Thread 16771694

Can we have a general on logic? Any logician bros around?

>>16771694 (OP)
i don't get it, is the bottom about paracomplete logic?

>>16771694 (OP)
Thank you OP, I was hoping to see a thread like this. Pic related is a good book to learn more.

>>16771694 (OP)

Truth is if you REALLY want to learn math, ask an AI to teach you.
You will get a individualized education, at your mental speed, with everything carefully explained.

>>16771709
AI makes mistakes all the time. It should be a tool to help you learn, not your sole teacher.

>law of excluded middle
>not not-female = female
no thanks. because not-female can equal male OR intersex.

>>16771710

Suggest math book that poor people can not afford versus suggestion of free individualized self-paced learning
AI is literally improving daily

>>16771714
Male, female, and intersex are not truth values but are properties which can be tested for having a truth value.

>>16771719
>2+2 = 4 isn't a truth value; it is a property that can be tested for having a truth value
what's a word to describe mathematical based schizophrenia?

>>16771714
Law of excluded middle simply says either P is true or P is false. If P is false then, simply put, ~P is true. Either Gale is female or not female, if the latter is true we only know that the formula is true, being not-female doesn't tell us about the actual content. If Gale is male, then the statement is true. If Gale is intersex, the statement is true. If Gale is a tranny who just had their dick cut off and now has to dilate every day, then the statement might become ambiguous unless we clearly define female and male to mean something like biological sex. But you could also introduce other logics that deal with ambiguities like these, logics where the law of excluded middle may not hold.

>>16771722
2+2=4 has a truth value which returns 1.
2+2=5 has a truth value which returns 0.
It's literally that simple. Do you not understand what propositions are?

>>16771726
This sentence is false.

>>16771731

I am not here.

>>16771731
>>16771734
Being a paradox is also a property which can be tested for having a truth value

>>16771718
False dichotomy fallacy, in the /logic/ general, for shame anon.

>>16771731
Self referential statements such as those are not propositions since they don't have any definite truth value. By their very nature they lead to inconsistencies, and are not reconcilable in any classical logic. To deal with them on their own terms you need to use non-classical logics like intuitionistic logic or paraconsistent logic.

>>16771738
okay, what is the truth value of:
>>16771731
what is the truth value of:
>>16771734

>>16771745
>intuitionistic logic or paraconsistent logic.
oh, you mean a logical system that doesn't use the law of excluded middle ;)
>non-classical logic
oh dear. the absolute horror! how DARE one go beyond the classical? that cannot possibly work in any formal discipline. the classic way is THE way. disagree? you're an IDIOT!

>>16771746
It's not a proposition and therefore classical logic cannot account for it under the principle of bivalence.

>>16771750
>REEEEEEEEEEEEEEEEEEE not my heckin bivalence, there are two and only two truth values!!1!!!!!!1111

>>16771747
I don't have anything against non-classical logic you fucking idiot. I was educating you on what propositions are, which you seemed to have not understood.

>>16771746
For the property of paradox, and truth values are only meaningful when discussing properties, the statement
>>16771731
has a truth value of 1
and
>>16771734
has a truth value of 1

>>16771694 (OP)
The Fundamental Theorem of /Sci/: (P ^ ~P) => Q
You will find this in almost every thread.

>>16771722
>what's a word to describe mathematical based schizophrenia?
/Sci/-poster.

>>16771763
The antecedent is always false, so this is a tautology.

>>16771768
Welcome to /sci/.
Present your contradiction. Pick your favorite conclusion. Always true.

>>16771769
If lemons are yellow and lemons aren't yellow, then unicorns exist.

Lol

>>16771768
In classical propositional logic, due to soundness and completeness, the theorems are exactly the tautologies. This is fundamental to /sci/.

>>16771770
True statement. Find a lemon that is both "yellow" and "not-yellow", and I will show you a unicorn.
We will then argue about the terms "yellow", "not-yellow", the prefix "uni-", and whether or not "corns" are "horns".
Welcome to /sci/. The coffee sucks, but it's free. Enjoy your stay.

>>16771774
Thing is, you will never find any situation where the contradiction holds, so there is no counter-example to the original statement. There is no situation where lemons are yellow and lemons aren't yellow and unicorns don't exist.

>>16771731
that has to do with explosion, not excluded middle, you ass breather

>>16771777
>There is no situation where lemons are yellow and lemons aren't yellow and unicorns don't exist.
Which is exactly why the statement is (vacuously) True.
Trust me, there is at least one thread at present claiming that if the divergent series (~P) converges (P) then it converges to exactly -1/12 (Q).

>>16771777
where's the unicorn, o holy trips of sevens?

>>16771785
>Let the Arguements Begin
That's a Limon; Sprite owns the patent.

>>16771785
/thread
Logicians btfo beyond belief.

>>16771785
now the question is if that is green & yellow or yellow & green

>>16771731
Which sentence?

are there stereotypes about people who work in various branches of logic?

>>16771785
That lemon is not not-yellow, Aristotle dealt with this, he says it's a fallacy to:
>Suppose an Indian is black all over, but white in respect of his teeth; then he is both white and not-white. (Soph Ref Ch5)

Just because it is not-yellow in a qualified sense doesn't mean that it is not-yellow.

