Propositional logic - translating sentences into statements and drawing conclusions

[r2vven]

I came across an exercise on the Internet which I have had real trouble solving. I have the following text:

> If you know you don't exist, then you don't exist. Obviously, the knowledge that you don't exist is sufficient to deduce that you know something. You exist, unless you don't know anything. Does this imply your existence or your nonexistence? Algebraically prove it.

So this is what I have come up with:

A - I exist, and B - I know something.

1. "If you know you don't exist, then you don't exist" - [MATH]\overline{A}B \Rightarrow \overline{A}[/MATH]This is a simple matter of an implication.


2. "Obviously, the knowledge that you don't exist is sufficient to deduce that you know something" -[MATH] \overline{A}B \Rightarrow B[/MATH]Again, this is a matter of an implication, but I am not so sure about the "the knowledge that you don't exist is sufficient to deduce that you know something", because [MATH]\overline{A}B[/MATH] doesn't literally translate to that.


3. "You exist, unless you don't know anything" - [MATH]\overline{A} \Rightarrow B[/MATH]It would have been a normal implication were it not for the word "unless".


Now here comes the hard part. How do I prove that I exist or that I don't exist? I have two ideas:

1.[MATH](\overline{A}B \Rightarrow \overline{A})(\overline{A}B \Rightarrow B)(A \lor B) \Rightarrow \textbf{A},[/MATH] where I would need to get [MATH]\top[/MATH] as final result

2[MATH].(\overline{A}B \Rightarrow \overline{A})(\overline{A}B , \Rightarrow B)(A \lor B) \Rightarrow \overline{\textbf{A}}[/MATH], where I would get[MATH] \bot[/MATH] as final result.

However, the first one yields [MATH]A \lor \overline{B}[/MATH], and the second one yields [MATH]\overline{A}[/MATH].

Can anyone please help? This would be crucial in understanding how this works.

 

[lex]

3. "You exist, unless you don't know anything" - ¯¯¯¯A⇒BA¯⇒B\displaystyle \overline{A} \Rightarrow B
It would have been a normal implication were it not for the word "unless".

I would interpret No. 3 as B[MATH]\rightarrow[/MATH]A

(I don't know much about this sort of thing), but in any case, I don't think I accept some of these 'propositions' as meaningful (never mind true). I think the concept of 'knowing that I don't exist' doesn't mean anything. It doesn't represent any situation, let alone one that may hold or not.

 

[Afive]

Try this: [please note: negation = -, disjunction = or, conjunction = and, implication = ->]
A = I exist
B = I know I exist
-B = I don't know that I exist
C = I know that I don't exist
-C = I don't know that I don't exist

1: translates as C-> -A
2: translates as C-> B
3: translates (A and B) or (A and C)
You have everything to calculate the conclusion (hint: Descartes)

 

[Afive]

Well...
read 2: translate as C->C (that is (-C or C) principle of the excluded third)