Veeky Forums, I'm having difficulty here.
Over the last few months I have developed a strong love for logic but have no fucking idea how the fields of logic relate to one another.
Philosophical logic, Philosophy of logic, Logic, Mathematical Logic, Philosophy of Mathematics, and possibly Philosophy of Language.
What in the fuck. Can anyone point me somewhere that definitively clears this up or is there a reason for this?
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W-why, Anonymous, I didn't know you felt that way about me! *kysses back with passion and gusto*
Mmm. That feels nice. *puts my hand on your inner leg*
IF YOU PURPOSEFULLY MISUDNERSTAND ME LE ONE MORE LE TIME I LE SWAER I WILL KILLLLLLLL YURUYITROTYOTOTO AHHHAHQHAHHAHHHC
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Logic or mathematical logic is the study of propositions, reasoning, predicates, axiomatic systems, syllogisms. Mathematics are a specific case of logic inasmuch as an arbitrary, symbolic, axiomatic system. Philosophy of mathematics is a meta discipline which might include discussions on epistemiology. All three of these are linked together mainly through Boole, Frege, Gödel, Łukasciewicz.
Then there's Peirce, one of the founders of semilogy, who coincidentally also made advances in propositional logic and mathematical logic (he was the first to formally define material implication (⊃)). Frege himself was a semiotician too. The discussion on which comes first, whether language or thought, DIRECTLY influences the other question, language or reasoning.
[corrected a typo]
Logic is understanding and coming up with arguments, given agreed upon modes of reasoning.
Argument:
"It rains. Also, if it rains it follows I'm wet. Therefore, I'm wet."
The mode of reasoning behind this arugment is what's called Modus Ponens:
"If a proposition hold and if from that proposition another follows, then this second proposition holds"
en.wikipedia.org
Argument:
"If I never met Obama, he couldn't know my name. If I met Obama, he wouldn't rember my name anyway. Either I met Obama or I didn't, but in both cases he wouldn't know my name. Thus, Obama doesn't know my name."
This uses the Law of Excluded Middle:
"A proposition is either true, or it's negation is true."
en.wikipedia.org
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Mathematical Logic is a formalization of that - you translate sentence to symbols and then you can effectively come up with arguments by manipulating the symbols according to agreed upon rules.
(in the follwing, the words forall, implies, and, or and not are usually represented by some symbols)
Modus ponens (second order logic):
forall P. forall Q. (P and (P implies Q)) implies Q
L. of E. M.:
forall P. P or not(P)
Another classical rule would be (example)
"if it's not the case that I'm both tall and pretty, then at least one is the case: I'm not tall or I'm not pretty."
In formal/mathematical logic
forall P. forall Q. not(P and Q) implies not(P) or not (Q)
Another classical rule would be (example)
"If from my being rich follows I have a gf, then if I have no gf, it means I can't be rich"
In formal/mathematical logic
forall P. forall Q. (P implies Q) implies (not(P) implies not(Q))
You can do math with the symbols, teach them to computers and so on.
All the above rules are takes as valid in
en.wikipedia.org
A logic that, for example, drops the Law of Excluded Middle is
en.wikipedia.org
Here is a logic with what's called modalities
en.wikipedia.org
Here more stuff
en.wikipedia.org
You may chose a logic and add axioms about things other than proposition (e.g. sets, lines, number,...), and the formal frameworks you obtain this way are called rigorous mathematics.
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Philosophy of logic (or Mathematics) is, one may say, thinking about the reality of all the different things we can do. People fight about logics like they fight about politics.
Philosophy of Language is what happened when Russel and Wittgenstein sort of failed their project of putting human reasoning on a formal level - the project that arguably worked for mathematics.
>developing love for a meta-narrative
Another one bites the dust
Maybe I should change the question.
I just finished studying, term, propositional, and I suppose the introduction to predicate, logic. I also have studied some set theory.
My problem is I dont know where to go with logic. I love studying it and plan on doing grad work on it but I dont know where to go. I can continue with predicate and then decide to go to a higher order logic or stay with predicate, or I can go to more set theory, or modal logic.