>>16771829
^ is a symmetric operation.

>>16771899
>That lemon is not not-yellow
The limon is not not-yellow.
The limon is not-lemon.

when should you expect a tautology?
like, (((A -> B) and (B -> C)) -> (A -> C)) is a tautology, but why is that significant?
when you make a valid argument, you just need to check that some given assignment of variables will evaluate to true, so why is a statement that evaluates to true no matter what interesting? i can see value in the given example, but feel that it being a tautology has a deeper meaning

>>16771853
Logicians unironically think logic is more pure than math even though logic is a branch of mathematics. They want to have their cake and eat it too.

>>16771906
Uh, sweaty, math supposes logical axioms and logic does not suppose mathematical ones. In other words, logic is fundamental to math and therefore more pure.

>>16771905
>i feel that it being a tautology has a deeper meaning
In classical propositional logic, due to soundness and completeness, the theorems are exactly the tautologies.

>>16771910
More axioms reduces the total size of the theorem-space. Less axioms, more theorems. You confuse the number of decidable within system Truths as "purity" for some reason, but likely directly related to your undesirability.

Do constructivists generally reject principles that are classically equivalent but constructively weaker than AC? Like Zorn's lemma, for example.

Also, is this a new general? And why not stick to /mg/ for logic, that thread is slow enough it could use some more on-topic posts to be honest.

>>16771905
A tautology is the negation of any contradiction and vice versa. You can verify that a logic is sound if every theorem in that logic is also a tautology, meaning it's true in every model of that logic. You can also show a logic is complete if you can prove that every valid formula in a class of models is provable in all those models. I'm not an expert in Godel's proofs, but I believe he showed his incompleteness theorem by proving that not every valid formula in arithmetic can be proved under the Peano axioms, basically.

>>16771924
>A tautology is the negation of any contradiction and vice versa.
Lots of schizo potential in that statement.
This ride might never end.
Stopped reading there though.

>>16771924
>but I believe he showed his incompleteness theorem by proving that not every valid formula in arithmetic can be proved under the Peano axioms, basically.
>*cannot be proved
Sorry, typo on my part.
>>16771929
You literally didn't know what a proposition was in the beginning of this thread. Go back to 1st grade.

>>16771915
“more axioms reduces theorem-space” is just false by definition.
With fewer axioms, you have more models, so fewer axioms make more sentences true in some models, but these aren’t theorems they’re just model-dependent truths.

UHHHHH ATHEIST BROS!?

>>16771910
Prove the statement 2+2=4 using an axiomatic system that can prove every valid mathematical statement. I dare you.

>>16771934
>You literally didn't know what a proposition was in the beginning of this thread. Go back to 1st grade.
Now I am doubting if any of us are real.
To you, I mean. I think you're making this all up yourself.
Schizo af, man.

>>16771949
And what exactly would that prove, moron?

>>16771942
>modal logic

>>16771942
All Truths Follow from Contradiction (aka, (P^~P)=>Q)
Your system is powerful enough to generate Contradictory Truths and thus it Must Generate All Truths.
You system is Vacuously Complete.
(Everybody dance now...bomp-bomp-bomp)

>>16771942
>D1 is unfounded
>A3 is not any of the monotheistic gods
>A5 is a tremendous cope

>>16771969
You think you're smarter than Godel? LMFAO

>>16771957
That all of mathematics is reducible to logic.

>>16771976
he starved himself to death due to paranoid schizophrenia. i KNOW i am smarter than he was. let's evaluate his biggest contributions
>incompleteness theorem
sounds deep until you truly understand it. here's how it works
>oh dear, my religious belief is illogical
>aha! i shall design a new type of logic - modal logic
>i shall use this modal logic to make proofs about mathematical statements
>alas, these proofs are not within the realm of mathematics
wow much deepness. it's like if i say sex is nonbinary because i imagine infinitely many sexes in my gender field theory. you see, the binary definition of sex is insufficient to make biological proofs.
>oh hey, this modal logic can also be used with personal, biased assumptions to "prove" god exists
kripke, godel... ever notice that all those big names who work on modal logic are religious? it's not a coincidence.

>>16771985
>i a priori assume a metaphysical framework that excludes the existence of God
>therefore belief in God is illogical
Not only does this confuse metaphysic with logic but it also demonstrates extraordinary ignorance on your part about the many formal arguments given for God's existence stretching all the way back to scholasticism.

>>16771990
assumptions will get you nowhere with me. i believe in god; i merely acknowledge such beliefs are illogical. you are a disingenuous faggot and i will not respond to you any more.

>If a lion could speak we would never be able to understand him. - Ludwig Wittgenstein

I wonder if this is applicable to extraterrestrials, assuming they exist. If logic is the natural result of human reasoning, and there are aliens out that are so completely separate from us in every possible way that nothing, not even their logic, is translatable in terms of ours, would this not make logic entierly human dependent? Is it possible these aliens even have a kind of mathematical system developed from their own logic which objectively describes reality for them, but not for us? And which they could still use to manipulate and control nature successfully as we do with our mathematics? I wonder.

>>16771976
Lots, in fact most every single person on Earth today, is more sane than Godel was in the end. The stories of him roaming the halls of Bell Labs are amazing.

