Anonymous at Thu, 13 Feb 2025 08:08:22 UTC No. 16584433

X/0 = infinity

Anonymous at Thu, 13 Feb 2025 08:13:02 UTC No. 16584437

>>16584428
>IF P
>NOOO BUT WHAT IF NOT P
That's not the question. If not P then make a new statement. An implication is just asking if Q is keeping its end of the bargain, so to speak

Anonymous at Thu, 13 Feb 2025 08:19:17 UTC No. 16584440

>>16584437
>If not P then make a new statement
Then Why was it defined anyway? Not only that, Why define it as truth, when it may as well be defined as false, or better yet, undefined?

Anonymous at Thu, 13 Feb 2025 08:30:54 UTC No. 16584451

>>16584433
If x/0=1, then 2(1/0)=2
This is true by definition.
QED.

Anonymous at Thu, 13 Feb 2025 09:18:16 UTC No. 16584476

Coq defines X/0 = 0 because it can't into partial functions. Still better than working in ZFC.

Anonymous at Thu, 13 Feb 2025 11:22:43 UTC No. 16584587

>>16584428
The [math] F \to T \therefore T [/math] evaluation is sometimes referred to as "vacuously true".

"Every unicorn is immortal" is true because there doesn't exist a single mortal unicorn. This is true there doesn't exist a single unicorn of any kind.

Anonymous at Thu, 13 Feb 2025 12:32:26 UTC No. 16584689

>>16584428
>How come they don't call this undefined like they do division by zero?
They do in non-classical logic. They don't in classical logic to keep it binary, because it's useful.

Anonymous at Thu, 13 Feb 2025 12:33:37 UTC No. 16584691

>>16584587
>"Every unicorn is immortal" is true because there doesn't exist a single mortal unicorn.
It is also false because there doesn't exist a single immortal unicorn.

Anonymous at Thu, 13 Feb 2025 15:03:10 UTC No. 16584810

>>16584689
>to keep it le useful

This is by far the most bullshit thing I've come across in math so far. This is more bullshit than complex numbers, it is completely made up, puts on a self-aggrandizing name like "Classical Logic" as if it had come from Plato and it's plain and simple a goddamn arbitrary convention.

You know what, I retract my positon that math is discovered. Math is fucking invented.

Stop guessing start learning at Thu, 13 Feb 2025 15:29:21 UTC No. 16584829

>>16584587
Wasn't this goodles whole completeness theorem? First order logic or some shit?

Anonymous at Thu, 13 Feb 2025 18:43:14 UTC No. 16584967

>>16584691
The existence of an immortal unicorn is not required for "if a unicorn exists, it must be immortal" to be true.

Anonymous at Thu, 13 Feb 2025 18:44:26 UTC No. 16584969

>>16584810
Hey dude, just admit you suck at math and leave it at that. Nothing to be ashamed about.

Anonymous at Thu, 13 Feb 2025 18:57:41 UTC No. 16584981

>>16584829
Material conditionals (statments like [math]U(a)\toI(a)[/math]) are just propositional logical statements, which is a subset of first order logic. The "All unicorns are immortal statement" is just a material conditional wrapped in a first order. You'd write it out formally like [math]\forall a U(a) \to I(a) [/math]. Literally for all things "a" if "a is a unicorn" then "a is immortal".

Godel's incompleteness theorem has nothing to do with vacuously true material conditionals, but says something about the assumptions we make before we start doing logic. Basically any set of assumptions we make at the start will either yield a statement which cannot be proven true or false, or the assumptions will yield a contradiction and they're meaningless.

Anonymous at Thu, 13 Feb 2025 18:58:43 UTC No. 16584983

>>16584981
> wrapped in a first order **quantifier**
oops

Anonymous at Thu, 13 Feb 2025 22:40:08 UTC No. 16585209

>>16584428
@OP I'm sorry for being mean earlier. Here's another way to think about it.

The statement "My lawn is wet whenever it rains" is true whether you say it during a rainstorm or the middle of a drought. But the lawn could just as easily be wet when it's not raining in the event that I spray said lawn down with a hose. I'm still technically telling the truth if I say "my lawn is wet whenever it rains" while hosing down my lawn in the middle of a drought.

"rain [math] \to [/math] wet lawn" says nothing about the particular truth value of whether it's currently raining now. It only describes the *relationship* between rain and wet lawns. So when we say "rain [math] \to [/math] wet lawn" is true, we are only stating that this line of reasoning is sound.

Anonymous at Thu, 13 Feb 2025 23:29:08 UTC No. 16585268

>>16584691
Why is it false? Can you show me a unicorn that is not immortal?

