page. Curly braces have a different, more vague, meaning. Curly braces pick something out for a generic comment, like circling something. The comment might or might not be a substitution of something equal to the marked bit, whereas H-notation means equality. Just because it's Lazy, it's still an equals sign. Just because it's partial, it refers to a part of an expression, it still asserts equality between the above and below subexpressions that it connects.
Here: Draw a super-wide H below the part you want to say something about, and write what is equal to that part below the H.
Thus H-notation lets you substitute equals for equals within any expression or equation.
F = m * a |---| |------------| Fg 32'/sec/secFor example, above, Newton's Law F = m * a, in the context of an object under gravity, becomes the statement that the force of gravity on that object is the mass of that object times 32 feet per second per second.
You can use H-notation as often as you want on either side of the equals sign, and you can do it in columns too.
I find that about half of my math homework problems go away, when I don't do the recopying and just use H notation. Try it, you'll like it!
Incidentally, the way real mathematicians seem to solve this is to skip or hide all the step-by-step reasoning and substitutions, and just write the last equation. If you ask them they say that part is "obvious". Because it's job security for them, you see, the more obscure the reasoning is, then the harder everyone has to work to keep up with them, and then they seem to be so much smarter than everyone else. That method works in a bunch of professions, like law, medicine, and even plumbing. But I think it's obnoxious, especially when you're teaching, and math is (or should be) nothing else but teaching people. Write it down, see what comes out, that's how you actually teach yourself, and that's how you also teach others. Surprise! Language works! Who knew!
Dearie, we never told you this but Grandma Della had a secret baby named Robert with that roughneck Joe Smith before she met and got married to Grandpa Fred and had Helen and Fred Junior and John. But Grandma never told Grandpa she had a boy already, and made Joe raise him on the other side of town. So then Bob grew up and met your Aunt Helen, and fell in love and asked her to marry him. But after we explained the facts, she told him No. Because Bob's your uncle.
Tautology and its UsesAnother way to say "H notation" is to say "Tautology". Like A = A is a tautology, you can also say
A |---| ABecause it's the same thing. H.
Tautologies are statements that are true automatically. If something is true by definition, like The sky is above or The earth is below, then they don't actually say anything. Of course it's true: Duh!
Actually it's not duh!, you can discover things that are very surprising using tautological reasoning, or H notation, or mathematical substitution, for example that those three things are the same thing.
You philosophers will recognize H as a Kantian analytic judgement: the consequent is contained in the antecedent, as contrasted with synthetic judgements in which the consequent contains something more that is added to the antecedent.
Axiom, Definition, Observation, and Tautology
Axioms provide supposedly solid ground (you have to evaluate them to see if you believe them, yourself) which then let you reason about stuff.
Sets of axioms together define the logical basis of domains of math or knowledge such as geometry or number theory, etc.
Sets of axioms compete to be the minimal set of axioms from which the knowledge in some topic can be derived. (See my essay contra Hilbert Negative Dimensionality, for a list of redundant and excessive geometrical axioms which, I point out there, can be reduced and simplified.)
Sets of axioms fail when they are found to be mutually inconsistent. Inconsistency is when you can derive both a proposition and its negation, within the system. For example, Schopenhauer destroyed Hegel's philosophy when he proved that any proposition and its opposite could be derived from accepted axioms using canonical, correct procedures of Hegelian reasoning.
Veatch's Conjecture: I have long thought that the same as Schopenhauer did to Hegel could be done to the deep abstractions combined with standard structuralist methodology as used in theoretical syntax and phonology in the field of Chomskyan linguistics. I conjecture as follows: Any assertion about a deep abstraction in formal linguistics might be proven, including its opposite, by using the standard structuralist methods while simply modifying enough supporting assumptions about the meaning of other elements of assumed theory so that that assertion can be made to follow consistent with the data. It was this apparent freedom of conceptual replacement of basic ideas with their opposite which I observed on the part of great formal linguists like Paul Kiparsky, which repelled me from those subfields of linguistics. In syntax and phonology in particular, the distance between the abstract primitives of theory and the concrete actual observations is so very great. If any assertion, inclusing its opposite, may be derived from the same set of facts using this method, then nothing is trustworthy, there, all is pointless. This might force such linguists to make fewer assumptions in their various theoretical approaches.Inconsistency is a bitch. From a contradiction can be derived any proposition. Proof: Let B be any proposition. Then:
Therefore beware of inconsistency, or you will spout every nonsense imaginable. Do I do so? It is your job to evaluate this.
Definition. A definition is an Axiom that asserts substitutability of two expressions. It is really a domain-specific rule of inference; given a definition A ≝ B, and an expression ...A..., you may infer ...B.... A definition is like a perspective. If you look at something from a certain perspective, such as using a different label for that thing, which might free you from non-essential connotations and let you see a deeper insight, maybe you can have a simplified understanding of things which could be complicated without that definition.
For example, using polar coordinates, radius R and angle Theta, is a different way of thinking about the same geometrical things as using Cartesian, rectilinear coordinates X and Y. You can define R sqrt(x^2+y^2) and theta ≝ arctan(X/Y), and for some purposes everything will be simpler, like understanding systems of gears, like bike gears or automobile transmissions, where rotation is more important than translation in the X or Y dimension, for example.
Often a definition gives a simple name to a complex configuration. This will give you mental rest and an easier time of representing that configuration as part of even larger and more complex configurations. Chomsky refers to this as "Merge" when used on the fly in the creating and understanding of complicated sentences. "The pitcher who blah blah blah pitched to the batter who blah blah blah." There can be a lot of detail in the blah blah blah part, but you get reason with your limited mental capacity at different levels first by submerging the details in the higher level relationship, and once you have that squared away, you can descend into the details, and then all the things that were said in the complicated sentence, you can have thought about in some comprehensive and easy order, so that you can end up thinking very complicated thoughts after all. Definitions provide this service in a general and reuseable way.