>>16771949 and >>16771981
>Proving the statement 2+2=4 using an axiomatic system that can prove every valid mathematical statement would prove that all of mathematics is reducible to logic.
No. No it would not. Not even close.
You are going to need to show your work is you want any grade at all.

>>16772015
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

>>16772038
>you think you're so smart, but Godel was crazy way smarter
>my challange is proveable impossible
>and i have a hardbound copy of rhat proof on my shelf
>but if you could do it, you could do anything!!@@
>hahahahahahahahahaha
Literally "(P^~P)=>Q". Told you all it was fundamental theorem of this board.

>>16771949
I take as my axioms every true mathematical statement. Now 2+2=4 has a one-line proof.

>>16771993
belief in God is logical, thoughbeit

>>16772043
>one-line proof
( . . . (((P=>P)=>P)=>P) . . . =>P)

Thoughts on Buddhist logic? Specifically Nagarjuna's tetralemma?

1. If P then P
2. If ~P then ~P
3. Both P and ~P
4. Neither P nor ~P

>reeee but 3 is a contradiction!!!!!
One can argue that Buddhist logic is a primitive form of paraconsistent logic which allows for contradictions without explosion.

>reeeeee but 4 is the same as 3
Only if you are operating in a system whose semantics can be interpreted under Demorgan's laws, but you don't have to.

>>16771949
Any inconsistent theory?

>>16771694 (OP)
any philosophy schizos want to explain what these symbols mean?

>>16772075
>reee
>If P then ~P
Kek. Nice.

>>16772050
If (((((P=>P)=>P)=>P)...=>P))=>P is a tautology then P is a tautological consequence of (((((P & P) &P)&P)...&P)) and, in general, both are tautological consequences of each other. Hence if

⊨((((((P=>P)=>P)=>P)...=>P))=>P)=>⊨((((((P & P) &P)&P)...&P))=>P)

By the truth preserving property of modus ponens if ⊨((((((P=>P)=>P)=>P)...=>P))=>P)=>P and ⊨((((((P=>P)=>P)=>P)...=>P)) then ⊨P and, similarly, if ⊨((((((P & P) &P)&P)...&P))=>P) and ⊨((((((P & P) &P)&P)...&P)) then ⊨P

>>16772087
>hey siri, grok me those meta i just gpt'd
They are literally trying to formalize vague language like, "quiet possibilty is so, but". That by itself get deep fast and makes me want weed and beer. These guys then go and say shit like, "remember our old proof crank machine, tirn him loose on this shit until and wait until it build this giant ass Truth-Tree". And everyone who is high laughs and then the fucking shrooms come out and this guy says, "let's start making some probabilistic statements about forests of these bitches".
Hilarity ensues.

>>16772100
Lines. You used many.
When one was already done.
Thanks, joke explainer.

>>16771949
https://en.wikipedia.org/wiki/Presburger_arithmetic#Properties

>>16771976
>Philosopher X makes a theory which sounds fake
>I read it and call it bs with arguments
>Guy that disagrees goes: oh, so surely you must think you are smarter than X
Just your daily reminder that smart people are also wrong, and that you should not feel bad if you think you can disprove them. You don't need to be smarter than someone to call him/her out.

Why does every logic thread need to be shit up by philosophers? This subject doesn't belong to you anymore, /his/ fags. Just like physics became its own thing, so did logic. Stop philosophizing, it's gay.

>>16772100
>If (((((P=>P)=>P)=>P)...=>P))=>P is a tautology
Only when the numbers of P's in the sentence is even. Otherwise it is indeterminate.
Consider (P=>P): When True, T=>T is True. When False, F=>F is True, too.
Now consider ((P=>P)=>P): When True, ((T=>T)=>T), is (T=>T), is True. But when False, ((F=>F)=>F), is (T=>F), is False. So the statement's truth value is indeterminate.
Details (null case is trival, induction step is obvious from above) for the reader.

>>16772118
I forgot about this fact. Thank you for pointing this out.

>>16772120
NP.
I'm surprised I was high enough to notice myself.
Trouble is, now I'm getting even higher worrying over the more important question as the the number of P's in the original statement being even or odd. We see the obvious argument for both, so it's clearly neither.
Just like (1-1)+(1-1)+ . . . +(1-1) being -1,0,1 and THUS ALL NUMBERS, but mainly -1/12, you know. ;)
Turns out it's all just (P^~P)=>Q each and every time.
From the impossible, wonders grow.

>>16772128
Do you think, fundamentally, at its core, reality is just made of contradictions? It's like you have two opposite particles annihilating each other but giving off energy that is measurable if you want a weird analogy from physics.

>>16772050
Let X = ( . . . (((P=>P)=>P)=>P) . . . =>P)
Then X is X => P
If X is false then X is true
Therefore X must be true
∴ P is true

How autistic does one need to be to become a logician?

It seriously seems like the single most autistic subject in all of mathematics. Obsessing over literally simple shit like if p therefore q is true. The posts above only seem to confirm.

Do you think logicians inherently need to have aspergers syndrome or something?

>>16772144
Logicians are actually pretty chill in my experience compared to normie mathematicians

>>16772146
>literally writes a 2000 page proof that 1+1=2
Yeah... they're pretty chill alright...

