I started to read the first chapter of your book, it looks nice. If I may ask you, could you also provide answers to the questions you ask? Somewhat hidden, e.g. behind a button "Show me the answer". I often I think I know the answer, but A) my answer may be wrong or B) I'm right but my reasoning is wrong. (So I'd like to compare my answer with yours.) Thanks.
What are good introductory books for logic (classical and intuitionistic)? For category theory? Do you know any similar representation of paraconsistent logic?
For logic, it really depends of what you are searching for, for classical logic you can read the the classics, for example Russell and Tarski.
For constructive logic I cannot think of a good introduction (besides mine ;) ), I personally picked it up from books about category theory and computer science.
I think the general program of categorical logic, the work of Lambek and Scott, and J. Bell on topos theory and local set theory really make clear the relationship between category theory and logic, as well as lambda calculus.
A topos is essentially a cartesian closed category with a subject classifier. In Set this is the two element set of 1/0 which is a Boolean algebra and thus the internal logic of the category Set is classical.
In general though the subobject classifier is a heyting algebra which expresses the semantics of intuitionistic logic.
There is also a very good, but introductory, book by Goldblatt on Topoi that covers this logical aspect
So in terms of logics the category of Sets is the exception.
By internal logic I mean that for every topos one builds up a theory using it's objects and function between them. An equivalence theorem (see J. Bell) states that a given topos is essentially equal to the category generated by this internal theory.
This program began with Lawvere who noticed that conjunction and implication were really adjoints, the same one as between the product and hom functors in a cartesian closed category.
I'm still learning Cat theory myself (through the Category for Programmers book). I have a couple of questions/observations (that I would love your opinion on).
First observation: True and False are the limit and co-limit of the Bool Category.
Question 1: About ordering, would you say that ordering is a requirement (as in a necessary property) of Cat Theory? It was mentioned in Bartosz Milewski's book but it wasn't as strongly emphasized as in your article.
Question 2: You mention how you can't express "A or not A" using intuistic logic. Since it is expressible in Set Theory, could we not use an Adjoint between the Bool and Set Categories respectively? Specifically Kan extensions?
The bool category is not involved in this. True and False are the initial and terminal object (related to limit and colimit, but not the same thing) of all logical categories as I call them, of which there are many (one for each set of axioms that you might construct).
1. Ordering is not required for a category to be a category, the necessary requirements are just the ones listed in the beginning of the book. It is just that orders can be seen as categories.
2. You can express "A or not A" in intuitionistic logic it is just that it is not necessarily true. Also, not sure how would you express that or any logical relation in set theory.
I've picked up those things peace by peace form Wikipedia to be able to understand the slang in Haskell land. But it was a long and puzzling process. This great summary offered here will hopefully help other people in the future get a coherent picture more quickly. (I hope the SEO is good so people will find it. I'm at least going to recommend it form now on whenever someone asks related questions).
It could be extended with type-theory I guess.
Also I would be interested to know more about the relation of those things described with abstract geometry and/or topology.
I've really been wanting a friendly-like-what-you've-made category theory treatment of those two papers is what I'm trying to say.
I've been assuming most/all things in life are uncertain. It's had a profoundly helpful impact on my life, so I'm trying to come to a deeper understanding of how to reason about an uncertain universe, which I think may be something a lot of people are needing these days. The first paper helped a bit, but I haven't really dug into and understood most of it.
Hi, would it using different shapes altogether, instead same circle with different colors, be a better choice? Colors are soft too which makes the contrast between any two circles very low and hard to differentiate.
I'm curious in intuitionist logic what is the concept for "not proven", like neither true nor false? It's not "not-true" because going by your article it seems that would be "disproven" or bottom. Is it literally "not-false" ?
For a side project of mine, I've started to use "True" to mean proven and "False" to mean not-proven, under the argument that if it were disproven, that's the same as a true proof for a counterargument.
There is no equivalent to "neither true nor false" in classical logic, because in classical logic there are no propositions that are neither true or false.
Actually, there is no "neither true nor false" in intuitionistic logic as well, because there is no True and False in a first place. There is only Proven and not Proven.
Don't think in terms of true and false, think in terms of proofs
But in intuitionist logic, "not proven" is bottom or disproven. So what is the concept for "neither proven nor disproven"? Is that literally "not disproven"?
“Not proven” isn’t the same as implies bottom/Falsum?
As I understand it, one can have a proof, or one can have a disproof (I.e. a machine that takes as input a proof of the statement and produces a proof of Falsum), or one can just, not have either of those things.
You never have a “I don’t have a proof” with which to do things with, even if you don’t have a proof.
Regarding the truth of a given statement, you can either say that there is a proof of it, or you can stay silent about it (while possibly saying something about another statement, e.g. saying that there is a proof of the negation of the original statement).
> "Not proven" isn't the same as implies bottom/Falsum?
The article states: ¬A is A → ⊥.
But how would you logically express "A is neither proven nor disproven"?
It seems to me that if "A" is proven, and "~A" is disproven, then maybe "~~A" is neither proven nor disproven. Is that right? Since intuitionist logic doesn't have the double negation elimination axiom?
Indeed, that is what ¬A means.
"¬A" does not mean "not proven". It means that A implies a contradiction. I.e. It means not A. To have a proof of ¬A is to have a disproof of A.
"A is not proven" is not a statement in the language. You can't express it in the language. (if you want to add on some provability logic on top of intuitionistic logic, you can do that, but the basic language of intuitionistic logic does not have any way of expressing "it hasn't been proven that A".)
The "either it has been proven, or it hasn't been proven" isn't a statement made in the language, but a statement about, how to reason using the language.
"~~A" does not mean "neither proven nor disproven", it means --
-- well, it means what it says. It means not(not(A)) .
If you have a proof of A, you can use that to produce a proof of ~~A , but not the other way around.
A proof of ~~A is, a disproof of ~A, essentially saying "if it could be shown that A implied a contradiction, that implication itself would imply a contradiction".
> Logic is the science of the possible. As such, it is at the root of all other sciences, all of which are sciences of the actual, i.e. that which really exists.
Not the focus of the article, but a good view of logic is that it is the art of consistency. A powerful, single-purpose tool to tell its user if they hold mutually inconsistent beliefs. I can believe that the sky is blue, the sky is green and that the sky is exactly one colour - but if I use logic I can detect that something in my worldview is woefully broken.
I'm not really sure what you mean by higher level, but I'm still not sure that's the case if one is assuming you mean works that are more introductory in nature.
For example, from the introduction in the first chapter, titled "Consistency", in Wilfred Hodge's Logic:
> Logic can be defined as the study of consistent sets of beliefs; this will be our starting-point. Some people prefer to define logic as the study of valid arguments. Between them and us there is no real disagreement, as section 11 will show. But consistency makes an easier beginning.
> Logic is about consistency – but not about all types of consistency. ...
> The type of consistency which concerns logicians is not loyalty or justice or sincerity; it is compatibility of beliefs. A set of beliefs is consistent if the beliefs are compatible with each other. To give a slightly more precise definition, which will guide us through the rest of this book: a set of beliefs is called consistent if these beliefs could all be true together in some possible situation. The set of beliefs is called inconsistent if there is no possible situation in which all the beliefs are true.
Note that this is identical to the parent comment, and this book is originally from the 1970s.
By higher level, I mean big picture. Not necessarily introductory, but ones that don't employ logical concepts such as "proof", "formal" etc, because you cannot define a field of study using concepts from that field. This example is good, but not sure what are you trying to say by posting it (I didn't say that noone ever wrote anything on what logic is.)
Does a field of study have to be defined in order for the concepts in it to be defined? I don’t see why.
Can’t you define concepts x,y,z, and then define the field of study as being “about x,y,z and the things related to them”?
I’m guessing you mean something different by “defining a field of study” (different from my default/initial interpretation of the phrase I mean), but I’m not sure what else you mean by it.
You can do that, but that's surely not the best way to do it, unless the concepts don't have anything to do with each other aside from being part of that field of study.
If it lets me avoid a circular definition, I’m willing to tolerate a bit of ugliness in my definitions (at least, provided that it can be shown to be equivalent (under assumptions of some results that can also be proven under the awkward form) to a nicer framing).
