If A then B
Hello 👋
happy new year!
So, I'm somewhat running against a wall in my brain with the If A then B logic and I was hoping someone has the time to explain it a bit further, especially regarding this problem: https://brilliant.org/practice/if-a-then-b/?problem=logic-problem-119327&chapter=puzzles-and-riddles
I simply don't understand. If door A is safe, then door B is deadly. Then: if the statement is false, what is true?
For my mind, if the statement is false, I turn around the said: If door A is deadly, then door B is safe/deadly.
Why is the solution, that both doors are safe?
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3 solutions
Thank you. I understand the metaphor with the pathway and the destination. In this puzzle, "If door A is safe, door B is deadly" means that as long as door A is safe, door B is deadly. If door A was deadly, it doesn't matter what door B is. What my mind now does not comprehend is what the statement "the statement is false" does to the conditional statement. Does that mean door A is deadly and therefore clearly door B can be deadly or safe (obvious to the answer that's not the case). Basically, how does door A stay "safe" if the statement is false?
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"The statement is false" refer to the whole conditional statements : if we're talking about my metaphor, then it translates to ”the signboard is fake", meaning that somebody had TRULY went through A (the journey is true) to THEN ended up somewhere that is NOT B (the destination is false).
Now I'm going on to your problem.
"If {door A is safe}, then {door B is deadly}" as a WHOLE is false, not just the if nor only the then. What's false is the whole premise, this is what I believe you haven't figured out all this while.
"If {door A is safe}, then {door B is deadly}" ==> FALSE
Now you can refer to my table and look up for false if-then Boolean, there's only one right? It says A to be true and B to be false.
A = "door A is safe" ==> TRUE ==> go to concluded truth #1
B = "door B is deadly" ==> FALSE ==> go to concluded truth #2
THE CONCLUDED TRUTHS BY PART :
1) Door A is safe.
2) Door B is NOT deadly. Door B is safe, too.
Quoting you, "In this puzzle, "If door A is safe, (missed a then here) door B is deadly" means that as long as door A is safe, door B is deadly.”. Tbh, what you thought as its meaning is incorrect.
| DOOR A | DOOR B | IF-THEN | INTERPRETATION |
| SAFE | DEADLY | TRUE | You're right to trust the warning |
| SAFE | SAFE | FALSE | Faulty warning, said one thing but another happened |
| DEADLY | SAFE | TRUE | Unrelated to the original warning |
| DEADLY | DEADLY | TRUE | Unrelated to the original warning |
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So, the statement "the statement is false" is simply stating the fact of the only possibility of if-then = false, not as the statement is false and we put another false on it with the statement. (I hope you know what I mean :D) So to make it true we have three other options (in your table). Deadly - safe and deadly - deadly are not relevant because...? Going back to the metaphor, you're signboard needs to stay true otherwise it'd be completely new if-else (because of new signboard)? (God, I'm sorry if I explain too confuse, thank you for your help anyway!!)
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@Jil Sonea
–
I'm gonna use my signboard first. I place it at a junction / fork and you saw it passing by. My signboard could be legit and true in 3 ways, but might only be fake (false) in 1 way.
3 true signboards :
a) you went the A way and you arrived at B, or
b) you went the OTHER way, and still managed to arrive at B, or
c) you went the OTHER way, and didn't get to arrive at B.
If you have an imaginative mind, let's have roads A, C, D and F and destinations B and E. My true signboard is standing at the FIRST fork, say the left one is road A and right one road C. Road A is a direct route to B (explaining the possibility of a)), no other junctions or distractions along the way. But say that you take road C for some reason (like you managed to read the signboard but the car was too fast for a lane change for example), and decided it's just too much of a hassle to make a U-turn for it, kept driving on C, encountered a SECOND fork onto road D to the left and road F to the right. Say you take road D since you remembered the last mistake was missing a left turn at my signboard, and somehow ended up at B. This is how my signboard is true despite the fact that your arrival at B doesn't involve a journey through road A. My signboard didn't mention that another way to B is impossible. It's just showing you one of the possible routes, and as a standard signboards show us the legally shortest / easiest to follow / mainstream way to reach a destination.