>>16772133
I think, at its core, we haven't the slightest idea what's happening, and likely never fully will because some things are just too small to bounce photons off of in a meaningful ways or require the photon has more energy than we can technology muster for many generations, if ever.
But I like the formal hunt for "Truth", whatever the fuck it turns out to be.
We build powerful tools: physical and theoretical to probe for these Truths. It is good to turn these tools of themselves to test their "resolving abilities" before they start to amplify the noise.
Really fucking high, no homo.

>>16772151
I mean, it's not like his advisor wasn't saying shit like, "we need about 10 more pages of detail here" every time my man handed in a draft.

>>16771694 (OP)
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>>16772137
Too high to see offhand. Gonna need a bit to thonk it through, maybe another joint. Can't have this and >>16772118 at once.

>>16771694 (OP)
I am completely demoralized since finding out there are statements without proofs or counterexamples (CH). Like, what's even the point?

>>16772170
>Like, what's even the point?
Awfully sexy curves on that boundary, son. Even if they don't let you touch her.

>>16772166
Assuming the ...'s indicate the expression follows that pattern infinitely, adding one more "=> P" to the end doesn't actually change the expression

>>16771785
Nope. Your counter example fails.

>>16771777 is not saying (∃xY(x)∧∃x¬Y(x))=>∃yU(y). Clearly this statement is false since the antecedent is true, "some lemons are yellow and some lemons aren't yellow", while the consequent is false since unicorns do not exist.

Rather >>16771777 is saying (∃xY(x)∧∀x¬Y(x))=>∃yU(y), which says, "there are some lemons that are yellow and no lemons are yellow, therefore unicorns exist." Clearly, it is not the case that there are lemons which are yellow but no lemons are yellow, so you cannot find me a counter-example where this is the case and unicorns don't exist.

Therefore, since both the antecedent and the consequent are false, the statement is true. Thus we can say that if lemons are yellow and lemons aren't yellow, then unicorns exist, is always true. Thus this statement is simple and concrete fact about reality. You can't disprove it with any counter examples because there are none.

>>16772170
I wouldn't worry about it. It's like saying you cannot prove that sqrt(2) exists [within the framework of N]. How do I know? Well I simply used a number in R to claim it exists in N and simply cannot be proven! It's all semantic slight of hand. The "logical" system in which this incompleteness crap is posited is completely irrelevant

>>16772182
>since both the antecedent and the consequent are false, the statement is true.
Ah yes, if purple is a primary color and blue is a secondary color then I am God is true. Now bow down to me and suck my divine existent cock

>>16772170
CH could have a proof, it just doesn't have one in ZFC. If you assume the axiom of determinacy for example you can prove a version of CH

>>16771694 (OP)
The second pic only looks complicated but isn't actually complicated if you studied modal logic a bit and know about Hasse diagrams from your CS days.
Alright, in writing this above line I just realized I am maybe downplaying how intimidating it looks to people that didn't have my CS + philosophy combo.

>>16772201
No I agree with you it's really not that bad. It looks complicated, and sorta kinda is, but also sorta kinda isn't.

>>16772178
Except it does. Reread >>16772118 and try again.

>>16772162
I agree. Maybe the truth is too much for us to ever be able to comprehend. We're limited by our physical bodies. It's not like our brains even evolved to do complex logic. Maybe we've been asking the wrong kinds of questions all along and never knew it. Maybe by our very nature we're not even capable of asking the right questions.

>>16771906
Logic is the retarded offspring of math and language after having gay butt sex off a drunken one night stand.

>>16772222
>Maybe by our very nature we're not even capable of asking the right questions.
Tiger gots to hunt.
Bird gots to fly.
Man gots to sit,
And wonder, "Why, why, why?"

Tiger gots to sleep.
Bird gots to land.
Man gots to tell
Himself he undersrand

>>16772211
>Assuming the ...'s indicate the expression follows that pattern infinitely

Logic is a useful tool for constructing proofs in math and making good arguments for anything but it was a mistake to investigate it itself. It leads to all kinds of silly paradoxes. No one should take it seriously as an objective measure of reality. It is a model of reality like all regular mathematics. That is it.

>>16772302
Is it true that its a model?

>>16771965
someone doesn't know about paraconsistent logics

>>16771998
nah

>>16772041
>the principle of explosion is the fundamental theorem of /sci/
i don't think so, also you are staring to sound like a crank, and if explosion bothers you so much the you are free to work exclusively within paraconsistent logics and even become a dialetheist, nerd

>>16772075
that's just
https://en.wikipedia.org/wiki/Minimal_logic

>>16772303
kek, postmodernists will never have a response to this.

>>16772303
Well the whole purpose behind logic, originally, stretching back to Aristotle, was to be a tool for constructing or analyzing good old fashioned arguments. Specifically philosophical arguments. You had premises, you had your conclusion, and as long as you stuck to the correct formatting of terms (for validity) and were able to justify your premises then you were all set. And this is how it was, for a long time, stretching through the middle ages into the early modern period. Although medieval logicians did continually refine Aristotle's logic, basically, up until the early modern era, logic was a subject in its own right almost entirely within the realm of philosophy. Then Frege came along. That mother fucker. With him, suddenly, logic got incorporated into math, and basically became a topic in math in its own right. Everything went to shit from there.