> 6.2.1 Let us start with a look at the original rationale for intuitionism. Consider the sentence ‘Granny had led a sedate life until she decided to start pushing crack on a small tropical island just south of the Equator.’ You can understand this, and indefinitely many other sentences that you have never (I presume) heard before. How is this possible?
I don't think that logic is at the root of most sciences, if at all. The statistical tools which science employs may have logical underpinnings, but the theories formed by science aren't axiomatic. For example, there is hardly any enthusiasm for axiomatizing physics by either mathematicians or physicists.
Science is also very comfortable with holding competing theories, whereas in conventional logic that would be "explosively" destructive.
Not necessarily, you have competing theories based on the same kind of logic - it is just that every theory would be based on different set of postulates and will reside in a different logical space (i.e. a different category). But the fact that that two theories don't agree with each other does not mean that one of them is illogical. Actually, I'd argue that they have to both be logical in order not to agree with each other at all i.e. logic is the common ground which enables you to express the differences.
Also, I do think that physics is pretty axiomatic right from it's roots. Newton's main inspiration for forming his theory is the work of Euclid, i.e. first formal system that was ever created.
For science, competing theories exist within the same universe of discourse. That's because scientific theories find their basis in their ability to generate new or better predictions from existing empiricism.
And in terms of modern physics, I reiterate that there is no enthusiasm from either mathematicians or physicists to axiomatize physics.
Science can't be logically inconsistent. If a system is logically inconsistent then you can prove anything from it, so there would be nothing for science to explain - everything would follow from the inconsistency.
Science is comfortable with uncertainty, but not with things that are illogical. It has to be logical, and the scientists have to accept some sort of logic to give theories a better-worse hierarchy.
Except most of sciences take experiment as the criterion of truth rather than logical inference. Real-world science can almost always be expected to be inconsistent - simply because each scientific theory is a "local" approximation, and theories developed for different areas of the same science are not required to be mutually consistent in order to be useful. One (in)famous example is quantum mechanics and general relativity.
There seems to be a markdown error (and a grammatical error) in:
> This is it about axiom schemas and *rules of inference are almost the same thing except they allow us to actually distill the conclusion from the premises.
Question about a claim in the article:
"If classical logic is based on set theory, intuitionistic logic is based on category theory and its related theories."
I'm no expert, but as I understood it, category theory can be used to construct a semantics for certain constructive logics, such as type theory, but to claim that intuitionistic logic is "based on" category theory seems off. A logic can exist all by itself, regardless of which theory is used to give semantics for it. Right?
Well, “classical logic is based on set theory” seems off too…
In fact, a theory can always be formulated using a minimum number of axioms, so none requires some other theory as its base. For example, the system of real numbers can be built starting from the natural numbers, but what this achieves is, this just gives a “proof of existence” of the reals, i.e of consistency of the definition of the reals as a complete ordered field.
Yes, this seems completely off to me too. It’s perhaps worth noting in this connection that intuitionistic logics predate anything that could reasonably be called “category theory”.
Author here. Yes. Texts says "If classical logic is based on set theory", that is I am comparing the relations to set theory and category theory to that between classical logic and intuitionistic logic. I will try to fix the ambiguity.
"If we view classical logic as based on set theory, then intuitionistic logic would be based on category theory and its related theories."
And regarding the fact that CT is not "needed" for algebraic logic what I say is "category theory and its related theories." where I consider orders related to categories.
Orthodox Set theory is a "standard library" for classical logic.
It just so happens that the people who like one tend to like the other, and so explorations of the semantics of classical logic (mostly, model theory) use Set theory.
The Tarski-style algebrization of logic which is "logics as orders" stuff from this article is an alternative to that one can do without category theory. Heyting algebra vs boolean algebra is sufficient to distinguish classical and intuitionistic logic.
Category theory is this not necessary for intuitionism, but is nicer, because the point is to compute things. Otherwise we just speak abstractly of what can be computed, which is like a pessimist compromise between realism and idealism.
So
Logic : order :: type theory : category
Roughly, and the right side makes intuitionism a lot more exciting and applicable.
Curiously, Category Theory appears to be the most “illustrated” mathematical theory by its very nature (because of the wide use of diagrams in any text on it) and, at the same time, the most misunderstood one (or, often, not understood at all) - because its conceptual content and the patterns it considers are so deep that no amount of illustration could convey that depth, to the point that even drawing diagrams, however complex, is a skill comparable to being able to write numbers on a piece of paper - which is just as far from what we think what the understanding of the subject is.
I guess you are talking about limits like in analysis, but even then I don't think this is really true. I'm not really sure if you intend to distinguish between limits and colimits; I'll assume not, but I would certainly agree that infinite (inverse) limits are harder to understand with diagrams.
On a simple level, limits in category theory don't need to involve infinite things at all, you can take the limit over a small category like (x --> x <-- x).
With regard to derived concepts, one can draw pictures of "projective resolutions" or draw level by level pictures of the nerve of the category you want to work with. So I think you can in fact make useful pictures even in this case.
> conceptual content and the patterns it considers are so deep that no amount of illustration could convey that depth
My experience is different - for me category theory abstracts / generalizes / encodes ideas that we understand well in a particular setting. Intuition comes from particular examples and is then "categorified" to make more general arguments. Of course some times it goes the other way ("hey you really need to do X because the category you are working with doesn't have property Y").
And sometimes the imported understanding is not quite accurate, like always there are issues. I don't think that impacts the main line of this argument, though it may be an example of the depth that category theory brings.
> As such, it is at the root of all other sciences, all of which are sciences of the actual, i.e. that which really exists.
I once read somewhere that theology is the science of the unknowable. So for how much theology is a science then is discussed within theology itself ie one cannot know how much of the unknowable is existing in an ontological sense.
I once read that philosophy and theology are similar to each other, in that “philosophy is seeking answers to questions that do not have answers; theology studies that which does not exist.”
(Incidentally, “science of the unknowable” seems like an oxymoron, because “science” means “knowledge”…)
In my personal opinion, philosophy is about understanding what the questions even are and creating the tools needed for this understanding. So philosophy seeks to codify the questions and provide frameworks for answers to be framed in.
All the sciences have questions that do not have answers.
A good book that actually knows what it is talking about is The Logic of Information.
> Logic is the science of the possible. As such, it is at the root of all other sciences, all of which are sciences of the actual, i.e. that which really exists.
Logic describes our universe well but even here it starts falling apart pretty fast. The idea of axiomatic systems as we know them is flawed. Mainly due to Godel's and partly from what we know of the psyche.
Gödel's incompleteness theorem is very similar to the halting problem, in that they are both really profound results, but they have almost no bearing on the practical benefits of logic / computation.
In the halting problem example, how often do you create infinite loops as a programmer? Even in the software verification space, proving termination of an algorithm is fairly simple. So, the halting problem doesn't really affect us in daily programming.
The same with the incompleteness theorem. The fact that we can't say _every single thing possible_ (completeness) has no bearing on our ability to say _an innumerable amount of very practical things._
The amount of things that formal logic can express is so vast and useful that, to call it "flawed" is a pretty big misunderstanding on the incompleteness theorem.
Extrapolating logic to other realities requires no gaps in the system though. If it doesn't encompass our own reality how can we assume that it would even exist in others?
Yeah, I don't see how the colours are helping. It does make it look friendly and inviting, but switching back and forth between coloured balls and the usual symbolic mathematics (algebra notation) is jarring and confusing to me. Using colours instead of letters is a superficial trick and basically a step backwards. It's like those algebra memes where they use fruit instead of x, y, z etc. What's the point? But anyway I don't want to sound too dismissive because the mathematics literature definitely needs more of this kind of creativity!
Honestly, the entire part of the post before getting to categories is what I think is most useful for the broadest range of people. We often look at "logic" and "mathematics" as one thing, and even more than that, we look at them as absolutes - if something is mathematically proven, it is certainly a fact, no?
Well, it turns out even here there are still shades of gray, and the difference between classical and intuitionistic logic is the perfect example. Not everyone even agrees on the foundation of what logic and math are. And, each has their own benefits and strengths (intuitionistic logic tends to go hand in hand with type theory as the basis for theorem provers / proof assistants for mostly practical reasons).