Thing is I dont know where to start. I dont have a structure on what to study since I dont know how the field develops.
What are you saying? Meta-narrative?
>Thing is I dont know where to start. I dont have a structure on what to study
just b urself :)
Instead of diving into each aspect separately, you could read an introductory text on the field of logic in general. Evandro Agazzi has a good book called "Symbolic Logic" in which he defines exactly what is and is not logic, the historical development, etc.
You could also start at the very beginning with Aristotle and work your way to the present state of affairs.
>I dont know how the field develops.
Pal logic is a such a borderline-esoteric field that the notion of "advancing" it is kinda funny. There really is no battle plan. At this point the concerns are mostly on the metalogical, the foundations of mathematics and autistic system building.
So does this cover everything I mentioned here besides philosophy of language?
I'm still not sure about philosophical logic or philosophy of logic
Mathematical Logic consists of Axiomatic Set Theory, Model Theory, Recursion Theory (it's a more rigorous version of Computability Theory of computer science although the two names are used interchangeably in some contexts), and Proof Theory. All four fields assume (and use) First-Order Logic and its metatheoretic results and limitations. All four disciplines interconnect intricately and so do with other disciplines of modern mathematics. Model Theory, for example, has applications in Algebraic Geometry.
Philosophy of X (Mathematics/Logic/Language, etc.) as the names imply, are concerned less with proving theorems within a well-defined logico-mathematical framework of axioms, grammar, and rules of inference, but with philosophical aspects of X which, depending on the discipline, usually entails giving philosophical accounts for thorny, controversial issues. In all essence it means that a bunch of dudes are arguing for their views, which academic Philosophy is all about: giving arguments for views you find most persuasive. For example, philosophers of Philosophy of Mathematics ask (and in effect argue) whether or not abstract objects exist (Platonism, nominalism, naturalism, structuralism, etc.), and if they do, how we can know about them (Benacerraf); how, if they do exist, they relate to our best physical theories (Putnam-Quine), and so forth. Philosophy of Logic is about questioning the orthodox semantics of First-Order Logic (Tarskian definition of truth, logical consequence, etc.), about proposing different semantics for a wide variety of quantifiers that can be found in natural languages, whether or not Second-Order Logic should serve as the foundational framework, etcetera.
Philosophical Logic is about non-classical logics mostly used in Analytic Philosophy, Computer Science, and Linguistics, not so much in Mathematics: temporal logic, dynamic logic, alethic logic (= logic of necessity and possibility), provability logic, deontic logic, epistemic logic, and so on and so forth.
Don't try to understand them all at once, it's impossible. The fact that you're even asking this means that you're clueless about their similarities, differences and the contemporary geography of those disciplines. Mind you, it goes way deeper than what I and others ITT had just described. Nailing down the basics (First-Order Logic) should be your priority now.
And the final thing, which goes without saying, is that if you want to work in, or contribute to, these fields, you should get a BSc in maths first. Coming to Mathematical Logic without already being mathematically mature/comfortable will only end in a trainwreck which you will realize soon after working through the very first pages of most grad-level textbooks.
>I just finished studying, term, propositional, and I suppose the introduction to predicate, logic.
WTF? Are you a time-traveler from the middle ages?
>My problem is I dont know where to go with logic. I love studying it and plan on doing grad work on it but I dont know where to go.
Read the PDFs here: logicmatters.net
^ This guy nailed it, OP. You should pay him for his comprehensive answer.
Thank you very much.
I don't have a BSc in maths but I do have a year to study on my own before doing a terminal MA, then two more years to get ready to do real grad work.
I should probably get looking into that then. Veeky Forums seems to have a pretty decent guide to getting into undergraduate mathematics.
Have an opinion on this or anything like it?
amazon.com
Linguistics
Psycholinguistics
Structuralism and post structuralism
The innateness theory
Read about this
It's called comedy.
Consider applying for the logic year or a masters at the ILLC in Amsterdam.
I'm not going to drop 14k to do that for one year.
>amazon.com
Does anyone have a pdf to it even though it seems like a fresh, 2016 release?
Judging solely by the blurb it doesn't seem like much of a math textbook.