/sci/ - Science & Math

>>11842344
P -> Q I meant to write. Unicode didn't show

>>11842344
To prove [math]P \rightarrow Q[/math], you assume P and you have to show Q. Since in this case you know that [math]\neg P[/math], then you reach a contradiction immediately. This is often described as Q being vacuously true.

>>11842344
definition. I don’t like it either.

>>11842344
What you meant to type was [math]P \rightarrow Q [/math]

It’s just a quirk of our system of symbolic notation, really, something to differentiate the conditional from the biconditional (“if and only if”). It’s done for the practical purpose of simplifying logical statements rather than any semantic purpose relating to what the words represent.

>>11842344
Think of it this way: if it is raining, you will for sure use an umbrella. But you can use an umbrella even if it’s not raining (that doesn’t go “against” your conditional). That’s why the only “false” statement is the one where it rains but you’re not taking an umbrella, because you literally said you would if it rains. Hope this didn’t sound too confusing.

>>11842344
"P Implies Q" is only false for P True and Q False because the implication of P would reach the contradiction that Q is false when it is expected to be true, therefor the statement is false.
when P is false, no such implication holds, and Q can assume any value without influencing the value of P implies Q

>>11842344
if evidence doesn't matter
then anything goes

because logic is fake bullshit. just treat it as random systems used to do stuff. implying nothing more than the arbitrary rules set.

>>11842344
>if carrots are orange, then Trump is great


this is how retarded logic is.

>>11842353
How is the biconditional different?

>>11842391
>logic
material conditional

>>11842363
but if it’s not raining you have no information whether you carry an umbrella or not. undefined would be much more sensible linguistically with “implies”.

>>11842366
>therefore the statement is false
but it return true

>>11842344
Being French implies you're European, but just because you're not French it doesn't mean you can't be European.

>>11842406
the statement ''p implies Q'' is nor the statement not ''P'' nor ''Q''

Also the intelelctuals who made up the material implication said that what they wanted was that the only requirement was that something true cannot imply something false, so P=>Q must be false when P is true and Q false.
that's it.
Also learn history of logic.
There are lots of other ''logical implication'' created by intellectuals.

>>11842391
snowing implies it’s cold
not snowing implies it’s cold

>>11842406
for P True and Q False, P implies Q is false because P (True) is expected to imply Q to also be True, but reaches a contradiction with Q being false, thus returns False
As the frog said >>11842419 it is equivalent to the statement "(~p)Vq"

>>11842412
but you can be American.

>>11842419
i meant for the implication P to be false. It shouldn’t return True when it most definitely can be false.

>>11842392
Bi-conditional has the truth table
[eqn]
\begin{matrix}
P & Q & P \iff Q \\
T & T & T \\
T & F & F \\
F & T & F \\
F & F & F
\end{matrix}
[/eqn]
note that [math]P \iff Q[/math] is equivalent to [math](P \to Q )\wedge (Q \to P)[/math], hence the name.

>>11842392
Bi-conditional has the truth table [eqn]
\begin{matrix} P & Q & P \iff Q \\ T & T & T \\ T & F & F \\ F & T & F \\ F & F & T \end{matrix} [/eqn] note that [math]P \iff Q [/math] is equivalent to [math](P \to Q )\wedge (Q \to P)[/math], hence the name

>>11842481
Thx

>>11842412
That's more set theory (French \subset European) than logic. But I see how it applies.

>>11842344

"False" is the statement that does not have a proof. If you have "False => True," then you are allowed to assume "False," which means that False has a proof.

Here is the important point:

The definition of False is the null definition. If you have proven the statement "False," You have proven the statement " ". Formally, you have the propositional equality "False := ".

This definition makes sense because how can there be any theorem which is harder to prove than the theorem which gives you nothing to work with?

In order to prove "False," you must come up with a proof that can prove a theorem independently of anything that you are given. Hence, a proof of "False" also suffices to prove any other theorem.

>>11842481

> LEM

incredibly blue-pilled

>>11842534
>>
>"False" is the statement that does not have a proof.
i don't think this is true
with the LEM you can get a proof of false with P & not P.

>>11842504
It’s syllogistic logic, when the set theory is applied to form a formal argument

>>11842344
When in doubt,
Type P - > Q
As
(Not P) Or Q
Read one proof that theese are equivalent, convince yourself of it and just whenever you feel confused operate with the second equivalent expression.
If you want to negade implication just negate the second one.
I could get into the philosophical argument why theese can be conceptually understood as the same but people here did that before so here is just my tip for practical purposes.

>>11842614

which is why LEM simply does not work

What's a good book on logic?

>>11842403
You either carry an umbrella or you dont carry an umbrella.