Classical logic has always made the most sense to me, because accepting the law of excluded middle leads to a neater world of arguments to me. If you can't prove something, it is treated as false. Might be harsh, but that harshness yields simplicity.
I have come to at least understand the intuitionistic perspective though, because of how prevalent it is in the verification space - I have been super into F* recently (https://www.fstar-lang.org/), and this comes from a line of logics and programming languages that owe their life to intuitionistic logic. I like Andrej Bauer's position of being a "mathematical relavist," where he says "we cannot ignore the many worlds of mathematics" (http://math.andrej.com/2012/10/03/am-i-a-constructive-mathem...).
Math and logic are vast spaces with different systems and perspectives, and rather than treat one as universal truth, it is practically more beneficial to be familiar with the different systems and their perspectives, and to be able to switch between them when practical.
> If you can't prove something, it is treated as false.
Maybe I'm being excessively literal in interpreting the phrasing here, but that doesn't seem like how I would describe classical logic? There are plenty of cases where neither a thing nor its negation can be shown in some system which is based on classical logic (I'm sure you are well aware of this fact, so I'm probably just misinterpreting you by being too literal).
But yes, I also share the sentiment that, for ordinary/day-to-day reasoning, classical logic is substantially more intuitive than intuitionistic logic, and makes things simpler.
Can I think of a precise way to say the thing that I imagine you must have meant by that?
In both classical logic and intuitionistic logic, if by assuming something you can derive False, then you can prove the negation of the thing. So that's not the distinguishing thing. If by assuming the falsehood of something you can derive False, in classical logic you can use that as a proof of the thing. (double negation elimination), while in intuitionistic logic you only get the double negation.
This, I think at least mostly describes the difference (though not all the like, implications of it), but I'm not sure how to make it into something similar to "If you can't prove something, it is treated as false."
Though, that statement does remind me of the idea of a/the Constructible universe in set theory..
Yes, knowing that ~P and not knowing P are clearly and fundamentally different things in the world of classical logic. (I'm not an expert, but I know this much.) It's hacky computing systems like Prolog and the relational model and/or SQL which erase the distinction. https://en.wikipedia.org/wiki/Closed-world_assumption The law of the excluded middle only says that at least one of P and ~P must be true, and doesn't at all promise that we will necessarily be able to conclude which one.
> Classical logic has always made the most sense to me, because accepting the law of excluded middle leads to a neater world of arguments to me.
What use is a neat argument if it is divorced from reality?
LoEM tricks us into viewing ontologies as static, leading to subtle but drastic errors in thinking.
LoEM/classical logic should always be employed as a conscious assumption, with awareness of its drawbacks. Leave room for dialectical reasoning as well.
By far the bulk of pure mathematical results in the last 100 years have no demonstrated real world function. They are logically sound--and utterly divorced from reality.
And any useful innovations tend to come about from dialectical reasoning, because classical logic is incapable of genuine ontological generation.
For example, infinitesimal calculus was not "embedded" somewhere in the existing mathematics, waiting to be logically deduced by an expert logician; Newton developed new methods of mathematically modeling unquantifiable quantities to address contradictions in the application of traditional mathematics to physical systems.
> By far the bulk of mathematical results in the last 100 years have demonstrated no real world function
How absolutely arrogant and unappreciative of you to say. You are very divorced from reality. That much is clear - normally I don’t like to get personal, but really. I’m sure you hear this all the time which ferments your bitterness.
I have a degree in mathematics and (PhD) friends who are still in the system. Event they admit that pure mathematics is largely an illusion. But it pays, if you're lucky.
Mathematics in general, and logic, are extraordinarily useful tools when applied to real systems. I use them frequently when developing software. However, like every tool, they can metastatize beyond their use. In my opinion that describes the vast bulk of "pure" mathematics and "pure" logic (and "pure" programming).
Everything I’ve seen be called “dialectics” has either seemed like something trivial wrapped up in fancy language, or seemed to be something bunk [edit: or I didn’t understand it enough to have an evaluation of it]. I’m skeptical that it is a useful concept.
I mean, there’s game semantics, but as far as I’ve seen, there’s no reason to confuse the idea of that by appealing to “dialectics”.
Are there any examples of an insight which is uncontroversially regarded as an insight, and uncontroversially regarded as being first made as a result of viewing things in terms of “dialectics”?
Also, do you mean the law of the excluded middle, or the law of non-contradiction? [edit: I removed a part I said here because I realized I didn’t know what I was talking about with it. It referenced Kierkegaard, but, again, I didn’t know what I was saying.]
Refraining from affirming the law of the excluded middle can be useful at times, but I cannot accept the rejection (or non-affirmation) of the law of non-contradiction. If one thinks that one has a true contradiction, one is simply confused.
> If one thinks that one has a true contradiction, one is simply confused.
Consider the time just before the concept of negative numbers was widely accepted. There were natural numbers, perhaps zero as well, and the operation of addition. Someone realized you could reverse the process of addition and yield subtraction. Yet when you applied this "reverse addition" to some pairs of numbers, you would get an impossible result, a number that doesn't exist. This represents a contradiction in the method. A reasonable logician would stop there and simply avoid doing contradictory subtraction. A super smart and reasonable logician would go further and reject subtraction altogether since its "not closed over N", "not safe", etc.
However, a "confused" accountant might see the value in such impossible values. He's balancing payments for dozens of deals and finds that his calculations are much smoother if he temporarily allows the impossible values, as long as he marks them as "impossible", like with a small horizontal line.
His increased efficiency brings him success and his methods spread, and eventually "negative numbers" becomes a core component of virtually all mathematics. All because this one accountant was "confused" and pragmatic.
Calling this a "contradiction" is, imo, unjustified.
It is not the accountant who I am calling (more) confused (though if the accountant believes what they are doing to be "impossible" they are still somewhat confused).
It is the person who asserts that the thing which they know exists, "is impossible", who is confused.
If you have a partial function that takes in a beardnacle and a frompulin and (where it is defined) returns a brimwich, and one has some conditions under which, in the cases it is defined, certain patterns have been shown to hold, and one has found a number of cases in which, it seems as if, if the function were defined at some of the places it isn't, and the value fit with the patterns which have been shown to hold where it is defined, in that other things are consistent with that, even though no brimwich can actually satisfy those properties, a logician can perfectly well ask "well, can we define a concept which generalizes the concept of a brimwich, in order to make this partial function into a total function, in a way that keeps the patterns we have shown?" .
This does not involve any contradictions. There are no contradictions here. And yet, this is doing the same kind of thing as generalizing the natural numbers to the integers.
You are either using a notion of "contradiction" which is quite unlike the everyday concept (and, if it differs, also quite far from the mathematical concept), or you are mistaken in thinking that there is a contradiction.
Like, what is the alleged contradiction? "I want to subtract these two numbers, but there is no natural number which is the result of that subtraction"? That's not a contradiction in the sense that [people who aren't talking about dialectics] use.
If you couldn't use it in order to derive False in a theorem prover that allows the relevant statements to be expressed, then it isn't a contradiction-in-the-ordinary-sense.
"John is married" and "John is not married" are contradictory. "Smith has 6 apples" and "Smith has fewer than 3 apples" are contradictory.
"John has a banana, but Gerald really hates bananas and wishes John didn't have a banana." is not a contradiction-in-the-ordinary-sense. It may be a conflict, which may be resolved in some way, but it isn't a contradiction.
The pre-negative-number contradiction here is between the "obviously" true propositions:
A) "undefined quantities can't/don't exist" (practically a tautology)
B) "subtraction sometimes produces an undefined quantity"
The first proposition contradicts any proposition that asserts the existence of undefined/impossible/negative numbers. Keep in mind that, before negative numbers, there would be no valid symbols to express the result of the expression "one minus two"; it is irreducible.
The root of the problem is that classical logic doesn't have good tools for dealing with ontological deficits, in this case, the lack of a well-defined concept of "negative numbers".