Anonymous at Thu, 13 Feb 2025 23:50:43 UTC No. 16585286

>>16584428
>How come they don't call this undefined
The motivation is for the connective to be truth-functional, so it can be substituted for t or f for any subformula. You'd need more than truth tables to represent a subjunctive conditional

Anonymous at Fri, 14 Feb 2025 00:13:14 UTC No. 16585299

>>16585268
Occam's razor.

Anonymous at Fri, 14 Feb 2025 00:18:09 UTC No. 16585304

>>16585299
>pseud's razor

Anonymous at Fri, 14 Feb 2025 00:26:57 UTC No. 16585313

>>16584810
>if it had come from Plato
Plato did use inferences in his dialogues, IF you accept X proposition, then you also accept Y absurd proposition, and since you reject Y proposition, you also reject X.
>plain and simple
I don't know why you would think ancient greek philosophy would be simple

Anonymous at Fri, 14 Feb 2025 00:37:57 UTC No. 16585324

>>16585313
Yes, and Plato's logic also came to be little more than sophistry.

Anonymous at Fri, 14 Feb 2025 01:57:48 UTC No. 16585371

>>16585209
>>16584587
Ok. Let's keep it simple and self-evident.

The figure is red
Therefore, the figure is 71.

>Hurr durr! It's true! You are completely right!

There is absolutely no way of knowing what the figure is because it's not colored. It could be any figure, any number, as undefined as x/0. I know what the figure is for sure, and we can try every case until we guess the true one by mere chance. If I went with probability, I say you have more chance of guessing wrong than guessing right, therefore, I call the statement false.

Why was it defined true?

Anonymous at Fri, 14 Feb 2025 04:35:50 UTC No. 16585435

>>16585371
You are misreading the material conditional.

You're writing it "The figure is red. Therefore the figure is 71" But the material conditional doesn't say that the figure was red. The material conditional simply states that if the figure were red, it would be 71. This conditional could be true or false, so your inductive reasoning is useless.

Moreover, what do you think is more likely? That the last 2400 years worth of logicians used the wrong convention against all sense? Or that you are simply failing to understand the concept because you're slow, or because you picked a hill to die on before you studied the topic?

Anonymous at Fri, 14 Feb 2025 05:36:59 UTC No. 16585477

>>16585435
>Could be true or false
It's literally "true" according to the truth tables posted in OP.

Why is it defined as true when it's completely counterintuitive and pants on head retarded is what I want to know.

>You are wrong because tradition!
>Implying this shitty formal logic system was invented more than 150 years ago

Anonymous at Fri, 14 Feb 2025 08:18:33 UTC No. 16585547

>>16585371
It makes it possible to say things like "for every number in this picture, if the number is red, then the number is 71."

Anonymous at Fri, 14 Feb 2025 13:34:26 UTC No. 16585724

>>16584428
I make a promise to someone that if they prove OP is not a faggot, i will give then a million dollars.

OP not a faggot million dollars
p q

They did not prove OP is a faggot, obviously, you are a huge faggot. p is false. I did not give them a million dollars. q is false.

Did i break my promise? No, therefore it wasn't a false promise. Would i have broken my promise if i gave them a million dollars anyways? No. Therefore, the implication is always true if p is false.

Anonymous at Fri, 14 Feb 2025 13:35:27 UTC No. 16585725

>>16585724
4channel ate my fucking arrows i want a refund

Anonymous at Fri, 14 Feb 2025 16:27:58 UTC No. 16585816

>>16584587
You don't know that. One of your precious stars might have a planet rotating around it with unicorns and some might be immortal while others aren't

Anonymous at Fri, 14 Feb 2025 16:47:23 UTC No. 16585832

>>16585724
I like that explanation.

Anonymous at Fri, 14 Feb 2025 17:12:05 UTC No. 16585849

>>16584428
because propositional logic and natural language don't mix

Anonymous at Fri, 14 Feb 2025 18:57:34 UTC No. 16585921

>>16584428
Maybe a different perspective from type theory can help.

First, we need to clarify what an assertion is. An assertion is an statement of the form 'p is true', where p is just a formula. In order to be able to state that 'p is true' you need two things: the formula p (obviously), and _a proof_ that p is true. Keep this in mind.

Now, usually formulas are not atomic as in just 'p'. In practice, they are made up of smaller formulas put together through 'conectives': and, or, implies. There are rules that relate these conectives to assertions (read: proofs). Here are the rules.

* **Conjuction (and)**: When is the statement 'p and q' true? If you know your logic tables, you will know that 'p and q is true' holds when 'p is true' and 'q is true'. In terms of proofs, this means that to obtain a proof of 'p and q' you need **two** things: a proof of p _and_ a proof of **q**.

* **Disjunction (or)**: Similarly, in order to assert 'p or q', you need **either** a proof of 'p is true', _or_ a proof of 'q is true'.