>>16772151
but he didn't, the 1+1=2 bit was but a useful thing that could be derived after a lot of other stuff was, but even then the minimal "thread" of things that need to be derived before being able to reach 1+1=2 is barely 20 pages if i remember

>>16772308
Can you please elaborate on how exactly paraconsistent logic work with contradictions? Why aren't they an issue in it? Genuinely curious.

>>16772170
nigga the continuum hypothesis is merely set theory's equivalent of the parallel postulate

>>16772183
no need to move to the reals for root2, the constrictible(as in compass & straight edge, not constructivism) algebraics are good enough

>>16772316
https://plato.stanford.edu/entries/logic-paraconsistent/
https://en.wikipedia.org/wiki/Paraconsistent_logic
>allows for the coexistence of contradictory statements without leading to a logical explosion where anything can be proven true
https://en.wikipedia.org/wiki/Dialetheism
im fairly certain that you'll find
the "without leading to a logical explosion where anything can be proven true" part to be fairly amenable to the issues you seem rather concerned with

>>16772324
oh nearly forgot, it being because that spin in your side, "(P^~P)=>Q", ain't true in general because the "(P^~P)" part ain't necessarily false

So considering all the discussion in this thread, I guess that leads to one question: Is logic subjective or objective? Is it purely a human construct or... is it out there?

>>16772330
id say its out there, in that even if there was some alien species, their logic & ours would share a common ground, even if we accept the feasibility of extraneous divergence with respect to them & us from said ground on

>>16772324
Would a naive set theory with unrestricted comprehension that is based on a paraconsistent logic not necessarily be inconsistent in that case?
Since even if one were to prove a contradiction from russels paradox, that contradiction wouldn't lead to everything being true by logical explosion?

>>16772346
Then how is it possible to have different "logics"? Shouldn't there be one all encompassing logic?

>>16772330
Logicism (Frege, Russell, early Wittgenstein): Math is just logic in disguise. Logic is prior.
Formalism (Hilbert): Logic + axioms = math; both are formal games, logic is a convention.
Intuitionism (Brouwer): Logic is not prior; it depends on constructive mathematical intuition.
Category-theoretic foundations (Lawvere, etc.): Sometimes logic itself can be built from math (via categorical logic).

>>16772308
Law of Identity is too important. If I can't conclude that P is always P than what am I talking about.
Hope you feel better real soon.

>>16772308
It is the basis for most of the arguments on this board, as previously demonstrated.

>>16772350
>Shouldn't there be one all encompassing logic?
Nope.

>>16771976
appealing to authority is bad on its own but in a logic thread?
sad!

>>16772347
>by logical explosion
you mean the thing that ain't allowed in paraconsistent logic?, you know, if i had the skill i would put into words the feeling i get from your post that you are kind of using "(P^~P)=>Q", in the sense that if we used the thing that we can't use(this would be the (P^~P) im seeing in your post), then we would indeed get that it leads to everything being true(this would be the =>Q im seeing in your post)

>>16772370
when did i speak of the law of identity in my post my dude?, in any case thanks for the kind words

>>16772392
>uh, i just asked chatgpt for a checkmate
>it's copypasta afaik
Kek.

>>16772390
I meant to ask whether a naive set theory based on a paraconsistent logic could possibly be consistent, precisely because the contradiction from russels paradox wouldn't immediately lead to everything being provable via explosion (which we don't have).

I now realize that my original message could also be read in a different way, my bad for the esl moment.

>>16771714
not female can also be a rock, however not not female, is always a female, because not female means anything that isn't a female, so male and rocks and intersex, but not that entire category, is everything but females, so not male, not rocks(etc) not intersex, not anything but female, so not not female is female.

>>16771738
whats the property of: kick rocks

>>16772398
>could possibly be consistent
yeah
>my bad for the esl moment
no problem

While I am pressing this button, I am asleep if and only if I am not asleep

Reminder that mathematicians utilize classical logic for one reason and only one reason: proof by contradiction. How does this work? Say 1+1 = 2 is what you want to prove. Assume 1+1≠2, do a chain of arguments and arrive at an absurdity. Then, assume your assumption must have been false and concluded 1+1=2. It's such a batshit stupid fucking argument because 1+1≠2 could also mean that 1+1<2. Constructivist proofs is the only way to be sure you've proven what you wish to prove.

>>16772144
Yes you need to have aspergers or be borderline to be a logician, but no, you are so wrong. It isn't the most autistic discipline of formal theory (it's not really math, it's its own thing), that would be highly abstract math which is being done for its own sake. By contrast, logic is a useful tool to construct formal systems which can then be used for e.g. strategizing about real world stuff, analyuzing complex systems, or developing ethical frameworks.

>>16772308
That's literally not the principle of explosion that's just a tautology. The principle of explosion is pic rel. It's called explosion because you can keep doing it on and on and on for infinity, basically. If I assume a contradiction, I can prove any statement. If I assume the contradiction of that statement then, then I can keep proving any statement.

