Anonymous at Sun, 14 Jul 2024 04:04:22 UTC No. 16280993

>>16280954
Go ask philosophers, mathematicians and comp sci guys do rudimentary logic unless they’re profs. Scientists don’t do logic unless they’re strapped to a cross and forced by a domina with a whip to do so.

Anonymous at Sun, 14 Jul 2024 04:46:21 UTC No. 16281010

This is a total mindfuck because normally the vacuous truth of a single conditional (P->Q true when P is false) is something intuitively acceptable, but here the logical equivalence seems completely unaligned with intuition (mine, at least).
On the other hand, the equivalence between (P v Q) -> R and ((P -> R) ^ (Q -> R)) aligns with intuition, and at first I thought this is what was intended and there was some mistake in the image.

Anonymous at Sun, 14 Jul 2024 04:57:56 UTC No. 16281012

>they look the same when you ignore their differences
Both accurately can describe a situation.
Just pick the correct one, it's literally the easiest part of the whole logic thing.

Anonymous at Sun, 14 Jul 2024 05:04:48 UTC No. 16281015

>>16281012
they are the same in the classical logic.
shalom.

Anonymous at Sun, 14 Jul 2024 06:47:22 UTC No. 16281067

>>16281010
This isn't really about vacuous truths, it's about contrapositivity. The conclusion becomes more intuitive when you rephrase it as
>One of the following is true:
>1. If bulb R is off, then switch P is open.
>2. If bulb R is off, then switch Q is open.
The hint that you should be looking at negation lies in recognizing that the given equivalence is a simple generalization of de Morgan's law !(P v Q) = (!P)^(!Q), which is the special case of R=0.

Anonymous at Sun, 14 Jul 2024 07:16:56 UTC No. 16281090

>>16280954
The logical expression and the thought experiment that OP presents here are not equivalent.
1. If the equivalence was the case, the light bulb could still be turned on even if the switches aren't closed.
2. The technical diagram requires both p and q to be closed in order for the light bulb (r) to be turned on. There is no way for r to be true when p and q aren't true. So, what the diagram conveys isn't (p ∧ q) -> r
It's (p ∧ q) <-> r.

Anonymous at Sun, 14 Jul 2024 08:12:40 UTC No. 16281120

>>16281067
these paradoxes get eliminated in logics that have a conditional without vacuous truth but that retain contrapositive equivalence.
those contrapositive rewrites don't really help all that much. the natural understanding is "if bulb R is off, then switch P or switch Q (or both) is open," but that's not what those rewrites are saying. they're directly naming the switch that's open, which is different. if you give me "if bulb R is off, then switch P is open" as true, I can rightly ask "how do you know it's switch P and not switch Q that's open?" if you give me "if bulb R is off, then switch Q is open" as true, I can rightly ask "how do you know it's switch Q and not P that's open?" if you give me both "if bulb R is off, then switch P is open" and "if bulb R is off, then switch Q is open" as true, then by conjunctive elimination (X ^ Y -> X), I'm allowed to isolate one of the conjuncts as true and then ask the same kind of question.

>>16281090
you're arguing from the real world. no shit that it makes no sense in the real world. that's the point of a paradox.

they are equivalent in formal (classical) logic, which has a material conditional (i.e., one with vacuous truth). his series circuit is the typical demonstration of the paradox.

there are non-classical logics that attempt to be more consistent with the real world (which mostly means real-world uses of language), and one way they do this is by replacing the material conditional with one in which vacuous truth doesn't hold (or holds in only uncontroversial cases).

🗑️ Anonymous at Sun, 14 Jul 2024 08:32:25 UTC No. 16281127

>>16281120
>that's the point of a paradox.
This isn't a paradox. A paradox involves an inconsistent, self-contradicting and impossible result that is the logical outcome of some premises and an axiomatic system. This example here, however, is just poor modelling.
>they are equivalent in formal (classical) logic
Not exactly sure why this would or should be such a conundrum?
If r is true when p and q both are true, it follows that r can still be true, when, for example, q is false. All it says is that r simply can't be false when both p and q are true.
Vice versa, if r is true if either p or q are true, then r is true when p and q are true. Simultaneously, when p and q now happen to be true but r is false, then p or q would be false too.

Anonymous at Sun, 14 Jul 2024 08:33:58 UTC No. 16281128

>>16281067
I dunno. I'm admittedly rusty (I drew up the full truth table in mindfucked confusion lmao), but everything I can find online that reeks of academic masturbation pinpoints vacuous truth as the culprit.

Anonymous at Sun, 14 Jul 2024 08:34:07 UTC No. 16281129

>>16281120
>that's the point of a paradox.
This isn't a paradox. A paradox involves an inconsistent, self-contradicting and impossible result that is the logical outcome of some premises and an axiomatic system. This example here, however, is just poor modelling.
>they are equivalent in formal (classical) logic
Not exactly sure why this would or should be such a conundrum?
If r is true when p and q both are true, it follows that r can still be true, when, for example, q is false. All it says is that r simply can't be false when both p and q are true.
Vice versa, if r is true if either p or q are true, then r is true when p and q are true. Simultaneously, when p and q now happen to be true but r is false, then (p -> r) and (q -> r) are false too, hence (p -> r) or (q -> r) are false. Hence, the equivalence holds true.