All of Plato's classic "laws of thought" presume the existence of a well-defined universal set of things/ideas/truths that they can operate over, which is usually a flawed assumption. In reality, the baseline is that everyone has their own ontology derived from personal experience and it takes serious collective effort (or pedagogical browbeating) to establish shared ontologies on which systems of communication can be built.
On the other hand, a core feature of dialectical reasoning is resolving contradictions through the synthesis of novel orthogonal concepts.
“Subtraction [as defined from pairs of natural numbers to natural numbers] sometimes produces undefined quantities” is wrong (or at best phrased in a sloppy way which you in one part interpret in a way that would make it wrong).
Instead, for some inputs, there is no defined/prescribed output.
Anyone who thinks that example you gave is a true contradiction in-the-ordinary-sense is simply confused about about what they mean by those things. (Though, again, if you are using “contradiction” in a way other than the ordinary sense, this doesn’t apply to you)
This confusion would presumably come from being sloppy with phrasing.
If one simultaneously interprets “produces an undefined quantity” as synonymous with “does not have a defined quantity” (which is a true statement) and synonymous with “produces a quantity which doesn’t exist, [thereby establishing the existence of something that doesn’t exist]” (which is a false statement), it is no wonder that one would conclude that there is a true contradiction, and this is precisely the mistake.
Thinking that two things (e.g. statements) are the same thing (and therefore have the same properties, e.g. if they are statements, that if one is true so is the other) simply because one has given them the same name, is a mistake, and is a somewhat common source of confusion.
Now, in some cases, everyone (or nearly everyone) considering a topic will be confused about something in a way that is at least kind of similar to that way of being confused (or perhaps even in exactly that way). In these cases, one way that people might come to realize that they are confused is if they seem to be able to derive a contradiction. But this doesn’t mean that the contradiction is true, but that either (perhaps due to being confused) at least one of their assumptions was wrong, or one or more of their inference steps were invalid (e.g. unintentionally conflating two distinct things/ideas).
There are no true contradictions, only people (such as myself) who are wrong ( and/or confused ) about something.
> Logic is the science of the possible. As such, it is at the root of all other sciences, all of which are sciences of the actual, i.e. that which really exists
I really don’t think this is right. Logic is a discipline of consistent re-writing of expressions. However, there may be (even existing) things that are not expressable to sufficient degree in finite (or even countably infinite) strings, which I think means they are outside of logic. There are definitely “possible” things that are outside of logic (because not expressable).
Logic is one of the fundamental axioms of the universe. We assume logic is true and consistently applies everywhere throughout the universe. This "rewriting expressions" thing is just a symbolic representation of logic. It is not logic in itself. The symbolic representation of logic by writing down "expressions" works because logic is a inherit property of our universe and since you are writing those "expressions" in that same universe, it works.
There is no way to prove or verify logic is consistently real. We just recursively assume logic is real. We observe it to be real and assume that the observation will consistently apply across all time and space.
Another thing that I should mention that is an axiom of our universe is probability. WE have no way of knowing why rolling dice or random variables follow the rules of probability. These are just arbitrary rules and we assume that they're consistently true about our universe. Logic along with probability are two things that we have zero methodology of verifying the veracity of but we just assume these two things are fundamental properties of the universe.
A more elegant way to look at it is to just assume probability is the foundational axiom of our universe. Logic is just a special case of probability where all causal connections are 100%. Of course given inherit unreliability and limited knowledge of all things we never actually see or can verify 100% causality on anything. This effectively limits logic to mathematical and axiomatic games while science is the only available tool for the real world.
Science is a whole different beast. Given the assumptions that probability is real and that logic is real, science is an attempted methodology to verify theorems or statements about the universe using the axioms of probability and logic.
For example Newton guesses that a ball should travel a certain distance according to his made up laws of motion. Using science we perform several experimental observations of moving a ball and statistically correlate to a certain degree that yes the ball does indeed move according to newtons laws of motion. This is what science is. It is making a hypothesis and using statistics to sort of verify it. The term "sort of" is key here because science is limited to the fact that it can never prove anything.
One thing to note here is that EVEN when we assume logic and probability is true, science is unable to prove anything is true. You can make 10,000 observations about newtons law, it proves nothing because at any point in time a new observation can render the entire hypothesis as false. Thus falsification is possible with science [1] but proof is impossible. Proof is the domain of mathematics and logic games and cannot exist in the real world due to limited knowledge.
This is not some pedantic philosophy I'm making up. This is foundational to a true understanding of what science and logic is. To quote Einstein:
"No amount of experimentation can ever prove me right; a single experiment can prove me wrong."
There are a lot of intelligent people who don't understand the true depth of the above quote. But if you get it, then you truly understand what science is, and the differences between science and logic. Obviously both the OP and the parent poster don't fully get it... by combining logic and science into one thing and calling it the "science of the possible" it shows that they don't have a clear delineation of the two terms. Most people think of science as some kind of fuzzy "technical study" of a topic. No. This is wrong. There is a clearer definition of science that separates it from logic and mathematics.
[1] Note that technically total falsification is also impossible. Inherit unreliability of observation tools and limited knowledge makes it so that no observation can be 100% reliable. Thus even falsification is technically limited to the domain of logic and mathematical games.
Hard to say which came first. Probability can be defined in terms of axioms and classical logic just read any probability book. Classical logic can also be defined in terms of probability by setting all causal relations to be 100% (See bayes theorem).
I'm making a vague sweeping statement here that eventually gets a bit more cleared up later in my writeup.
> Logic is one of the fundamental axioms of the universe
This definitely a very opinionated view on what is logic. There is no reason to embed anything in universe/anything else. If you remove expressability, I’m not sure what properties remain. At the same time you can have rules outside of anything physical (in a sense of denoting something that subsists). Consistency of re-writing rules (in a sense that one can always write a propositional string that cannot be reduced to true or false), is an entirely separate beast.
> Logic is just a special case of probability
This is factually false. Probability is embeddable into predicate logic, but predicate logic is not embeddable into (Bayesian) probability. This is actually an open problem (how to define a probability theory that is equivalent to predicate logic).
> Science is a whole different beast
On this I do agree, I wouldn’t call logic a “science.”
It is neither the accepted modern nor historic definition of the word “logic.” Your definition is also unrelated to the word’s etymology. In a sense that people can have personal definitions of words I generously called it “opinionated.” I guess I should have said “wrong.”
>This is factually false. Probability is embeddable into predicate logic, but predicate logic is not embeddable into (Bayesian) probability. This is actually an open problem (how to define a probability theory that is equivalent to predicate logic).
Completely wrong. How can someone say I'm factually false when your own statement is completely and utterly incorrect. I don't think you're really aware of what's going on here:
It's not an "open problem" or aka "unsolved problem" in the sense you implied. Far from it. Rather it is just not a popular field of study.
>This definitely a very opinionated view on what is logic. There is no reason to embed anything in universe/anything else. If you remove expressability, I’m not sure what properties remain.
I don't think you get it. The nature of an axiom IS by definition an opinion. An axiom is an Assumption. Assumptions are just things that are assumed to be true, they are NOT things that are proven to be true. Thus the only thing an assumption can be if it is an unproven statement is that it is an OPINION. When anyone makes a statement about logic such as my statement "Logic is a fundamental axiom of the universe." Or "Logic is NOT a fundamental axiom of the universe" is by DEFINITION a statement of OPINION.
When I say it's a fundamental axiom it means that it is something all of humanity consistently assumes to be true. Our culture, our science, our mathematics, our interpretation of reality and the universe as we know it is founded on the assumption that logic is true. It is a general statement about the broad "opinion" of all of humanity.
Look at your own statement. You said and I quote: "There is no reason to embed anything in universe/anything else." What does this statement even mean? What is reason? Reason IS Logic. Look it up. You are literally saying there is no logical reason to embed logic into our universe. Yet here you are making a logical statement in a universe where you say we cannot assume logic is true.
We assume logic is true. You can't make a single argument otherwise. Literally all arguments you've ever made in your life hinge on logic being true. That is unless all your arguments are illogical. Are you saying your own argument is illogical?
There's really nothing left to argue about here. Our entire technical framework of reality is founded on the "Opinion" that logic real.
>On this I do agree, I wouldn’t call logic a “science.”