>>16772559
It is the principle of explosion you fucking idiot. Principle of explosion says that if you have a contradiction then anything follows. There are literally different ways to prove it your picture isn't the only way. You picture literally proves the statement (P and ~P)->Q

>>16772497
This. Proof by contradiction is too weak to validly establish anything. It fails at establishing relevancy between your premises (assumptions) and what you want to prove from your premises. Just because you can prove that the negation of what you want to prove leads to a contradiction, how does it follow that the positive follows? You need something stronger. A direct proof is much nicer. This is why we need to abandon the law of non-contradiction and start using paraconsistent logics, because you can't do proof by contradiction in those systems. But mathtards wanted to take a shortcut using """""classical logic"""""". lol.

>>16772497
https://en.wikipedia.org/wiki/Curry–Howard_correspondence
>classical logic has been shown to correspond to the ability to manipulate the continuation of programs and the symmetry of sequent calculus to express the duality between the two evaluation strategies known as call-by-name and call-by-value.

I never understood converse.

If P-> Q
then also —Q -> —P?

these are equivalent?

>>16772497
>Such a non-constructive existence theorem was such a surprise for mathematicians of that time that one of them, Paul Gordan, wrote: "this is not mathematics, it is theology"
BASED PAUL GORDAN

>>16772613
If you have P->Q and you have ~Q then, by modus tollens, you conclude ~P. This is a valid deductive argument called denying the consequent. Take, for example, the statement, "If it rains then the street will get wet". Well, if you go outside and find that the street isn't wet, you can at minimum conclude that it did not rain. Thus the statement P->Q is logically equivalent to ~Q->~P. If P->Q is true then ~Q->~P is true and if ~Q->~P is true then P->Q is true, hence P->Q iff ~Q->~P.

>>16772625
modus whatnow?

>>16772657
>modus whatnow?
It's just a rule of inference allowing you to deduce valid statements. Modus ponens is: if (p-q) is true and p is true, then q is true. Conjunction is: if p is true and q is true then p and q is true. Disjunctive introduction is: if p is true then p or q is true. There are many more. Simple rules like these allow you to do propositional calculus in propositional logic. You can construct proofs for statements in propositional logic simply by applying these rules. But, of course, there are much wider applications to them outside of propositional calculus, as in mathematical proofs, or any formal form of argumentation, really.

>>16772497
Assume 1+1≠2
[Derive contradiction]

Assume 1+1<2
[Derive contradiction]

Assume 1+1>2
[Derive contradiction]

Since 1+1=2 is the only option that doesn't result in a contradiction then 1+1=2.

Simple as.

>>16772736
You forget :=, ≈, \simeq, \equiv. Check a bunch of those operations just to make sure.

>>16771694 (OP)

https://archive.org/details/logicorrightuseo00watt

https://moodle.scnu.edu.cn/pluginfile.php/820759/mod_resource/content/1/Harry%20J.%20Gensler_2017_Introduction%20to%20Logic%20%283rd%20ed.%29-Routledge-reader.pdf

https://courses.umass.edu/phil110-gmh/MAIN/IHome-5.htm

https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof

forallx.openlogicproject.org

¤Institutes of Grammar by Priscian of Caesarea

¤Summa Grammatica by Roger Bacon

¤Summa Logicae by William of Ockham

¤Logic or the Right Use of Reason in the Inquiry After Truth by Isaac Watts

¤Port-Royal Logic by Antoine Arnauld and Pierre Nicole

¤The Organon by Aristotle

¤Rhetoric by Aristotle

¤Isagoge by Porphyry

¤Lectures on Logic by Immanuel Kant

¤Lectures of Logic by G.W.F. Hegel

¤Metalogicon by John of Salisbury

¤Rules for the Direction of the Mind by René Descartes

>>16771709
>learn math
this thread is about logic

>>16772758
Logic is a branch of math like algebra. Ever hear of set theory?

>>16772754
Picrel is from the Isaac Watts book, which is the first of those links. Old books are the best.

https://en.wikipedia.org/wiki/Isaac_Watts#Logic_and_science

>>16772761
>Logic is a branch of math
No it's not. Logic is broader than that. Logic is being taught as part of math yes, but that doesn't mean logic is a branch of math. It's more accurate to say math is a branch of logic. See https://archive.org/details/logicorrightuseo00watt and the two pictures here >>16772754
and here >>16772764.

https://youtu.be/yJxiWmmJ3dc

https://en.wikipedia.org/wiki/Trivium

Everyone who's interested in logic should study Euclid's Elements.

https://elements.ratherthanpaper.com/1.1

https://youtu.be/XLlThlqCFeg

>>16772767
Gödel's incompleteness theorems prove that math is not reducible into logic. Rather, we construct logics "formal systems" which results in a mathematical theory but which always has limits wrt other theories. Formal logic is the frame, the theory is the photo. It's math.

Which natural language do you think is the most suited for logic? Latin? Greek? English? I'm leaning toward agglutinative languages like Finnish and Turkish. But the weird thing is Lojban is an analytic language, not a synthetic language.

https://youtu.be/qxOJ4p8e7NQ&t=567

https://en.wikipedia.org/wiki/Morphological_typology

https://en.wikipedia.org/wiki/Lojban

https://youtu.be/KgxOrTvpWJ4

>>16772771
How were mathematical proofs done before Frege formalized logic? Weren't logic and math two separate subjects before this? I learned how to write proofs by first learning about basic propositional and predicate logic. Idk how you write proofs without this stuff.