>>11842698

Propositional and Predicate Calculus: A Model of Argument 2005th Edition
by Derek Goldrei

I just found this

Logic textbooks written by women

A conversation on twitter yesterday made me realize I couldn't name a single logic textbook written by a woman (other than my own draft book), where by "logic textbook" I intended to capture "book I could use as a primary text for an intro logic or advance logic course". Twitter to the rescue, I got lots of suggestions. So I've decided to collate them here. If there are any missing, please share in the comments.

Logic of Mathematics: A Modern Course of Classical Logic, by Zofia Adamowicz & Pawel Zbierski
Fundamentals of Symbolic Logic, by Alice Ambrose and Morris Lazerowitz
An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems, by Merrie Bergmann
The Logic Book, by Merrie Bergmann, James Moor, & Jack Nelson.
Philosophy of Logics, by Susan Haack (this couldn't quite serve as a primary textbook, but could definitely be a very heavily used secondary book).
Elements of Logic as a Science of Propositions, by Emily Elizabeth Constance Jones (Edinburgh: T. & T. Clark, 1890).
https://plato.stanford.edu/entries/emily-elizabeth-constance-jones/
Understanding Symbolic Logic, by Virginia Klenk
An Introduction to Symbolic Logic, by Susanne K. Langer
Structural Proof Theory, by Sara Negri & Jan van Plato
Proof Analysis, by Sara Negri & Jan van Plato
Dual Tableaux: Foundations, Methodology, Case Studies, by Ewa Orlowska & Joanna Golińska-Pilarek

>>11842735
An Algebraic Approach to Non-Classical Logics, by Helena Rasiowa.
Introduction to Logic and Critical Thinking, by Merrilee H. Salmon
The Logical Status of Diagrams, by Sun-Joo Shin
First Steps in Modal Logic, by Harold Simmons (nom de plume for Sally Popkorn)
A Modern Introduction to Logic (1930), by Susan Stebbing
Logic in Practice (1934), by Susan Stebbing
A Modern Elementary Logic (1943), by Susan Stebbing
Three Views of Logic: Mathematics, Philosophy, and Computer Science, by Donald W. Loveland, Richard E. Hodel, & S.G. Sterrett
What is Logic?, by Sara L. Uckelman

I also received a couple suggestions for linguistics books:

Mathematical Methods in Linguistics, by Barbara Partee
Semantics in General Grammar, by Irene Heim & Angelika Kratzer.
Introduction to Natural Language Semantics, by Henriette de Swart.

Finally, someone else mentioned this, which isn't quite logic, but since it's logic-adjacent I'll happily include it:

A Philosophical Introduction to Probability, by Maria Carla Galavotti

>>11842344
The law says that if you steal (P), you go to jail (Q). However, that doesn't mean you can't go to jail without having committed theft.
If you went to jail without stealing (say, for murder) that doesn't mean the law of P->Q isn't true.

>>11842735
>>11842736

I don't understand why you are posting this irrelevant information

Any book that is not "Homotopy type theory: univalent foundations for mathematics" is archaic

>>11842768
remind if the russian guy died of natural causes or suicide

>>11842792

voevodsky died of an aneurysm

>>11842743
>>11842363
>>11842481
Best explanations

If you ace the exam, you'll get an A

However, you can still not ace the exam and still get an A (by perhaps doing well in other assignments, midterms etc.)

If you go to college, you will be successful. But there are also other ways to be successful that do not require college.

In either case, because the promise was not broken (because you never achieved the premise of the promise to begin with) it doesn't matter what the outcome is.

>>11842713
and if it’s not raining that statement tells you I carry one, which is retarded.

>>11843555
you can ace the exam and still get an A. But if you don’t ace the exam you’re not sure you get an A.

>>11842352
/thread

>>11842344
-1=1 which is false implies (-1)^2 = 1^2 which is true

it makes sense

>>11842649
You're retarded, you don't need LEM to prove that [math]A[/math] and [math]\neg A[/math] imply False.

>>11843920

Lmao chief, clearly you lack experience

there is a difference between accepting LEM as an axiom and proving the statement LEM => False

>>11843920

Lmao chief, clearly you lack experience

there is a difference between accepting LEM as an axiom and proving the proposition "LEM => False."

>>11842542
what part of this post uses LEM? >>11842481
The construction of the truth table for "P => Q"?

>>11842344
What a brainlet.

>>11842344
There's a bunch of good answers in this thread already, but here some remarks.
You can take this truth table as just the definition of the implication and roll with it.
The fact that this implication (the material implication) is a bit odd is an acknowledged situation.
Here's one reference for a few it's consequences that put people at unease
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication
and there's some more.
My favorite is the following tautology of first-order logic
[math] \exist y.\,\left( B(y) \implies \forall z.\,B(z) \right) [/math]
In words (with semantic for B), this reads:
>(If there exist anything at all then) There is a thing, such that if that thing is a bird, then everything is a bird.