If you truly understood what's going on, you would agree with me on all points. Not just this.
Not going to address your weird attempt at wordplay but again just so you know, subjective logic, which you linked, is equivalent to propositional (in a sense you can express any statement in subjective logic via a set of propositions, eg Ap distributions make this connection), and does not embed predicate logic (eg any statement of this system can be expressed in predicate logic, but not vice versa)…
Also, axiom can be just a rule in a rewriting system. Doesn’t have to be an assumption/opinion as it is just a proposition (or higher order predicate) that is true within some system (not necessarily related to anything that exists).
Things can be outside universe yet discussed by people in-universe (subsisting vs existing without multiple worlds).
What I am saying isn’t controversial in analytic philosophy/ epistemology…
Edit: on a more reconciliatory note, a classic essay that I enjoy that touches upon the idea of denotion/subsistence/existence/difficulty with predicates is Bernard Russel’s “On Denoting”. I suspect you may enjoy it: https://www.uvm.edu/~lderosse/courses/lang/Russell(1905).pdf
>Not going to address your weird attempt at wordplay
There's absolutely zero need to be rude. There is no "wierd" word play here just misunderstandings and impoliteness by you.
>Also, axiom can be just a rule in a rewriting system. Doesn’t have to be an assumption/opinion as it is just a proposition (or higher order predicate) that is true within some system (not necessarily related to anything that exists).
This is obvious. You're just restating what I said and coming up with a circular argument. A rewriting system is a universe you create, not dissimilar to our own universe that we live in. An axiom in this system is an statement made by you stating something is true without any proof. A statement made without proof IS an assumption. Now this explanation may seem like word play to you but it is not. The exact identical words "assumption" and "axiom" serve to confuse your reasoning and blind you from the actuality of the concept at hand. What I am doing is unraveling your misunderstanding in attempt to explain your confusion but you are unable to see it and you view the argument as "word play." Axiom and Assumptions are two words that mean the exact same thing. There is one concept at hand and two words for that same concept that are confusing you.
>subjective logic, which you linked, is equivalent to propositional (in a sense you can express any statement in subjective logic via a set of propositions, eg Ap distributions make this connection), and does not embed predicate logic (eg any statement of this system can be expressed in predicate logic, but not vice versa)…
This system has axioms and theorems that use predicate logic. It is built off of predicate logic.
When you set the subjectivity to 100% the result is something equivalent to predicate logic. So logic is both a special case of probability as well as foundational building blocks for subjective probability itself. You can arbitrarily pick and choose which came first and which one is axiomatic. Probability or Logic.
You are indeed wrong here. Please politely admit it without calling my statements weird. If you disagree there is no need to resort to derogatory statements.
I read it. Science is not logic in the same way that a brick building is not a brick. Your description is inconsistent. You differentiate logical theories and scientific theories as different yet you call logic the science of what's possible.
Maybe you were trying to promote understanding over correctness but I would recommend that you do both as such fuzzy wording leads to people inconsistently and incorrectly using the word "science."
The definition of logic I retain comes from "Logique mathématique" from René Cori and Daniel Lascar, a pretty good book I have yet to finish ;)
Logic models the mathematical reasoning itself, like vector spaces model our world and differential equations model most of physics. At no point however we should confuse the model with the studied object. As such logic is a field of mathematics and not mathematics itself. The best illustration of this is that theorems in the field of logic use results from e.g. set theory.
Of course using a formal logic to prove theorems is valid. Still the formal logic has to be studied with "intuitive" mathematics. The book uses the metaphor of spiraling downstairs to illustrate this.
propositions, firstly, tell me what a proposition is for the purpose of this document - this is a common problem with maths documents/books, I come across something which I don't understand.
It says primary propositions, what is a primary proposition? Can you give an example?
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[ 2.9 ms ] story [ 178 ms ] threadFor logic, it really depends of what you are searching for, for classical logic you can read the the classics, for example Russell and Tarski.
For constructive logic I cannot think of a good introduction (besides mine ;) ), I personally picked it up from books about category theory and computer science.
A topos is essentially a cartesian closed category with a subject classifier. In Set this is the two element set of 1/0 which is a Boolean algebra and thus the internal logic of the category Set is classical.
In general though the subobject classifier is a heyting algebra which expresses the semantics of intuitionistic logic.
There is also a very good, but introductory, book by Goldblatt on Topoi that covers this logical aspect
So in terms of logics the category of Sets is the exception.
By internal logic I mean that for every topos one builds up a theory using it's objects and function between them. An equivalence theorem (see J. Bell) states that a given topos is essentially equal to the category generated by this internal theory.
This program began with Lawvere who noticed that conjunction and implication were really adjoints, the same one as between the product and hom functors in a cartesian closed category.
> Axiom schemas/Rules of inferene
First observation: True and False are the limit and co-limit of the Bool Category.
Question 1: About ordering, would you say that ordering is a requirement (as in a necessary property) of Cat Theory? It was mentioned in Bartosz Milewski's book but it wasn't as strongly emphasized as in your article.
Question 2: You mention how you can't express "A or not A" using intuistic logic. Since it is expressible in Set Theory, could we not use an Adjoint between the Bool and Set Categories respectively? Specifically Kan extensions?
1. Ordering is not required for a category to be a category, the necessary requirements are just the ones listed in the beginning of the book. It is just that orders can be seen as categories.
2. You can express "A or not A" in intuitionistic logic it is just that it is not necessarily true. Also, not sure how would you express that or any logical relation in set theory.
I've picked up those things peace by peace form Wikipedia to be able to understand the slang in Haskell land. But it was a long and puzzling process. This great summary offered here will hopefully help other people in the future get a coherent picture more quickly. (I hope the SEO is good so people will find it. I'm at least going to recommend it form now on whenever someone asks related questions).
It could be extended with type-theory I guess.
Also I would be interested to know more about the relation of those things described with abstract geometry and/or topology.
But it's fantastic already as it stands!
Would you do uncertainty logic next?
https://arxiv.org/abs/1506.03123 https://arxiv.org/abs/1810.01310
I've been assuming most/all things in life are uncertain. It's had a profoundly helpful impact on my life, so I'm trying to come to a deeper understanding of how to reason about an uncertain universe, which I think may be something a lot of people are needing these days. The first paper helped a bit, but I haven't really dug into and understood most of it.
For a side project of mine, I've started to use "True" to mean proven and "False" to mean not-proven, under the argument that if it were disproven, that's the same as a true proof for a counterargument.
Actually, there is no "neither true nor false" in intuitionistic logic as well, because there is no True and False in a first place. There is only Proven and not Proven.
Don't think in terms of true and false, think in terms of proofs
As I understand it, one can have a proof, or one can have a disproof (I.e. a machine that takes as input a proof of the statement and produces a proof of Falsum), or one can just, not have either of those things.
You never have a “I don’t have a proof” with which to do things with, even if you don’t have a proof.
Regarding the truth of a given statement, you can either say that there is a proof of it, or you can stay silent about it (while possibly saying something about another statement, e.g. saying that there is a proof of the negation of the original statement).
The article states: ¬A is A → ⊥.
But how would you logically express "A is neither proven nor disproven"?
It seems to me that if "A" is proven, and "~A" is disproven, then maybe "~~A" is neither proven nor disproven. Is that right? Since intuitionist logic doesn't have the double negation elimination axiom?
Indeed, that is what ¬A means. "¬A" does not mean "not proven". It means that A implies a contradiction. I.e. It means not A. To have a proof of ¬A is to have a disproof of A.
"A is not proven" is not a statement in the language. You can't express it in the language. (if you want to add on some provability logic on top of intuitionistic logic, you can do that, but the basic language of intuitionistic logic does not have any way of expressing "it hasn't been proven that A".)
The "either it has been proven, or it hasn't been proven" isn't a statement made in the language, but a statement about, how to reason using the language.
"~~A" does not mean "neither proven nor disproven", it means -- -- well, it means what it says. It means not(not(A)) .
If you have a proof of A, you can use that to produce a proof of ~~A , but not the other way around. A proof of ~~A is, a disproof of ~A, essentially saying "if it could be shown that A implied a contradiction, that implication itself would imply a contradiction".