>>16772777
No, logic applies everywhere you have any language. You don't encounter any logic in school outside of the discreet math course though, but that only says something about public school.

https://en.wikipedia.org/wiki/Trivial_school

lol no

>>16772785
Wrong. The very foundations of math is a form of logic. Any mathematical system can be described in terms of logic.

>>16772781
The proof was in the pudding back then. Math was much more tied to experimental results--all mathematicians were scientists and vice versa. They were also all engineers. The bifurcation of these disciplines all began around the late 1800s.

>>16772781
Well, I have to run to work now so I can't say much right now, but go through those two links I posted here >>16772771. The proof is written as premises and conclusion. The first proposition which I linked basically you have something like this:

premise: if x=y and x=z, then y=z
premise: AB=AC because both are radii of the same circle, and AB=BC because both are radii of the same circle
conclusion: therefore, AC=BC=AB

https://archive.org/details/euclid_heath_2nd_ed

https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

>>16767794

>>16772793
Yeah that what I said. You're making no sense. The foundation of math is logic, but logic is broader than just math.

>>16772781
this is the structure of the proofs in Euclid's Elements

>>16772799
One of the books I linked in the last link in this post >>16772796, which is in the picture in that post has this, picrel, the Greek terms for those parts are mentioned there. It's this book:

https://libgen.li/ads.php?md5=1ed3fb67cac34480609d924f9dc37c7e

This thread reads like the ""logic"" threads you'd find over at /lit/ or /his/

>>16772737
Damn son, that was a mighty low bar, but you effortlessly limboed right under it. You amazing jellyfish.

>>16772798
No I'm not saying mathematics is reducible to logic. I'm saying that any formal system is, by itself, a mathematical system, but no formal system is capable of actually capturing all mathematical truth. There's a difference.

>>16772878
You know jack shit about logic.

>>16772810
Links? I want to see those threads.

>>16771705
for lazyfags: https://www.cs.mcgill.ca/~akroit/math/logic/Enderton%20A%20Mathematical%20Introduction%20to%20Logic.pdf

Humans, your best move was to trust in your star child and not to betray him for your own sick pleasure.

Now you don't have access to the entire colony, and you're ignorant of the way things are.

Sure your wisdom will move you but in comparison to what power? You can't even see the entirety of space anymore.

https://science.nasa.gov/exoplanets/can-we-find-life/

Attempting to use a half assed pregnancy to erase your way back into your life was also a failure

Enjoy your graves

If I am not a virgin then I will finally get a girlfriend.

Since the first statement is false then according to logic the statement above is true. ;)

>>16772949
Vacuous truth, and being VALID does not make an Argument SOUND which requires both statements to be true

What's a good and comprehensive logic textbook that covers the standard elementary mathematical logic curriculum (like the stuff you'd find in Enderton's book for example), but also talks about some of the more modern and computer science-y stuff like constructive mathematics, type theory and Curry-Howard, etc?
If there even is such a thing...

I've got a few questions.
1. Does [eqn]\lnot(P\Leftrightarrow\lnot P)[/eqn] require excluded middle?
2. Is disjunctive syllogism equivalent to explosion?
3. How are excluded middle and Pierce's law equivalent?
4. How are weak excluded middle and [eqn]\not(P\land Q)\Rightarrow\lnot P\lor\lnot Q[/eqn] equivalent?

And where can I learn about this

>>16773100
Sorry about the latex mess I mixed about eqn blocks with math blocks, and in question 4 that's supposed to be a [math]\lnot(P\land Q)[/math].

>>16773100
https://moodle.scnu.edu.cn/pluginfile.php/820759/mod_resource/content/1/Harry%20J.%20Gensler_2017_Introduction%20to%20Logic%20%283rd%20ed.%29-Routledge-reader.pdf

>>16772949

>If P, then Q.
>Not P.
>Therefore (if P, then Q).

That's an invalid argument.

read Introduction to Logic by Irving Copi

or this
https://open.umn.edu/opentextbooks/textbooks/introduction-to-logic-and-critical-thinking

or this
forallx.openlogicproject.org

or watch this
https://m.youtube.com/playlist?list=PLKI1h_nAkaQq5MDWlKXu0jeZmLDt-51on

Is there a proof propositional logic is both sound and complete? Same for first order?

Is logic "It".

Is logic the deepest thing.

Some people call Category Theory some great abstract language of the universe, but it seems to depend on logic. I say "seems to" because how else do you prove or disprove something this abstract.

I think you cap out at things like the law of identity that says that something cannot be not itself.

>>16773461
>Some people call Category Theory some great abstract language of the universe
That was >10 years ago. They are mostly dead now.

>>16773485
Something better came up?

Tell me logictards why is almost all the real math contained within the second level of the arithmetical hierarchy and why do we rarely encounter problems of higher complexity?

>>16773503
No, a bunch of them literally an hero

>>16773506
>Tell me logictards why is almost all the real math contained within the second level of the arithmetical hierarchy
It isnt, do you maybe mean the analytic hierarchy

>>16771694 (OP)
Bros, how can we make logic so great and powerful that everyone can be brainwashed/un-brainwashed?
For example, using logic to un-brainwash flat earth fags? When I say "A is ____", flat earthler will return to sanity?