* **Implication (implies)**: This is where it gets funky. When is 'p implies q' true? Well, a proof of 'p implies q' usually goes like this: you assume that p is true, then apply some kind of reasoning to conclude that q is true. This means that if 'p implies q is true', you have a procedure to get a proof of q from a proof of p. Do you see that this is _basically_ a function (a computer function, if you prefer) that receives a proof of p to obtain a proof of q? Let me show you some examples.

Consider the proposition 'n is a prime number greater than two implies n is odd'. To prove this, we use the function analogy. As our arguments to our function, we receive an integer 'n', a proof that 'n is prime', and a proof that 'n is greater than two'. We need to return a proof that 'n is odd'.

Anonymous at Fri, 14 Feb 2025 18:58:36 UTC No. 16585922

>>16585921
(continue)
Another example: 'n is either 1 or 2 implies n^2 is either 1 or 4'. To prove this, as arguments we receive 'n', a proof that 'n' is either 1 or 2, and we need to return a proof P that 'n^2' is either 1 or 4. How would you construct this function? I think the easiest way is by cases (a piecewise function) since there are two cases for P: if P is a proof that n is 1, give one proof. If P is a proof that n is 2, give another proof. Easy.

The final example: 'p implies q is true' when we know that p is false. To prove this, we receive as the single argument a proof P that 'p' is true. Just like before, we can construct a piecewise function by cases on P. How many possible cases are there for P? None! Because we already knew that p is false, so we will never get the situation where we would receive a proof of p is true as an argument. So, our piecewise function has zero cases to check, so we are... done? Yes, we are, we have constructed our function.

All in all: you can always obtain a proof of 'p implies q' when p is false. But this should not bother you anymore: saying that 'p implies q' is the same as saying that you have a function that transforms proofs of p intro proofs of q. But you cannot ever use that function anyway, since there are no proofs of p. So no, 'p impiles q' when q is false does not mean that everything is true, because you would need a proof of p, and by that point you would already have that p is both true and false, a contradiction, so nothing makes sense anymore.

Anonymous at Fri, 14 Feb 2025 20:02:15 UTC No. 16585983

>>16584428
"p -> q" is just an abbreviation for "not (p and not q)" and the problem vanish.

Intuition about implication comes from *quantified* logic where (in common language) if A(x) and B(x) are prooperties depending on the variable x, "A implies B" means in fact * forall x, not (A(x) and not B(x))* (example: being human and pregnant implies being a woman: you set A(x):= "x is human and pregnant" and B(x):= "x is a woman").

On a side note, "forall x P(x)" is also an abbreviation, for "not (exists x, not P(x))" ("P has no exceptions"); thus no more incontrollable fears about so-called vacuum truth (the sentence "(forall x, not V(x)) -> (forall x, (V(x) -> W(x)))" (just removes abbreviations as above and see for yourself.)

Anonymous at Sat, 15 Feb 2025 18:52:48 UTC No. 16586972

>>16584433
>why can't [math]\frac{x}{0}=\pm{\infty}[/math]?
Because, even in non-standard analysis, where there are non-zero infinitesimals and infinitely large numbers, [math]\left|\frac{1}{\textrm{Large}}\right|[/math] still isn't [math]0[/math] but a strictly greater than [math]0[/math] infinitesimal and [math]\left|\frac{1}{\textrm{Infinitesimal}}\right|[/math] still isn't [math]+{\infty}[/math] but an infinitely large number strictly less than [math]+{\infty}[/math].

Anonymous at Sat, 15 Feb 2025 20:21:02 UTC No. 16587107

>>16584428
Close your mind, you arrogant freak, lest you become a schizo

Anonymous at Sat, 15 Feb 2025 20:50:09 UTC No. 16587167

>>16584428
It's also unrelated to causation implied by natural language.

Anonymous at Sun, 16 Feb 2025 06:10:12 UTC No. 16587627

In standard propositional logic ("zeroth" order logic) the only models are assignments of truth values to propositions. Since we need the axioms:

1. A -> A
2. A -> (B -> A)

to be true in all models, any assignment of truth values to A and B above must always evaluate to true. In particular, if A is false, then in order for 1 to be true, false -> false must evaluate to true. And if A is true, and B is false, then the only way to make axiom 2 hold is to make false -> true statements be true.

This is a consequence of the simple semantics we have for standard logic.

Systems of logic such as modal logic (with necessary implication), and relevance logic, exist and have more complicated semantics. In modal logic □(A -> B) need not hold when A is false and B is true. However if A is necessarily false, then such statements will hold. On the other hand, relevance logic requires that implications are *relevant* and this prevents even necessarily false statements from having arbitrary implications. Models for relevance logic are considerably more sophisticated than those of standard logic.

You will be amused to know that, typically, it is relevance logic that is considered "not real logic" because it allows dialethia (statements which are both true and false) without explosion.

Having said all that, in most cases standard logic is perfectly adequate and, due of its simple semantics, is easier to work with.

I hope this helps.