Say you gambled, took the F road and reached a rural ghost town E's dead end. My signboard remained truthful, because your arrival at E won't change the fact for everyone else who's going the A way, since THEY will arrive at B through A.
In conclusion, going by my signboard,
a) (You + Them + signboard) --A--> B
b) (You + signboard) --C--D--> B
(Everybody else + signboard) --A--> B
c) (You + signboard) --C--F--> E
(Everybody else + signboard) --A--> B
Attempt at a diagram
{You} -<≥- {B}
@Jil Sonea – When I said "unrelated", what I meant was that the warning does NOT say anything about a door B that have a deadly door A, BUT what the warning DOES say still stands true one way or another.
@Jil Sonea – "So to make it true we have three other options (in your table)". If you are referring to the safe-deadly table, it's not to make anything true. I was just showing the 4 possible situations with the doors, that's all.
The statement : If Door A is safe, then Door B is deadly.
We are told that "the statement is FALSE". Equivalent to someone (most probably a local) approached you and said, "Don't trust the signboard". A misleading signboard will make you end up somewhere NOT the promised place.
Misleading meaning :
1) even when you follow the LEAD (or condition or instruction) completely,
2) the result is totally the opposite of (nothing like) what's expected, that we MISSed the mark.
If Door A is safe, then Door B is deadly ==> FALSE
By understanding the full, big picture, we can now give the truth value to the FALSE STATEMENT BY PARTS.
False if-then ==> True If & False Then
@Jil Sonea – You may find more on the net. Brilliant.org has its own Wiki here, as with the Wikipedia (search for Material Conditional). Or https://brilliant.org/wiki/propositional-logic/ --> Connectives --> Conditional
@Jil Sonea – There are more conditional logic than one, but the two most widely recognised are
1) strict conditional (nearer to natural language, linguistics), this one is the more familiar "causal" if, like "If you don't study, then you will fail the test" where the reason / excuse in the front part caused the consequence in the back part.
2) material conditional (mathematical logic, that can be linked to Venn diagrams and the logic gates), where the front (after the if) and back (after the then) parts may not have anything in common, and even have independent truth values between them. This is why truth tables are very useful to know; you don't have to process the statements by its parts one by one and just refer to them and make comparisons using the tables. Here, things from the {if part} didn't exactly cause the {then part} to happen, but usually we are given information about the truthfulness or falsity of 2 out of 3 things : the {if part} or the {then part} or the {whole parts}.
I can see from your interpretation of the conditional statement that you have already assumed its truthfulness, and only taking the triple truths part (from the first line on that table) of it. But if-then can be true (in 3 different ways) or false (in ONE UNIQUE WAY). What you need to remember is this 'formula' :
"A TRUE IF AND A FALSE THEN TOGETHER MAKES FOR A FALSE IF-THEN". Anything else is true. You can equate it to anything to make it easier to remember, something like "an ugly man and a pretty woman together makes for a pretty baby".
The thing to keep in mind here is that the logical "If-Then" doesn't behave in the same way as "if" behaves in normal language. Logically "If A then B" is only false if A is true and B is false, as your truth table shows. One could get rid of logical implication by replacing every " A → B " statement with " ( ¬ A ) ∨ B "
For the statement to be true, both of the conditions need to be met, in this case A being safe and B being deadly. Anything else would make the said statement false. So if both are safe, both of the conditions are not met because one of them is wrong (B is deadly), so therefore the statement can still be false with both doors being safe.
a is true that it is safe since b is death so b is lying about this which is false and ends up being that it is true that both are safe so if it is not it ends in a fallacy or contradiction so is true A-> B.
For the if-then conditional statements, the ONLY way they can be false is when A is true BUT B is false. Why? Because it's a directional logic. It guaranteed a B event happening after an A event. Easier to imagine if you consider A a signboard appointed pathway and B your destination. What should happen is you go the A road and you arrive at B as a result of trusting the signboard / map / GPS. If that's what really happened in reality, then you can say that the signboard is being truthful. But it's easy to imagine an angry traveler when he go through A but not reaching B at the end of the road. Understandably he felt cheated by the lying, fake, false signboard. If you don't take the A road, it didn't matter whether you ended up at B or not, since you cannot blame the faultless signboard as it's innocent of any supposed cheating.