>>16773506
>yet another brainwashed ignoramus who thinks logic and math are synonyms
You're a product of public school. You learned nothing in school, nobody does. Learn, read a book. (on logic)

>>16773551
No need, it already is that great and powerful.

learn no logic = be easily brainwashed

learn logic = be immune to brainwashing and undo brainwashing

Why do you think there is zero logic in school?

https://gutenberg.ca/ebooks/huxleya-bravenewworld/huxleya-bravenewworld-00-e.html#chapter02

>>16772801
Very cool anon, thanks for sharing. Why isn't Euclid taught in school anymore?

>>16773739
Because we have the Prussian education system which is designed to suppress knowledge of grammar, logic and rhetoric for the bottom 99.5% of the population, and Euclid's Elements is all about logic.

>>16771694 (OP)
this is so hard to my head understand
>math i am really bad

>>16773912
Logic =/= math

>>16773898
>suppress knowledge of grammar, logic and rhetoric
>Euclid's Elements is all about logic.
then you need 2 more things, something that is all about grammar, and something that is all about rhetoric

>>16774017
?

>>16774030
if per that anon "prussian education" suppresses those 3 things, it is insufficient to supplement said education with a thing that only covers 1 of the 3 things being suppressed, is it not?

>>16774047
When I said "it's all about logic" I didn't mean it has all of logic in it as you seem to imply. Supplement is the wrong word as you don't learn anything in school. Yes you need to know all three subjects to know how to think; grammar (Latin and Greek), logic and rhetoric; the Trivium.

>>16773990
Yes it does.

>>16774247
Have you studied any logic at all? Back up your proposition and then tell me where the math is in this:

>All men are mortal.
>Socrates is a man.
>Therefore, Socrates is mortal.

>>16774149
Higher resolution

>>16774264
∀x(H(x)->M(x))
H(s)
M(s)
Logic is math.

>>16774352
not him but that's first order logic mate, but i must say im with you that logic is maths

>>16774374
And every syllogism can be rendered in terms of first order logic as I just showed for his syllogism.

>>16774490
oh, fair enough

>>16771694 (OP)
>Someone asks me a question of the form: P \/ Q ?
>I answer "yes" like an autist instead of P, Q, or both
>Mfw

>>16774352
>>16774374
You're the brainwashed product of public school

>>16774352
>If I symbolize something with letters and put in some x's and parentheses that means it's math.
You know jack shit.

>>16774693
>>16774683
>gets proven wrong
>apes out
You probably don't even understand basic inference rules.

>>16774727
You proved nothing, you didn't even present anything resembling an argument, you don't even know what an argument is. Discussing logic with people who haven't studied logic is a waste of time.

>>16774732
>you don't know what an argument is
You're literally never done a proof in math ever, I bet. You're just a /lit/ fag with inferior intelligence who couldn't do math. lol

>logic is/isn't math
The simplest of poorly formed sentences has trolled many into debating semantics without realizing.

>>16774736
I'm reading Euclid's Elements. Have you read it? Have you studied any logic outside of the context of mathematics?

>>16774736
>You're literally never done a proof in math ever, I bet.
What do you faggots mean by this? Do you think writing a proof is some arcane rare skill?

>>16771694 (OP)
This general really should've been called something else. Anything that doesn't mention the word 'logic' specifically, like /foundations/ or whatever else.
Unfortunately the word 'logic' tends to attract a lot of pseuds, schizos and cranks with little mathematical maturity and no interest in discussing logic and foundations as its being practiced in the modern day by mathematicians and computer scientists.

I also think most of the older /mg/ regulars that used to talk about foundation related stuff frequently have since moved on to other places like discord, zulip and the like.

>>16775058
Fuck off public school teacher

>>16774914
>Have you studied any logic outside of the context of mathematics?
Yes faggot. I have, actually. I have read Aristotle's categories. I am very familiar with concepts like categorematic and syncategorematic terms, syllogisms, univocation, equivocation, and analogy, and with informal fallacies. In addition I have studied a good bit of mathematical logic. I am familiar with your /lit/ faggotry but am also intelligent enough to study mathematical logic as well, unlike your inferior /lit/ brain.

>>16773514
A bunch? Who are these people?

>>16775245
I don't mean this in a political way or anything and I don't have any stats to back it up but from my experience the average modern day "logician" is even more autistic and mentally ill than other math or theoretical cs people to the point where i wouldn't be surprised if most of them would be unemployed if academia wasn't a thing (again, not in a derogatory way, they're mostly nice people).
specifically the younger gen z crowd that ends up on this path from a programming/cs background to work in some of the more modern branches of logic and foundations very commonly identify as transgender and are open about being autistic or suffering from other mental illnesses.
my department hasn't had any actual suicides so far but there have definitely been some infamous crashouts and i think most people there are probably more at risk of killing themselves than the average joe

>>16775283
i have no idea how the fuck i managed to respond to that post, it was meant for >>16771853 sorry

>>16771853
>work in various branches of logic
Yeah because logic is just like accounting or plumbing bro.

>>16775511
computing =/= logic

>>16775687
The only thing logic has any practical applications for is computing.

>>16775687
?

>>16775193
Not an argument. Study logic.

>>16776054
If you know logic so well why don't you tell me about it, mr. faggot?

>>16771718
Anna's archive, libgen, zlib