Not the focus of the article, but a good view of logic is that it is the art of consistency. A powerful, single-purpose tool to tell its user if they hold mutually inconsistent beliefs. I can believe that the sky is blue, the sky is green and that the sky is exactly one colour - but if I use logic I can detect that something in my worldview is woefully broken.
I’m not sure that’s the case. Just searching “philosophy of logic” yields oodles of resources. It’s an entire field of study.
For example, from the introduction in the first chapter, titled "Consistency", in Wilfred Hodge's Logic:
> Logic can be defined as the study of consistent sets of beliefs; this will be our starting-point. Some people prefer to define logic as the study of valid arguments. Between them and us there is no real disagreement, as section 11 will show. But consistency makes an easier beginning.
> Logic is about consistency – but not about all types of consistency. ...
> The type of consistency which concerns logicians is not loyalty or justice or sincerity; it is compatibility of beliefs. A set of beliefs is consistent if the beliefs are compatible with each other. To give a slightly more precise definition, which will guide us through the rest of this book: a set of beliefs is called consistent if these beliefs could all be true together in some possible situation. The set of beliefs is called inconsistent if there is no possible situation in which all the beliefs are true.
Note that this is identical to the parent comment, and this book is originally from the 1970s.
Why not?
Can’t you define concepts x,y,z, and then define the field of study as being “about x,y,z and the things related to them”?
I’m guessing you mean something different by “defining a field of study” (different from my default/initial interpretation of the phrase I mean), but I’m not sure what else you mean by it.
> 6.2.1 Let us start with a look at the original rationale for intuitionism. Consider the sentence ‘Granny had led a sedate life until she decided to start pushing crack on a small tropical island just south of the Equator.’ You can understand this, and indefinitely many other sentences that you have never (I presume) heard before. How is this possible?
One could also say that logic is the study of what is not disproven.
Science is also very comfortable with holding competing theories, whereas in conventional logic that would be "explosively" destructive.
Also, I do think that physics is pretty axiomatic right from it's roots. Newton's main inspiration for forming his theory is the work of Euclid, i.e. first formal system that was ever created.
And in terms of modern physics, I reiterate that there is no enthusiasm from either mathematicians or physicists to axiomatize physics.
And according to the first sentence of the summary the current formulation is axiomatic as well...
Science is comfortable with uncertainty, but not with things that are illogical. It has to be logical, and the scientists have to accept some sort of logic to give theories a better-worse hierarchy.
Except most of sciences take experiment as the criterion of truth rather than logical inference. Real-world science can almost always be expected to be inconsistent - simply because each scientific theory is a "local" approximation, and theories developed for different areas of the same science are not required to be mutually consistent in order to be useful. One (in)famous example is quantum mechanics and general relativity.
> This is it about axiom schemas and *rules of inference are almost the same thing except they allow us to actually distill the conclusion from the premises.
I'm no expert, but as I understood it, category theory can be used to construct a semantics for certain constructive logics, such as type theory, but to claim that intuitionistic logic is "based on" category theory seems off. A logic can exist all by itself, regardless of which theory is used to give semantics for it. Right?
In fact, a theory can always be formulated using a minimum number of axioms, so none requires some other theory as its base. For example, the system of real numbers can be built starting from the natural numbers, but what this achieves is, this just gives a “proof of existence” of the reals, i.e of consistency of the definition of the reals as a complete ordered field.
"If we view classical logic as based on set theory, then intuitionistic logic would be based on category theory and its related theories."
And regarding the fact that CT is not "needed" for algebraic logic what I say is "category theory and its related theories." where I consider orders related to categories.
It just so happens that the people who like one tend to like the other, and so explorations of the semantics of classical logic (mostly, model theory) use Set theory.
The Tarski-style algebrization of logic which is "logics as orders" stuff from this article is an alternative to that one can do without category theory. Heyting algebra vs boolean algebra is sufficient to distinguish classical and intuitionistic logic.
Category theory is this not necessary for intuitionism, but is nicer, because the point is to compute things. Otherwise we just speak abstractly of what can be computed, which is like a pessimist compromise between realism and idealism.
So
Logic : order :: type theory : category
Roughly, and the right side makes intuitionism a lot more exciting and applicable.
https://bayes.wustl.edu/etj/prob/book.pdf
https://arxiv.org/abs/1506.03123 https://arxiv.org/abs/1810.01310
https://www.cs.ru.nl/B.Jacobs/PAPERS/ProbabilisticReasoning....
Curiously, Category Theory appears to be the most “illustrated” mathematical theory by its very nature (because of the wide use of diagrams in any text on it) and, at the same time, the most misunderstood one (or, often, not understood at all) - because its conceptual content and the patterns it considers are so deep that no amount of illustration could convey that depth, to the point that even drawing diagrams, however complex, is a skill comparable to being able to write numbers on a piece of paper - which is just as far from what we think what the understanding of the subject is.
Not to speak about “derived” concepts…
On a simple level, limits in category theory don't need to involve infinite things at all, you can take the limit over a small category like (x --> x <-- x).
With regard to derived concepts, one can draw pictures of "projective resolutions" or draw level by level pictures of the nerve of the category you want to work with. So I think you can in fact make useful pictures even in this case.
My experience is different - for me category theory abstracts / generalizes / encodes ideas that we understand well in a particular setting. Intuition comes from particular examples and is then "categorified" to make more general arguments. Of course some times it goes the other way ("hey you really need to do X because the category you are working with doesn't have property Y").
And sometimes the imported understanding is not quite accurate, like always there are issues. I don't think that impacts the main line of this argument, though it may be an example of the depth that category theory brings.
Original: https://news.ycombinator.com/item?id=28660131
I once read somewhere that theology is the science of the unknowable. So for how much theology is a science then is discussed within theology itself ie one cannot know how much of the unknowable is existing in an ontological sense.
(Incidentally, “science of the unknowable” seems like an oxymoron, because “science” means “knowledge”…)
All the sciences have questions that do not have answers.
A good book that actually knows what it is talking about is The Logic of Information.
https://www.amazon.com/Logic-Information-Theory-Philosophy-C...
Logic describes our universe well but even here it starts falling apart pretty fast. The idea of axiomatic systems as we know them is flawed. Mainly due to Godel's and partly from what we know of the psyche.
Regarding the psyche part, I have actually written an article about that which might change your opinion https://boris-marinov.github.io/logic-thought/
In the halting problem example, how often do you create infinite loops as a programmer? Even in the software verification space, proving termination of an algorithm is fairly simple. So, the halting problem doesn't really affect us in daily programming.
The same with the incompleteness theorem. The fact that we can't say _every single thing possible_ (completeness) has no bearing on our ability to say _an innumerable amount of very practical things._
The amount of things that formal logic can express is so vast and useful that, to call it "flawed" is a pretty big misunderstanding on the incompleteness theorem.
Our current computers all have finite memory and so a trivial algorithm can tell if a real program on a real computer will halt (terminate) or not.
Of course the trivial algorithm has huge (but finite) memory requirement :)
0. https://bartoszmilewski.com/2014/10/28/category-theory-for-p...
Well, it turns out even here there are still shades of gray, and the difference between classical and intuitionistic logic is the perfect example. Not everyone even agrees on the foundation of what logic and math are. And, each has their own benefits and strengths (intuitionistic logic tends to go hand in hand with type theory as the basis for theorem provers / proof assistants for mostly practical reasons).
Classical logic has always made the most sense to me, because accepting the law of excluded middle leads to a neater world of arguments to me. If you can't prove something, it is treated as false. Might be harsh, but that harshness yields simplicity.
I have come to at least understand the intuitionistic perspective though, because of how prevalent it is in the verification space - I have been super into F* recently (https://www.fstar-lang.org/), and this comes from a line of logics and programming languages that owe their life to intuitionistic logic. I like Andrej Bauer's position of being a "mathematical relavist," where he says "we cannot ignore the many worlds of mathematics" (http://math.andrej.com/2012/10/03/am-i-a-constructive-mathem...).
Math and logic are vast spaces with different systems and perspectives, and rather than treat one as universal truth, it is practically more beneficial to be familiar with the different systems and their perspectives, and to be able to switch between them when practical.