Proof: Let's grant that there's these two options: Either everything under consideration is a bird, or, on the opposite, there is something which is not a bird. In the first case, it's true that everything is a bird by assumption, and the statement is wrong. Now, in the second option, if there something that's not a bird, then if that thing were a bird, we'd have a contradiction and anything following (in particular this statement)

What happens here is also what happens in your original question:
>Why is "P implies Q" true when P is false and Q is true?
It's true by definition, and what motivates it is the principle of explosion: If you can prove falsehood, then you can prove anything.
The mathematical motivation - and the reason why this weird material implication => still gets us far in mathematics, is that explosion comes for free in arithmetic:

Note that there are alternatives to =>, i.e. alternative logics.
e.g.
https://en.wikipedia.org/wiki/Relevance_logic
Such things are more studied in the philo department than in math ones.
If 0=1 (the most basic falsehood), then 0=0+0+0=1+1+1=3 etc.
If you have 0=1, then you have n=m for any n and m.

At one point I tried to motivate this whole game here

https://youtu.be/eeLa9tIhFMs

>>11842344
There's a bunch of good answers in this thread already, but here some remarks.

You can take this truth table as just the definition of the implication and roll with it.
The fact that this implication (the material implication) is a bit odd is an acknowledged situation.
Here's one reference for a few it's consequences that put people at unease
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication
and there's some more.

My favorite is the following tautology of first-order logic
[math] \exists y.\,\left( B(y) \implies \forall z.\,B(z) \right) [/math]
In words (with semantic for B), this reads:
>(If there exist anything at all then) There is a thing, such that if that thing is a bird, then everything is a bird.

Proof: Let's grant that there's these two options: Either everything under consideration is a bird, or, on the opposite, there is something which is not a bird. In the first case, it's true that everything is a bird by assumption, and the statement is wrong. Now, in the second option, if there something that's not a bird, then if that thing were a bird, we'd have a contradiction and anything following (in particular this statement)

What happens here is also what happens in your original question:
>Why is "P implies Q" true when P is false and Q is true?
It's true by definition, and what motivates it is the principle of explosion: If you can prove falsehood, then you can prove anything.
The mathematical motivation - and the reason why this weird material implication => still gets us far in mathematics, is that explosion comes for free in arithmetic:
If 0=1 (the most basic falsehood), then 0=0+0+0=1+1+1=3 etc.
If you have 0=1, then you have n=m for any n and m.

Note that there are alternatives to =>, i.e. alternative logics.
e.g.
https://en.wikipedia.org/wiki/Relevance_logic
Such things are more studied in the philo department than in math ones.

At one point I tried to motivate this whole game here

https://youtu.be/eeLa9tIhFMs

There's a bunch of good answers in this thread already, but here some remarks.

You can take this truth table as just the definition of the implication and roll with it.
The fact that this implication (the material implication) is a bit odd is an acknowledged situation.
Here's one reference for a few it's consequences that put people at unease
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication
and there's some more.

My favorite is the following tautology of first-order logic
∃y. (B(y) ⟹ ∀z. B(z))
In words (with semantic for B), this reads:
>(If there exist anything at all then) There is a thing, such that if that thing is a bird, then everything is a bird.

Proof: Let's grant that there's these two options: Either everything under consideration is a bird, or, on the opposite, there is something which is not a bird. In the first case, it's true that everything is a bird by assumption, and the statement is true. Now, in the second option, if there something that's not a bird, then if that thing were a bird, we'd have a contradiction and anything following (in particular this statement)

What happens here is also what happens in your original question:
>Why is "P implies Q" true when P is false and Q is true?
It's true by definition, and what motivates it is the principle of explosion: If you can prove falsehood, then you can prove anything.
The mathematical motivation - and the reason why this weird material implication => still gets us far in mathematics, is that explosion comes for free in arithmetic:
If 0=1 (the most basic falsehood), then 0=0+0+0=1+1+1=3 etc.
If you have 0=1, then you have n=m for any n and m.

Note that there are alternatives to =>, i.e. alternative logics.
e.g.
https://en.wikipedia.org/wiki/Relevance_logic
Such things are more studied in the philo department than in math ones.

At one point I tried to motivate this whole game here

https://youtu.be/eeLa9tIhFMs

>>11842344
If P implies Q, Q can't be false when P is true, while Q can still be true with P being false because something else may imply Q.

>>11844150
I'm not sure if it helps starting with
>If P implies Q
and not make use of it

>>11844162
What are you talking about?

>>11844166
You made a point for "P=>Q" being false in the case it read "true => false", but "something else may imply Q" while P is whatever doesn't sound like an argument related to P=>Q.

>>11843716
that has literally nothing to do with conditional. it is literally a promise. if p happens, then q will happen. thats all it says.

>>11844207
The fact that something other than P may imply Q is why P being false and Q being true does not contradict "P implies Q".

>>11844067

The whole idea of truth tables is based in LEM.

The implication symbol is not a boolean function. It does not have a truth table.