Definitions can be bad in two senses. First, in the context of a set of axioms a given definition might produce inconsistency in reasoning, which would throw the whole system in the garbage, so definitions, like axioms, should be carefully constucted to prevent inconsistency. Second, they may not be useful, because they don't have traction in the current conceptual world, or the real world; they might not make things simpler after all. "Witchcraft is defined as magic carried out by a witch or warlock" might make the world more complicated after all, and might not be that useful when you think about it and draw out what it means when combined with everything else you think you know. But some might enjoy imagining imaginary worlds in which an otherwise useless, tractionless, world-complicating definition provides a key, titillating, even merely fun perspective within that imaginary world.
I'm saying that a definition cannot not wrong or false. It is just a perspective in the universe of all conceiveable worlds. It might be a bad definition, producing inconsistency or complication rather than traction, simplicity, and insight, but if you want to take that perspective, who will deny your right to self-contradiction, to superstitious bullshittery. On the other hand, we value useful definitions, which are consistent with the rest of what we know, and help us to reason about things.
Observation. An observation is, let's say, an observed actual event reduced to a linguistic description or characterization of that event. Observations transmute the rich sensory reality experienced by a perceiving observer into one or another limited classification, perhaps in multiple categories. Observation becomes data in the abstract, but it is tied to reality through the sensory and classification capabilities of the observer. Here's what's new. Statements about observations might be true or false. So far nothing can be false. But observational statements might be true or false. We hope we can trust an observer, for the time being we assume the observation is true as stated. But they could have lied, and it could in principle have been otherwise: the event observed might have occurred differently for any reason in or out of the screen of our knowledge or recording. Reality being the mystery that it is, observations could be wrong. In the world of observation is the world of empirical truth and falsity.
Tautology. But there is another kind of truth, which is the undeniable, non-empirical, truth of tautology. When a tautology is asserted, we say, Of course, it cannot be otherwise. Because P = P no matter what P is, we are persuaded.
For a child who doesn't have conservation of number, 8 is not 8, but sometimes it is 12, just like for Dad's dog, treats are numbered (and complained about) as 0, >0, where if Dad's other dog has 3 and it only has 2, there is no complaint, but as soon as Ralph has 0 and Princess has 1 (or more), it's time for a fuss. So we constrain ourselves to the knowledge (capable of being known) by the particular knower, within their capacities of knowledge, and we still say that P = P, once we have the right category set (where 8 for the child without conservation of number is replaced by >3, or whatever they are able to correctly reason with). In this way, definitions for those trying to understand us must be within the cognitive capacities of the audience.
Modulo this, tautology is indeed always true. It is not exactly empirical truth, because it doesn't come from particular observation. But it has the solidity, the satisfaction of trustworthy persuasion, when we see that some communicated essential idea is just a different way of saying the same thing. A complex argument becomes persuasive if its steps have the force of tautology, the force of obviousness.
ConclusionI'm trying to emphasize the value and centrality of the simple insight. A simple insight comes from a useful perspective that captures a true, perhaps general essence, out of what might be a complicated forest of observations and claims, along with axioms, definitions, and intermediate steps in the proof. It's the simplicity in the end that gives the power of thinking. And this is largely, highly, mostly achieved by steps that are tautological. You get to skip forward once you see how a step makes sense, and making sense usually means that it is equivalent to some previous step or steps, or to an understanding that you have already come to in considering the domain. Following an argument is a process of recognition of the previously known at each step, and dismissal of the novelty and unknownness of each step, one after the other, until everything falls, and you discover you know what you thought you didn't know. This is the way we learn what we can really trust to be true. The tautologies including axioms, definitions and restatements of previous steps in the chain of reasoning, along with the observations that you trust the truth of, bring you to persuaded understanding. And in the end, if it is a useful argument, reuseable later, it will conclude with a simple insight that you can hold, carry, and apply with your limited mental capacities: something that now you can use. Stand then on the shoulders of giants, and see! That's where we want to be.
What is science? Is it having statistical support for an assertion? It is a well-designed experiment? Yes, those, true, but science is reliable, useful, that is, trustworthy knowledge: true insight. What we want is the true insight. Not the P value, or the N, or the particular statistical distribution or model or test. And it's a combination of observation and logic, the supported, tautology-hard combination of axioms, definitions, rules of inference, and chains of reasoning, that bring us trustworthy insight, truth.
So I conclude by pointing out that in all kinds of areas of human knowledge, this applies. Logic is not a mere isolated and boring technical field, but a universal and encompassing system. Logic is what we use in cognitive science to prove insights which other sciences are too timid and logically unequipped to assess. If observation can be brought to bear within a system of logic on a question, more can be known than we might have thought.
I say so again in the other essays in this sequence of essays. I apply this aggressively to evolution, to biology, to the development of intelligence and language in humans, to the hierarchy of all knowledge. In a surprising number of cases, the truth is inevitable because it is tautology-hard, and what seems to be unknown or mere speculation, because one kind of direct observation is not available, still conclusions can be arrived at which are strong, reliable conclusions, using reasoning which is unquestionable, undeniable, where simple trustworthy insight is obtained by just taking the right perspective on what we do know, and following what is obvious to its necessary conclusions which might be far from obvious.
Mohammad, have I set you on the mountain? Have I opened vistas to you? I hope I have pulled you from boredom and technical mental strangulation in the Symbolic Logic world and given you both power and rest, given you a path of true insight.
So empowered, read on, I invite you!