Maybe I'm being excessively literal in interpreting the phrasing here, but that doesn't seem like how I would describe classical logic? There are plenty of cases where neither a thing nor its negation can be shown in some system which is based on classical logic (I'm sure you are well aware of this fact, so I'm probably just misinterpreting you by being too literal).
But yes, I also share the sentiment that, for ordinary/day-to-day reasoning, classical logic is substantially more intuitive than intuitionistic logic, and makes things simpler.
Can I think of a precise way to say the thing that I imagine you must have meant by that?
In both classical logic and intuitionistic logic, if by assuming something you can derive False, then you can prove the negation of the thing. So that's not the distinguishing thing. If by assuming the falsehood of something you can derive False, in classical logic you can use that as a proof of the thing. (double negation elimination), while in intuitionistic logic you only get the double negation. This, I think at least mostly describes the difference (though not all the like, implications of it), but I'm not sure how to make it into something similar to "If you can't prove something, it is treated as false."
Though, that statement does remind me of the idea of a/the Constructible universe in set theory..
What use is a neat argument if it is divorced from reality?
LoEM tricks us into viewing ontologies as static, leading to subtle but drastic errors in thinking.
LoEM/classical logic should always be employed as a conscious assumption, with awareness of its drawbacks. Leave room for dialectical reasoning as well.
a) far too extreme of a position to be useful to anyone b) not true in any sense of the word
Classical logic has been behind the bulk of mathematical results in the last 100 years or so. It may not be perfect, but no system is.
And any useful innovations tend to come about from dialectical reasoning, because classical logic is incapable of genuine ontological generation.
For example, infinitesimal calculus was not "embedded" somewhere in the existing mathematics, waiting to be logically deduced by an expert logician; Newton developed new methods of mathematically modeling unquantifiable quantities to address contradictions in the application of traditional mathematics to physical systems.
How absolutely arrogant and unappreciative of you to say. You are very divorced from reality. That much is clear - normally I don’t like to get personal, but really. I’m sure you hear this all the time which ferments your bitterness.
“It is not the critic who counts”
Mathematics in general, and logic, are extraordinarily useful tools when applied to real systems. I use them frequently when developing software. However, like every tool, they can metastatize beyond their use. In my opinion that describes the vast bulk of "pure" mathematics and "pure" logic (and "pure" programming).
I mean, there’s game semantics, but as far as I’ve seen, there’s no reason to confuse the idea of that by appealing to “dialectics”.
Are there any examples of an insight which is uncontroversially regarded as an insight, and uncontroversially regarded as being first made as a result of viewing things in terms of “dialectics”?
Also, do you mean the law of the excluded middle, or the law of non-contradiction? [edit: I removed a part I said here because I realized I didn’t know what I was talking about with it. It referenced Kierkegaard, but, again, I didn’t know what I was saying.]
Refraining from affirming the law of the excluded middle can be useful at times, but I cannot accept the rejection (or non-affirmation) of the law of non-contradiction. If one thinks that one has a true contradiction, one is simply confused.
Consider the time just before the concept of negative numbers was widely accepted. There were natural numbers, perhaps zero as well, and the operation of addition. Someone realized you could reverse the process of addition and yield subtraction. Yet when you applied this "reverse addition" to some pairs of numbers, you would get an impossible result, a number that doesn't exist. This represents a contradiction in the method. A reasonable logician would stop there and simply avoid doing contradictory subtraction. A super smart and reasonable logician would go further and reject subtraction altogether since its "not closed over N", "not safe", etc.
However, a "confused" accountant might see the value in such impossible values. He's balancing payments for dozens of deals and finds that his calculations are much smoother if he temporarily allows the impossible values, as long as he marks them as "impossible", like with a small horizontal line.
His increased efficiency brings him success and his methods spread, and eventually "negative numbers" becomes a core component of virtually all mathematics. All because this one accountant was "confused" and pragmatic.
It is the person who asserts that the thing which they know exists, "is impossible", who is confused.
If you have a partial function that takes in a beardnacle and a frompulin and (where it is defined) returns a brimwich, and one has some conditions under which, in the cases it is defined, certain patterns have been shown to hold, and one has found a number of cases in which, it seems as if, if the function were defined at some of the places it isn't, and the value fit with the patterns which have been shown to hold where it is defined, in that other things are consistent with that, even though no brimwich can actually satisfy those properties, a logician can perfectly well ask "well, can we define a concept which generalizes the concept of a brimwich, in order to make this partial function into a total function, in a way that keeps the patterns we have shown?" .
This does not involve any contradictions. There are no contradictions here. And yet, this is doing the same kind of thing as generalizing the natural numbers to the integers.
You are either using a notion of "contradiction" which is quite unlike the everyday concept (and, if it differs, also quite far from the mathematical concept), or you are mistaken in thinking that there is a contradiction.
Like, what is the alleged contradiction? "I want to subtract these two numbers, but there is no natural number which is the result of that subtraction"? That's not a contradiction in the sense that [people who aren't talking about dialectics] use.
If you couldn't use it in order to derive False in a theorem prover that allows the relevant statements to be expressed, then it isn't a contradiction-in-the-ordinary-sense.
"John is married" and "John is not married" are contradictory. "Smith has 6 apples" and "Smith has fewer than 3 apples" are contradictory.
"John has a banana, but Gerald really hates bananas and wishes John didn't have a banana." is not a contradiction-in-the-ordinary-sense. It may be a conflict, which may be resolved in some way, but it isn't a contradiction.
A) "undefined quantities can't/don't exist" (practically a tautology)
B) "subtraction sometimes produces an undefined quantity"
The first proposition contradicts any proposition that asserts the existence of undefined/impossible/negative numbers. Keep in mind that, before negative numbers, there would be no valid symbols to express the result of the expression "one minus two"; it is irreducible.
The root of the problem is that classical logic doesn't have good tools for dealing with ontological deficits, in this case, the lack of a well-defined concept of "negative numbers".
All of Plato's classic "laws of thought" presume the existence of a well-defined universal set of things/ideas/truths that they can operate over, which is usually a flawed assumption. In reality, the baseline is that everyone has their own ontology derived from personal experience and it takes serious collective effort (or pedagogical browbeating) to establish shared ontologies on which systems of communication can be built.
On the other hand, a core feature of dialectical reasoning is resolving contradictions through the synthesis of novel orthogonal concepts.
Instead, for some inputs, there is no defined/prescribed output.
Anyone who thinks that example you gave is a true contradiction in-the-ordinary-sense is simply confused about about what they mean by those things. (Though, again, if you are using “contradiction” in a way other than the ordinary sense, this doesn’t apply to you)
This confusion would presumably come from being sloppy with phrasing.
If one simultaneously interprets “produces an undefined quantity” as synonymous with “does not have a defined quantity” (which is a true statement) and synonymous with “produces a quantity which doesn’t exist, [thereby establishing the existence of something that doesn’t exist]” (which is a false statement), it is no wonder that one would conclude that there is a true contradiction, and this is precisely the mistake.
Thinking that two things (e.g. statements) are the same thing (and therefore have the same properties, e.g. if they are statements, that if one is true so is the other) simply because one has given them the same name, is a mistake, and is a somewhat common source of confusion.
Now, in some cases, everyone (or nearly everyone) considering a topic will be confused about something in a way that is at least kind of similar to that way of being confused (or perhaps even in exactly that way). In these cases, one way that people might come to realize that they are confused is if they seem to be able to derive a contradiction. But this doesn’t mean that the contradiction is true, but that either (perhaps due to being confused) at least one of their assumptions was wrong, or one or more of their inference steps were invalid (e.g. unintentionally conflating two distinct things/ideas).
There are no true contradictions, only people (such as myself) who are wrong ( and/or confused ) about something.
I really don’t think this is right. Logic is a discipline of consistent re-writing of expressions. However, there may be (even existing) things that are not expressable to sufficient degree in finite (or even countably infinite) strings, which I think means they are outside of logic. There are definitely “possible” things that are outside of logic (because not expressable).
