/sci/ - Science & Math

>>12213341
[math]A \rightarrow B[/math] is the equivalent of the operation ~A ∨ B. You can have a truth table for the last one, right?

>>12213341
Why WOULDN'T it have a truth table?

truth tables are problematic semantics

>>12213355
This
A therefore B is equivalent to Not(B and Not (A)) is equivalent to not(B) or B
In other words, A therefore necessarily B

If not has a table, or has a table, and has a table, then therefore must also have one

>>12213370
Not(A*) or B

>>12213355
I know its truth tables match, but how does that make sense? Why is "not A or B" equivalent to "if A then B"? Is there any intuition that can help harmonize the two?

>>12213359
Because it represents a proposition. "If A then B" is not something that inherently has a truth value to me. Or I just don't get why it makes sense.

>>12213376
Because you’re trying to apply natural meanings to logical symbols
The truth table is exactly the meaning of ->, the table for /\ is the exact meaning of /\
These are not representations of natural language, these are tools to represent particular boolean relationships
Once you practice them a little the equivalence of the tables will come to you naturally

>>12213364
What do you mean by that?

>>12213370
So is the implication operator just a made up operator as a shortcut for a specific combination of AND and NOT that is sometimes useful/convenient? Does it have no meaning beyond that?

>>12213341
because A and B can take on two states and they are related with a proposition?

>>12213404
Ex falso quod libet

>>12213403
>The truth table is exactly the meaning of ->, the table for /\ is the exact meaning of /\
>These are not representations of natural language, these are tools to represent particular boolean relationships
Then why is it called "implies", when an implication in the real world is quite different? We call it the AND operator because it behaves like a natural language "and". We call it the XOR operator because it behaves like the natural language "either or". Why do we call it the IMPLIES operator when apparently it does not behave like what we expect an implication to be in natural language? It just doesn't make sense to me.

>>12213423
Different anon
Have you considered that the natural usage of implies is actually a bastardisation of the logical usage of implies?
And that you encountered the “natural” version first, so the logical one seems incorrect
Sort of how retards say “it’s just a theory”

>>12213422
A: "2+2=5" (aka false)
B: "my dog is gay" (either true or false)

A and B <=> true if B is true, false otherwise
Makes sense because both statements have to be true in order for AND to be consistent with its natural language meaning

A -> B <=> "my dog is gay"
How does this make ANY sense at all?

>>12213435
>Have you considered that the natural usage of implies is actually a bastardisation of the logical usage of implies?
Natural language existed before Boolean logic was formalized. We have assigned the "implies" operator a truth table and a name that hopefully represents its meaning in an intuitive way. It just doesn't click for me.

>>12213449
>Natural language existed before Boolean logic was formalized
Yes. And in 1900 a computer was a human doing arithmetic, not a machine

>>12213376
>Because it represents a proposition. "If A then B" is not something that inherently has a truth value to me. Or I just don't get why it makes sense.
You can think of it as
I tell you, if you have a, then you must have b. Did I say the truth? Well that depends on A and B. If you have A and then you have B, then I told the truth. If you don't have A but you have B, then I still said the truth, since I never told you, you cannot have B without A, just that if you have A, then you must have B. Now if you don't have A and don't have B, did I say the truth? Of course I did, since I didn't say anything about B if you don't have A, only that if you do have it, then B must be true. Finally, if you have A but you don't have B, then did I say the truth? No, I lied since I promised you that if you have A you must have B, but that didn't happen.

>>12213498
Okay, that makes a lot of sense. Thank you, anon!

>>12213506
np, now just to go a bit further, imagine if I were to put all that babble into mathematical form. the four case presented become:

A and B (if a is true and b is true, then I said the truth)
~A and B (if A is not true and B is true, then I said the truth)
~A and ~B (if A is not true and B is not true, then I said the truth)
and finally
~(A and ~B) (If A is true and B is not true, then I lied, represented by the ~ in front of the parenthesis)
Now you combine all of those conditions:

(A and B) or (~A and B) or (~A and ~B) or ~(A and ~B)

Now, if you were to simplify the above expression, what do you end up with? you end up with >>12213355

>>12213513
This all ties it up really nicely. Thanks! I dont know why such a thorough explanation isn't featured in more books/learning resources.

