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I got stuck while reading the Discrete Mathematics book of Grimaldi. Say there are two primitives.

p: I read physics.
q: I pass physics.

Now consider below lines.

(p $\to$ q) $\Leftrightarrow$ ($\neg$p $\lor$ q) which can be read as "I don't read or I pass"

Now read this "I don't pass or I read". which can be written as (p $\lor$ $\neg$q). Both mean the same right? (Or only I perceived it like that?) then (p $\to$ q) $\Leftrightarrow$ ($\neg$p $\lor$ q) $\Leftrightarrow$ (p $\lor$ $\neg$q)

But I know both are not equivalent because I have written truth table for it. But can you convince me using English language sentence (not truth tables) the above non equivalence.

Madhusoodan P
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  • For instance: if in the second one you "read and don't pass", then this is different from if you "don't read and pass", as a possibility in the first one. – Dave Mar 08 '17 at 00:18
  • Why do you think they are the same? "I don't pass or I read" means you can both fail and read, but you can't pass and not read. "I don't read or I pass" means you can pass and not read, but you can not both fail and read. They are complete opposites. – fleablood Mar 08 '17 at 00:18
  • In general it is a very bad idea to think on formal logic using examples of the natural languages. This is the reason that approximately the 99% of introductory books to logic are terrible bad. – Masacroso Mar 08 '17 at 00:22
  • I disagree with that. But you need to be logical in you language. You must know and think A or -B means A, -B or possibly both is completely different than -A or B which means -A, B or possible both. – fleablood Mar 08 '17 at 00:31

2 Answers2

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Your question basically is:

Why is $\neg p \lor q$ not the same as $p \lor \neg q$?

OK, take:

p: $1+1=3$

q: $1+1=2$

Then: $\neg p \lor q$ i:s '$1+1\not=3$ or $1+1=2$' .. which is true

But: $p \lor \neg q$ is: '$1+1=3$ or $1+1 \not = 2$' .. which is false

Bram28
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"I don't read or I pass". means 1) It's possible that I don't read AND I pass and 2) It's impossible that I read AND fail. iii) Everything else is possible as either I won't read or I pass.

"I don't pass or I read" means 1) It's POSSIBLE that I read and fail 2) It's IMPOSSIBLE that I don't read AND I pass. iii) Everything else is possible as either I won't pass or I will read.

You are thinking of the exclusive OR where Either A or B but not both. In that case "either I don't read or I pass but not both" (So I better read!) and "either I don't pass or I read but not both" (So I better read!) are the same.

But with the normal OR, where Either A or B or both can happen they arent' the same because the "both" are contradictory.

"I don't pass or I read" (Then I better read because i might pass but if I don't I WILL fail) is not the same as "I don't read or I pass" (Then I better read because I might pass if I don't but I WILL pass if I do) are not the same.

fleablood
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