Logic is one of the fundamental axioms of the universe. We assume logic is true and consistently applies everywhere throughout the universe. This "rewriting expressions" thing is just a symbolic representation of logic. It is not logic in itself. The symbolic representation of logic by writing down "expressions" works because logic is a inherit property of our universe and since you are writing those "expressions" in that same universe, it works.
There is no way to prove or verify logic is consistently real. We just recursively assume logic is real. We observe it to be real and assume that the observation will consistently apply across all time and space.
Another thing that I should mention that is an axiom of our universe is probability. WE have no way of knowing why rolling dice or random variables follow the rules of probability. These are just arbitrary rules and we assume that they're consistently true about our universe. Logic along with probability are two things that we have zero methodology of verifying the veracity of but we just assume these two things are fundamental properties of the universe.
A more elegant way to look at it is to just assume probability is the foundational axiom of our universe. Logic is just a special case of probability where all causal connections are 100%. Of course given inherit unreliability and limited knowledge of all things we never actually see or can verify 100% causality on anything. This effectively limits logic to mathematical and axiomatic games while science is the only available tool for the real world.
Science is a whole different beast. Given the assumptions that probability is real and that logic is real, science is an attempted methodology to verify theorems or statements about the universe using the axioms of probability and logic.
For example Newton guesses that a ball should travel a certain distance according to his made up laws of motion. Using science we perform several experimental observations of moving a ball and statistically correlate to a certain degree that yes the ball does indeed move according to newtons laws of motion. This is what science is. It is making a hypothesis and using statistics to sort of verify it. The term "sort of" is key here because science is limited to the fact that it can never prove anything.
One thing to note here is that EVEN when we assume logic and probability is true, science is unable to prove anything is true. You can make 10,000 observations about newtons law, it proves nothing because at any point in time a new observation can render the entire hypothesis as false. Thus falsification is possible with science [1] but proof is impossible. Proof is the domain of mathematics and logic games and cannot exist in the real world due to limited knowledge.
This is not some pedantic philosophy I'm making up. This is foundational to a true understanding of what science and logic is. To quote Einstein:
"No amount of experimentation can ever prove me right; a single experiment can prove me wrong."
There are a lot of intelligent people who don't understand the true depth of the above quote. But if you get it, then you truly understand what science is, and the differences between science and logic. Obviously both the OP and the parent poster don't fully get it... by combining logic and science into one thing and calling it the "science of the possible" it shows that they don't have a clear delineation of the two terms. Most people think of science as some kind of fuzzy "technical study" of a topic. No. This is wrong. There is a clearer definition of science that separates it from logic and mathematics.
[1] Note that technically total falsification is also impossible. Inherit unreliability of observation tools and limited knowledge makes it so that no observation can be 100% reliable. Thus even falsification is technically limited to the domain of logic and mathematical games.
Is it the classical logic of the quantum logic you are talking about?
I'm making a vague sweeping statement here that eventually gets a bit more cleared up later in my writeup.
This definitely a very opinionated view on what is logic. There is no reason to embed anything in universe/anything else. If you remove expressability, I’m not sure what properties remain. At the same time you can have rules outside of anything physical (in a sense of denoting something that subsists). Consistency of re-writing rules (in a sense that one can always write a propositional string that cannot be reduced to true or false), is an entirely separate beast.
> Logic is just a special case of probability
This is factually false. Probability is embeddable into predicate logic, but predicate logic is not embeddable into (Bayesian) probability. This is actually an open problem (how to define a probability theory that is equivalent to predicate logic).
> Science is a whole different beast
On this I do agree, I wouldn’t call logic a “science.”
It is by definition, not opinion.
Completely wrong. How can someone say I'm factually false when your own statement is completely and utterly incorrect. I don't think you're really aware of what's going on here:
https://en.wikipedia.org/wiki/Subjective_logic
It's not an "open problem" or aka "unsolved problem" in the sense you implied. Far from it. Rather it is just not a popular field of study.
>This definitely a very opinionated view on what is logic. There is no reason to embed anything in universe/anything else. If you remove expressability, I’m not sure what properties remain.
I don't think you get it. The nature of an axiom IS by definition an opinion. An axiom is an Assumption. Assumptions are just things that are assumed to be true, they are NOT things that are proven to be true. Thus the only thing an assumption can be if it is an unproven statement is that it is an OPINION. When anyone makes a statement about logic such as my statement "Logic is a fundamental axiom of the universe." Or "Logic is NOT a fundamental axiom of the universe" is by DEFINITION a statement of OPINION.
When I say it's a fundamental axiom it means that it is something all of humanity consistently assumes to be true. Our culture, our science, our mathematics, our interpretation of reality and the universe as we know it is founded on the assumption that logic is true. It is a general statement about the broad "opinion" of all of humanity.
Look at your own statement. You said and I quote: "There is no reason to embed anything in universe/anything else." What does this statement even mean? What is reason? Reason IS Logic. Look it up. You are literally saying there is no logical reason to embed logic into our universe. Yet here you are making a logical statement in a universe where you say we cannot assume logic is true.
We assume logic is true. You can't make a single argument otherwise. Literally all arguments you've ever made in your life hinge on logic being true. That is unless all your arguments are illogical. Are you saying your own argument is illogical?
There's really nothing left to argue about here. Our entire technical framework of reality is founded on the "Opinion" that logic real.
>On this I do agree, I wouldn’t call logic a “science.”
If you truly understood what's going on, you would agree with me on all points. Not just this.
Also, axiom can be just a rule in a rewriting system. Doesn’t have to be an assumption/opinion as it is just a proposition (or higher order predicate) that is true within some system (not necessarily related to anything that exists).
Things can be outside universe yet discussed by people in-universe (subsisting vs existing without multiple worlds).
What I am saying isn’t controversial in analytic philosophy/ epistemology…
Edit: on a more reconciliatory note, a classic essay that I enjoy that touches upon the idea of denotion/subsistence/existence/difficulty with predicates is Bernard Russel’s “On Denoting”. I suspect you may enjoy it: https://www.uvm.edu/~lderosse/courses/lang/Russell(1905).pdf
There's absolutely zero need to be rude. There is no "wierd" word play here just misunderstandings and impoliteness by you.
>Also, axiom can be just a rule in a rewriting system. Doesn’t have to be an assumption/opinion as it is just a proposition (or higher order predicate) that is true within some system (not necessarily related to anything that exists).
This is obvious. You're just restating what I said and coming up with a circular argument. A rewriting system is a universe you create, not dissimilar to our own universe that we live in. An axiom in this system is an statement made by you stating something is true without any proof. A statement made without proof IS an assumption. Now this explanation may seem like word play to you but it is not. The exact identical words "assumption" and "axiom" serve to confuse your reasoning and blind you from the actuality of the concept at hand. What I am doing is unraveling your misunderstanding in attempt to explain your confusion but you are unable to see it and you view the argument as "word play." Axiom and Assumptions are two words that mean the exact same thing. There is one concept at hand and two words for that same concept that are confusing you.
>subjective logic, which you linked, is equivalent to propositional (in a sense you can express any statement in subjective logic via a set of propositions, eg Ap distributions make this connection), and does not embed predicate logic (eg any statement of this system can be expressed in predicate logic, but not vice versa)…
This system has axioms and theorems that use predicate logic. It is built off of predicate logic.
I recommend reading this: https://projecteuclid.org/journals/statistical-science/volum...
When you set the subjectivity to 100% the result is something equivalent to predicate logic. So logic is both a special case of probability as well as foundational building blocks for subjective probability itself. You can arbitrarily pick and choose which came first and which one is axiomatic. Probability or Logic.
You are indeed wrong here. Please politely admit it without calling my statements weird. If you disagree there is no need to resort to derogatory statements.
Maybe you were trying to promote understanding over correctness but I would recommend that you do both as such fuzzy wording leads to people inconsistently and incorrectly using the word "science."
Logic models the mathematical reasoning itself, like vector spaces model our world and differential equations model most of physics. At no point however we should confuse the model with the studied object. As such logic is a field of mathematics and not mathematics itself. The best illustration of this is that theorems in the field of logic use results from e.g. set theory.
Of course using a formal logic to prove theorems is valid. Still the formal logic has to be studied with "intuitive" mathematics. The book uses the metaphor of spiraling downstairs to illustrate this.