Anonymous at Sun, 14 Jul 2024 08:36:18 UTC No. 16281130

>>16281129
>This academic stupidity is probably false.

Does anyone actually trust this poster?

Explain it in English.

Anonymous at Sun, 14 Jul 2024 08:37:40 UTC No. 16281131

>>16281127
>This isn't a paradox.
go google
paradoxes material implication
and then shoot off emails about your findings. get a ouija board too.

Anonymous at Sun, 14 Jul 2024 08:57:32 UTC No. 16281142

>>16281129
That's a lot of redditcore energy, anon. No doubt you're always the smartest guy in the room and enjoy reminding inferior specimens that their centuries-old conundrums don't apply to you.

Anonymous at Sun, 14 Jul 2024 08:59:33 UTC No. 16281145

>>16281131
I'm sorry but I won't throw tantrums over your material implications not being representative of how the real world works. I don't consider the equivalence of the two logical expressions to be weird at all. Neither is the real-life example weird in any way. It clearly says "therefore, AT LEAST one of the following is true" which it is. Switch q, for all purposes and intents, could be not connected to the network at all and the statement would still hold true.

Anonymous at Sun, 14 Jul 2024 09:00:32 UTC No. 16281147

>>16281129
>Not exactly sure why this would or should be such a conundrum?

Anonymous at Sun, 14 Jul 2024 09:04:11 UTC No. 16281148

>>16281145
>material implications not being representative of how the real world works.
>[...]
>Neither is the real-life example weird in any way.
what kind of world-class pilpul is this?

Anonymous at Sun, 14 Jul 2024 09:34:35 UTC No. 16281162

>>16280954
Dunno if I agree with that diagram.

Anonymous at Sun, 14 Jul 2024 09:38:51 UTC No. 16281166

>>16281090
>The logical expression and the thought experiment that OP presents here are not equivalent.
Huh? It's a decades-old thought experiment.
Let p' = switch named p is closed
Let q' = switch named q is closed
Let r' = bulb named r is on.
If you accept that (p'^q')->r' is logically equivalent to (p'->r')v(q'->r') (hint: it is), then if (p'^q')->r' is true, (p'->r')v(q'->r') must be true also.
Is (p'^q')->r' true? That is, is it true that IF (switch named p is closed AND switch named q is closed), THEN bulb named R turns on? Well, yes.
So it follows that (p'->r')v(q'->r') is true. For this OR to be true, at least one of (p'->r') and (q'->r') must be true, which is to say that at least one of the following is true:
1. IF switch named p is closed, THEN bulb named r turns on.
2. IF switch named q is closed, THEN bulb named r turns on.
Not sure what you're missing, Broseph.

Anonymous at Sun, 14 Jul 2024 09:40:52 UTC No. 16281168

>>16280954
Translation: derp, I made a retarded thread.

Anonymous at Sun, 14 Jul 2024 09:41:42 UTC No. 16281170

>>16281168
low-iq bluff-seethe post

Anonymous at Sun, 14 Jul 2024 09:46:06 UTC No. 16281177

>>16281168
>unironic fednecking in 2024
that's basically a suicide. op has obviously rattled you, else you'd skip over the thread entirely. lemme guess: you "need" to get that word in.

Anonymous at Sun, 14 Jul 2024 09:48:08 UTC No. 16281180

>>16281120
>that's not what those rewrites are saying. they're directly naming the switch that's open, which is different.
Sure, they're different in constructive logic, but the topic of the thread is clearly classical logic, as evidenced by the failure of OP's equivalence in the nonclassical setting.
In this case, the disjunction is classical and nonconstructive, though if you insist on a computational interpretation, it can be variously implemented as an oracle (nonconstructivity as omniscience in the sense of Bishop), an amb operator (which can be built from call/cc), or a parallel play of the disjuncts (if you prefer game semantics). Any of these will yield an answer to your queries, although nonconstructively (and hence out of reach for constructive interpretations like BHK, but this is to be fully expected since in BHK the equivalence fails right out of the gate).

>>16281128
That's because I'm starting from the problem of how to explain OP's scenario intuitively, whereas most of the academic literature is starting from one idea that they came up with years ago (truth should be non-vacuous) and leaving it to you to apply it to the OP scenario. But it isn't always a good fit to your intuition, as evidenced by the fact that no one ITT so far has come up with an explanation in terms of vacuous truth (probably because the antecedents, at least in the form presented by OP, aren't vacuous in the first place).

By the way, are you looking at academic literature from philosophy or CS? I'm inclined to suspect the former, since the one big idea that CS came up with isn't really that truths should be non-vacuous (0->P is an intuitionistic theorem) but that truths should have constructive proofs.

