I'm learning logic. I had a discussion on another board. Can someone who knows a lot about logic comment on this discussion? Specifically how I (the OP), the Moldovan anon, and another Swede discussed how the fallacy can be either denying the antecedent or affirming the consequent. Yesterday I felt like I understood it but looking at it again today I'm confused.>>>/pol/494305518https://archived.moe/bant/thread/21913276/https://archived.moe/bant/thread/21913276/#21913359https://archived.moe/bant/thread/21913276/#21916009It seemed to make sense that it could be either denying the antecedent or affirming the consequent. This made me wonder if denying the antecedent can always be rephrased as affirming the consequent and vice versa. But when I was going over the reasoning again to figure out if that's the case I got confused.
Maybe I'm getting it again.If you don't eat your vegetables, then you won't get any dessert.denying the antecedent:If you don't eat your vegetables, then you won't get any dessert.if you do eat your vegetables, then you get dessertmodus tollens:If you don't eat your vegetables, then you won't get any dessertIf you did get dessert, then you did eat your vegetables.affirming the consequent:If you did get dessert, then you did eat your vegetables.If you do eat your vegetables, then you get dessert.If P, then Qdenying the antecedent:If P, then Qif not P, then not Qmodus tollens:If P, then Qif not Q, then not Paffirming the consequent:If (not Q), then (not P)if (not P), then (not Q)
starting statement: If P, then Qroute 1 (denying the antecedent)denying the antecedent:If P, then Qif not P, then not Q (end result)route 2 (affirming the consequent)modus tollens:If P, then Qif not Q, then not Paffirming the consequent:If (not Q), then (not P)if (not P), then (not Q) (end result)The end result is the same for both paths. I.e. you can see it as either denying the antecedent or affirming the consequent, they're the same thing.
I'm going to try another example.If you are a ski instructor, then you have a job.route 1, denying the antecedentIf you are a ski instructor, then you have a job.You are not a ski instructor.Therefore, you have no job. (end result)route 2, affirming the consequentmodus tollensIf you are a ski instructor, then you have a job.If you don't have a job, then you are not a ski instructor.affirming the consequentIf you don't have a job, then you are not a ski instructor.If you are not a ski instructor, then you don't have a job. (end result)Same end result.
Now I'll take an example of affirming the consequent instead and see if I can go the route of denying the antecedent with it and get the same end result, just like I took examples of denying antecedent and then went the route of affirming the consequent and got the same end result in previous examples.If someone lives in San Diego, then they live in California.Joe lives in California.Therefore, Joe lives in San Diego.route 1, affirming the consequentIf someone lives in San Diego, then they live in California.Joe lives in California.Therefore, Joe lives in San Diego.route 2, denying the antecedentmodus tollensIf (someone lives in San Diego), then (they live in California)If someone (does not live in California), then (they don't live in San Diego)denying the antecedentIf someone (does not live in California), then (they don't live in San Diego)If someone lives in California, then they live in San Diego (end result)Same end results.We can see that denying the antecedent and affirming the consequent are the same thing, just expressed differently.
>>16549094This would be:If P, then Qmodus tollensIf P, then QIf not Q, then not Pdenying the antecedentIf not Q, then not PIf Q, then P (end result)
So, to summarize.To go from denying the antecedent to affirming the consequent you do this.starting statement: If P, then Qroute 1 (denying the antecedent)denying the antecedent:If P, then Qif not P, then not Q (end result)route 2 (affirming the consequent)modus tollens:If P, then Qif not Q, then not Paffirming the consequent:If (not Q), then (not P)if (not P), then (not Q) (end result)And to go from affirming the consequent to denying the antecedent you do this.If P, then Qmodus tollensIf P, then QIf not Q, then not Pdenying the antecedentIf not Q, then not PIf Q, then P (end result)
>>16549130Maybe this can be written more eloquently, but I have to sleep now, and I think I got my point across. This can now be the basis for a discussion.
>>16549130>If P, then Q[.]>If not Q, then not P[.]The foregoing two statements are logically equivalent or interchangeable.
>>16549463what?
>>16550307it's the contraposition, but ofc only in classical logichttps://en.wikipedia.org/wiki/Contraposition?useskin=vectorwhen studying logic, ie classical logic, don't forget that the arrow => behave weirdly, it's not the perfect embodiment of ''logical entailment''if you want good books about logic i recommend:Craig DeLancey A Concise Introduction to LogicDerek Goldrei, Propositional and Predicate Calculus A Model of ArgumentLOGIC The Laws of Truth NICHOLAS J. J. SMITHthey are on libgen or even freely on internet
>>16550318You haven't addressed the topic. I'm saying denying the antecedent and affirming the consequent are really the same thing expressed differently, anytime you have denying the antecedent you can restructure the fallacy to affirming the consequent and vice versa. So whenever someone says the fallacy committed is denying the antecedent and someone else says it's affirming the consequent, and they disagree, they are both right. This is not something I read, it's something I discovered myself.I'm reading thishttps://forallx.openlogicproject.org/html/
>>16550307plug it in herehttps://programming.dojo.net.nz/study/truth-table-generator/indexthe truth tables are identical
Here's an example:t = Product[p[n]^x[n], {n, 1, Infinity}]p[n] is the nth prime numberx[n] is the nth exponentoriginal statement:If every exponent is an integer, then t is rational.contrapositive statement:If t is irrational, then at least one exponent isn't an integer.If t = 3^(2/7), then:x[1] = 0x[2] = 2/7 <> integerx[3] = 0x[4] = 0x[5] = 0et ceteraWhich suggests, that t = 3^(2/7) = 9^(1/7) is irrational.
>>16548964That's undoubtedly a delicious salad.In Europe, most persons eat rice with a spoon.In East Asia, most persons eat rice with chopsticks.In the Russian region which borders the NE-most province of China, most persons eat salad with a... miniature m*n*r*h?
>>16550410Still haven't addressed the topic
>>16550318Are those books better than this book?https://forallx.openlogicproject.org/html/
S1: If P, then Q.S2: If not Q, then not P.S1 <==> S2S3: If Q, then P.S4: If not P, then not Q.S3 <==> S4
