There is something deeply cultural in how Russians are falling in love with abstract bullshit, which they call theories to signal that they are smarter than they really are. It must be related to almost a century of brainwashing with Marxism, Leninism and Freudian and Hegelian bullshit.
Like any consumers of abstract systems they love to "stretch" the terminology a bit, forgetting that words are supposed to have precise meaning (being associated with actual aspects of reality) and that terminology should have been defined unambiguously before being used.
Of course, this is not necessarily required if you just want to get some hype by shitposting about some trendy abstract bullshit, which, like any esoteric and theosophical subject, never get critically reviewed by adepts of the sect.
By the way, so called Category Theory is the single biggest virtue-signaling opportunity, second only to religions, and, ironically enough, like Haskell, it is a trap for pseudo-intellectual narcissistic idiots.
Well, I will break the rule and do partial peer-review of this piece:
I know I will be down-voted into oblivion and banned for this post on HN (which has been evolved into a virtue-signaling platform), so I will publish my review on my own website.
So, lets begin.
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.
The very first passage raises some eyebrows, to say the least. What does "the science of possible" even mean? As far as I know, science is an evolved (from merely speculation) methodology to establish the truth about certain aspects of reality.
It consists of a systematic and well-structured process, which require careful observation of a phenomena in question, formulation of a hypothesis regarding some well-defined and carefully measured aspect of it, design of a replicable (reproducible) experiment to test the hypothesis, and then refining or reformulating the hypothesis depending on the results of experiments.
Science, therefore, implies only something possible - a science of impossible is a plain nonsense. The whole methodology (which is what the world "science" stands for) is to be applied to actual possibilities, and an absence of a result is a signal that some theorizing is, perhaps, went too far.
Logic studies the rules by which knowing one thing leads you to conclude (or prove) that some other thing is also true, regardless of the things’ domain (e.g. scientific discipline) and by only referring to their form.
Well, logic is not the study of the rules of inference. It is the study of a sound argumentation (including formal reasoning), to which the rules of inference (or of simplification of proofs) are just one technical aspect.
Being sloppy in terminology leads to this "to conclude (or prove) that some other thing is also true" and even to more horrifying "regardless of the things’ domain". This sloppiness is the sign of a conceptual mess.
First of all, a proof is not in the same category as conclusion. Applying an operator (OR in this case) to both of them is a type-error, plain and simple.
A proof is a reproducible (like an experiment) and verifiable sequence of terms, from a set of premises to a conclusion, which can be verified by the whole formalism - the particular system in question.
The use of logic to verify its own terms is the fundamental and definitive quality and is reflected in the recursive nature of any general (or concrete) simplifier (or evaluator).
Recursive, convergent processes are fundamental to any system of logic, of which mathematical induction is a specialization. Such view of a process (recursive and convergent i.e. spiral shaped) is fundamental to the Universe itself.
Second, there is no such thing as logic regardless of the domain. This statement should be re-read a few times.
There is nothing ephemeral or god given out there which is valid (or even exist) regardless of the domain. Literally nothing.
Although there are rules of inference based on the form of terms (compound terms with connectives), such terms cannot be formed abstractly, without domain knowledge. Unless you are Hegel, of course.
The principle here is that your premises has been proven valid (by the very same recursive process, which relies on what is already known, including the rules of inference) before they are being used as terms in new compound propositions.
That "Socrates is a Man" deduction reflects a few deep facts about biological evolution (that what we classify as species have common traits or qualities (attributes) based on which the mind classify them as "an external observer").
There is no justification to assume validity or existence of Modus Ponens as an entity disconnected from a domain.
So what Modus Ponens is? Well, it is an expression in a meta-language, a captured generalization, which is being observed again and again in domain-specific languages.
It captures the law of causation, that everything has its causes, and, it also generalizes to categorical thinking, which, in turn, is possible only because there are physical laws and the law of causation in the first place (everything which mind categorizes is because of causation).
Again, disconnected from the domain it loses meaning and ceases to exist.
And yes, there is a fundamental problem with the classic truth-tables, in part where falsehood implies anything. They cannot be universal because they contradict the [Multiple] Causality Principle. One of them must be wrong, and it is not the causality, of course.
instead of the word “formal” we used another similar word, namely “abstract”, and instead of “logical system” we said “theory”.
This is bullshit in its essence. Formal and abstract are unrelated, unless they are "purely formal" (which is nonsense) and "purely abstract" (which under-educated virtue-signalers love so much). And, of course, a logical system is not a theory.
today most people agree that every mathematical theory is actually logic plus some additional definitions added to it.
No, it is not. Formal logic is a sub-discipline (actually, a meta-language, the one above domain-specific languages), which includes a methodology for formal reasoning (reasoning, based on the form of compound expressions with involves certain logical connectives). Again, the premises must be valid apriory (or being form a valid, non-contradictory set of axioms) which connects and grounds it in reality.
Logical connectives and predicates operate at the level of the meta-language (or a particular logical system). It has been discovered that this meta-language must be typed (the conceptual space must be be partitioned).
A mathematical theory is, by definition, uses mathematical logic (which is what makes mathematics what it is), but using mathematical formalism and notation does not promote bullshit into realm of scientific disciplines.
rules of inference are almost the same thing except they allow us to actually distill the conclusion from the premises.
What the fuck is "distill"? Is this some vodka-induced logic, Ivan? Similar to Hegelian, but from Russia with love?
I am sorry, I cannot continue to read this bullshit. As one may imagine, it will not get any better down below.
What is important is to show what a fucking bullshit this virtue-signalling witting really is. Remember, that a single logical flaw is enough to discard the whole line of argument.
And no, this "theory" is nothing but a graphical representation of structures of some similar concepts, which, in turns, are generalizations, sometimes too abstract to remain real. Monoid, perhaps, where to stop.