>>12213341
literally anything can have a truth table. they are not special

>>12213376
>any intuition that can help harmonize
Real example?
If it's raining then the ground is wet.
That means combinations of rain and wet ground allowed are:
1. Raining and wet ground (A&B)
2. dry ground and not raining (-A&-B)
3. wet ground and not raining (-A&B)
which together are truth equivalent to "the ground is wet or it's not raining" (B or -A)
and raining but dry ground (A&-B) is false.

>>12213376
f -> f :
2>5
2+1>5+1

f -> t :
-3 = 3
(-3)^2 = 3^2

>>12213341
Because if you cannot make a truth table for the simplest case, you cannot work with truth tables. Also, this one in particular is rather confusing for starters because they have misconceptions about A implies B when A is false.

>>12213341
Material implication is not identical to the meaning of "implies" in natural language. For example, the word "implies" may suggest a causal relationship in natural language. Material implication does not require any causal relationship.
In short, material implication has a very specific technical definition, and misunderstandings of it are sometimes called "paradoxes of logical implication."

>>12213438
If A then B has a hidden phrase for the precedent. Its, if A is true, then B is true. So if something false is true, then everything can be true. I do agree with you that it is arbitrary and could easily be the other way (if something false is true, then everything is false). Thats why we make a distinction between classical and nonstandard logic

>>12216477
Gibberish.

>>12216477
>So if something false is true, then everything can be true. I do agree with you that it is arbitrary and could easily be the other way (if something false is true, then everything is false)

These are classically the same.
If everything becomes true true, then P is true and notP is true.
If everything becomes false, then P is true and notnotP is true as well.

>>12215131
Basically what this guy said. There are [math]2^{2\cdot 2}=16[/math] binary boolean truth functions and one of them we happen to call "implies".
See
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication

There are many other alternatives for the logic of entailment, see
https://en.wikipedia.org/wiki/Relevance_logic
But these of course all have much harder semantics than plain boolean predicate logic.

>>12216477
>So if something false is true, then everything can be true. I do agree with you that it is arbitrary and could easily be the other way (if something false is true, then everything is false)

These are classically the same.
If everything becomes true, then P is true and notP is true.
If everything becomes false, then not P is true and notnotP is true as well.

>>12215131
Basically what this guy said. There are 22⋅2=16 binary boolean truth functions and one of them we happen to call "implies".
https://en.wikipedia.org/wiki/Boolean_function

There are many other alternatives for the logic of entailment, see
https://en.wikipedia.org/wiki/Relevance_logic
But these of course all have much harder semantics than plain boolean predicate logic.

See also
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication

>>12213498
Nice analogy. To make it easier to go through these statements, you can think of the implication, If you are drinking tea then you must have a container that contains the tea.

>>12213341
"p implies q" is equivalent to "not(p and not q)"
You can interpret that as "don't allow to derive falsehoods from truths"
You can expand it as "not p or q", which if we are given that p true, then q has to be true, if we want the expression to be true.

>>12213341
"If the moon is made of cheese, then Paris is the capital of Oklahoma" is apparently true due to this definition of material implication
I think there's no further proof needed that propositional logic is a complete joke

>>12213341
You should remember that it doesn't also means that B -> A, which a lot of people, especially ones larping as the doctor helpers often forget to notice.

There's also inverse implication, which means if there is no A, there's B, but if there's B there can also be A, which most people doesn't clearly get without truth table.

Even I was so smart I expected everybody on the planet to know this, but planet has failed me, but it was not her fault.

>>12217511
Pardon, what's in Olejar's textbook as inverse impication is arrow back, and it means that if there's A, there was B, if there was B, there haven't got to be A.

I'm doomed but I don't really... In that book it's not completely well written, because that's just implication with swapped members, whatever.

In implications we have 3 different inverse implications which forms triangle somehow to be itself inverses.

Maybe I haven't seen truth table.

But in logic that takes maybe into account, how's implication solved?

>>12217522
Determining that problem if A -> maybe B is still implication is important. IMHO it doesn't implies,... If maybe then yes, is proper in implication in those case to not being rude and not make false advertising.

>>12213341

When you have "p implies q," what that really means is "everything besides p = T and q = F is sound."

>>12213341
[math] p \rightarrow q [/math]

>>12218144
[math] F \rightarrow T [/math]
is true