🗑️ Anonymous at Sun, 14 Jul 2024 09:52:29 UTC No. 16281182

>>16281177
Bro, it's basically a world of retards. Take Joe Biden for example and his wonderful wacky left, trying to assassinate trump. You - jibe and let it pass - fumble and get lost in fabricated confusion. Timeless beauty, home on the simplest map. Trust me bro you're retarded.

Anonymous at Sun, 14 Jul 2024 09:52:35 UTC No. 16281183

>>16281166
>Not sure what you're missing, Broseph.
The fact that, given the expression (p ^ q) -> r, the light bulb can be on, no matter what the specific values for p and q are, with the one exception that p and q can't both be true and r false.
This guy >>16281120 isn't wrong in pointing out that the problem lies with "vacuous truth values". My interpretation is that the nice little diagram doesn't actually show (p ^ q) -> r, but (p ^ q) <-> r. It's easy to get mixed up because (p ^ q) -> r is a necessary condition, but it isn't sufficient. The other condition that no one apparently wants to acknowledge is r -> (p ^ q).

Anonymous at Sun, 14 Jul 2024 09:55:21 UTC No. 16281186

>>16281180
>(probably because the antecedents, at least in the form presented by OP, aren't vacuous in the first place).
nta, but they are mostly vacuous. Of the 7 cases in which (p'->r')v(q'->r') is true, 6 involve a vacuous case. The one case that involves no vacuous truth is P=Q=true, R=true, but this case presents no problems to intuition whatsoever.

Anonymous at Sun, 14 Jul 2024 10:21:49 UTC No. 16281217

>>16280954
Once you realize what this 'vacuous truth' circle jerk is about (I'm German and didn't know the English name, desu), it becomes obvious how the paradox works. Quite simply,
>if switch P is closed, then bulb r turns on
Once you realize that this whole line is defined as true if switch P is open (and likewise for the switch Q statement), the bait image loses its magic. Although only one switch needs to be open for bulb r not to turn on, the "Aha!" idea is easiest to see when both p and q are open and r fails to turn on.
>if switch p is closed, then bulb r turns on
>if switch q is closed, then bulb r turns on
Then both of those are true in mathematical logic even though the bulb r fails to turn on. Remember that in mathematical logic, the interest is in when
(P^Q)->R is true, not in when R per se is true.
(P^Q)->R is also true when P and Q are both false.
Luckily, this kind of thing won't present any problems when studying math outside mathematical logic, because general mathematicians pretty much ignore the 'vacuous' cases when working with implications. When trying to prove something like
>if x and y are real numbers, ...
They don't really give a shit about the 'vacuous' case in which x and/or y is actually a throbbing German penis.

Anonymous at Sun, 14 Jul 2024 10:33:27 UTC No. 16281225

>>16281217
>They don't really give a shit about the 'vacuous' case in which x and/or y is actually a throbbing German penis.
I'll add that maybe they'll care about my throbbing German penis if a biconditional if-and-only-if is involved rather than a plain if-then (because in P<->Q, you can do something like ~P->~Q, which is of interest because it's actually the converse Q->P in contrapositive form, and unlike with the plain if-then, this converse is of interest in the biconditional case), but then these biconditional cases in which I get to introduce my throbbing German penis are usually pretty uneventful from a paradox perspective too.

Anonymous at Sun, 14 Jul 2024 10:54:32 UTC No. 16281243

>>16281186
Ah right, I should have been clearer, I was thinking about the two antecedents in my reply to the other anon ("if switch P is closed, then bulb R turns on"), not the antecedent of the classical identity. Personally, I wouldn't go around calling a sentence "mostly vacuous" (in my mind, sentences are either vacuous or not, depending on whether they're satisfied in a given model), but it's fine since you've explained what you meant by that.

Anonymous at Sun, 14 Jul 2024 10:56:40 UTC No. 16281246

>>16280954
This is just a paradox to people who do not understand implication

Anonymous at Sun, 14 Jul 2024 11:13:31 UTC No. 16281257

>>16280954
I give this one a 7/10. (I can't remember giving higher, fwiw.) The audience is kind of too targeted for mass damage:
1. some compsci freshman/sophomore faggot taking his discrete math class (MAYBE some math major doing an intro proofs class), and
2. knows enough symbolic logic to check the equivalence but hasn't completely appreciated the nuances of implications when conditionals are false, and
3. doesn't just ignore the image, assuming there's some brazen lie in it such (there's not), and
4. doesn't fuck up when drawing his truth table or doing (p -> r) v (q -> r) ≡ (~p v r) v (~q v r) ≡ (~p v ~q) v (r v r) ≡ (~p v ~q) v r ≡ ~(p ^ q) v r ≡ (p ^ q) -> r, and
5. is confident enough not to assume he fucked up somewhere, and
etc., etc.

Anonymous at Mon, 15 Jul 2024 19:03:22 UTC No. 16282707

>>16280954
> say something nice about the most commonly taught system of logic, /sci/
It’s both consistent and complete, unlike first-order ar*thmetic