brrly babbles about how to prove it (ch. 1.2)

hello everyone! this brrly babbles post marks the beginning of a series of blog posts about a math book that i'm reading AKA how to prove it!! i want to learn how to read and write mathematical proofs because i love math and i want to do more with math and in the future i will probably major in something very math-heavy like physics or just math itself so i might as well get ahead while i can! also i have practically nothing to do at home for math homework because i complete it all at school when i don't feel like paying attention during class. ANYWAYS enough babbling let's get into chapter 1.2 which is truth tables!!! i am very confused on this one example so i'm writing out my thoughts on this blog so i can sort them and eventually understand the example because i cannot wrap my head around this until i get it out into some coherent sentences!!!!!

the example is basically about drawing conclusions based on information that we already know while using logical connectives (and, or, not). i'm just gonna write out the example with english words because i'm too lazy to actually find the symbols online and copy and paste every single time i need to use it.

the following are the variables for each statement

B = the butler is innocent, C = the cook is innocent, L = the butler is lying

the argument is:

not(B and C)
L or C

the arguments together mean... not(B and C) and (L or C)

the conclusion drawn is:

L or not B

now this is the part where i try to make sense of how they got to that conclusion! time to babble AAYYYY

not(B and C) means that both the butler and the cook cannot be innocent at the same time, so either the butler is not innocent or the cook is not innocent because both of them can't be innocent!!!!!

L or C means that the butler might be lying, the cook might be innocent, or both of these statements could be true. 

let's go back to not(B and C) yay! if the butler is innocent, then the cook cannot be innocent and therefore the butler is lying because the second statement says that the cook is innocent but the butler and the cook cannot be innocent at the same time therefore...

B has the following conclusion drawn: not C and L
aka just L

if the cook is innocent, then the butler cannot be innocent. the cook's innocence does not change the fact that the butler could be lying therefore...

C has the following conclusion drawn: not B and L or not B and C
aka L and not B or not B

therefore the overall conclusion drawn is:
L or not B, because both can be true statements but at least one of them has to be true if the other is false

okay i think i get it now but let me practice with an easier example to see if my thought process works for all problems

variables:

S = john is smart, L = john is lucky

the argument is:

(not S and L) or S
S

everything in an argument must evaluate to true because that is what we are given

the arguments together mean... ( (not S and L) or S ) and S

the conclusion drawn is:
not L

since everything in an argument must evaluate to true, that means that if john is lucky then john is not smart because then ( (not S and L) or S) would evaluate to true since (not S and L) is true

OR john can be smart, so then  ( (not S and L) or S ) still evaluates to true

in this case, both statements of the first line of the argument cannot be true at the same time because if john is not smart, then john is lucky since that is the only outcome in which the entire statement would be true

next we are given that john is in fact a smartie poopoo pants. this leaves out the other outcome, which is that john is not smart but is lucky. remember that john cannot be smart and lucky at the same time because either john is not smart and lucky or john is smart. 

if not S is true then L must also be true because we are given that the entire argument of ( (not S and L) or S ) is true, therefore if S is not true aka (not S) then the other part of the statement must be true aka (not S and L) meaning that L must be true to make the entire statement true. 

if S is true then L must not be true because we are given that the entire argument ( (not S and L) or S ) is true, therefore if S is true then one of the statements of the entire argument is fulfilled, but when we plug it into the other statement aka (not S and L) we get false because not S is false since S is true, therefore L cannot be true 

guys.

you will not freaking believe what i just found out.

i should've just read the freaking book because i've been confusing myself this entire freaking time. 

the conclusion drawn is not always right.

this is so freaking stupid.

i did all of this for what.

if john is both lucky AND smart which is possible because i was thinking about it and i was like "well if L is true but S is also true then not S would be false and not S and L would still be false making ( ( not S and L or S ) true because S is still true" SO I WAS LIKE WAIT LET ME READ THE FREAKING BOOK

AND THEN I READ IT AND THE CONCLUSION DRAWN ISN'T ALWAYS CORRECT BECAUSE DEDUCTIVE REASONING OR WHATEVER

ARE WE SERIOUS

the purpose of the examples was to show that the conclusion drawn is either valid or invalid based on the premises. for example with the butler thing i was going through all the combinations to see why the conclusion drawn would be valid but i didn't realize that sometimes the conclusion drawn ISN'T valid but with this newfound knowledge because i read one paragraph of the book instead of skipping over everything as usual, i shall look over the first example we did together in this blog post and try to make it make sense to me

both the butler and the cook can't be innocent together, and either the butler is lying or the cook is innocent.

if the butler is innocent and the butler is lying, then the conclusion drawn is valid

the premises cannot be true if the conclusion drawn isn't true so that meannnsssssss

okay i give up i think im just thinking about this wrong because we're just deciding whether or not the conclusion is valid i thought we were trying to figure out how to get a conclusion based on the premises given of an argument but we have not gotten that far. i think it's time i settle down and read the next chapter instead of confusing myself any further. thank you for going on this confusing journey with me because i am spiraling and spiraling. sigh.