P=>q, q=>r, then p=>r (proof assumes p why?)

  • Thread starter chisser98
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In summary, the conversation discusses a logic problem where it is stated that whenever p is true, q is true and whenever q is true, r is true. The question is to prove that whenever p is true, r is true. The answer involves using the Implication Introduction step, but the person is confused about why they can assume p is true when it is not explicitly stated as a premise.
  • #1
chisser98
2
0
hi guys,

I'm struggling to learn logic and I'm stuck on what I'm sure is an easy answer. Here's the question:

whenever p is true, q is true. Whenever q is true, r is true. Prove that, whenever p is true, r is true.

Here's part of the answer:

p=>q Premise
q=>r Premise
(q=>r) => (p => (q=>r)) Implication Introduction

...etc

I can follow the proof just fine from this point. However, why can we assume p is true? The Implication Introduction step is allowing us to assume p is true..even when it's not a premise (or at least they don't state it as a premise explicitly).

Anyway, like I said, I'm sure this is easy, I'm just not wrapping my head around it.

Any help would be appreciated. Thanks guys!
 
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  • #2
chisser98 said:
The Implication Introduction step is allowing us to assume p is true..
No it's not. S is not a consequence of S => T.
 
  • #3
Thanks for the reply Hurkyl. Ok - since S is not a consequence of S=>T, we can say it's true? I'm saying this because the implication introduction scheme is:
M => (N => M)

in this example, we've set M = (q=>r) and N = p, so we get:
(q=>r) => (p => (q=>r))

I guess I'm just not understanding why we can assign N=p when p isn't an explicit premise?
 

1. What is the meaning of the symbols in this statement?

The symbol "=>" represents the logical operator for implication, meaning "if...then". The "p=>q" statement means "if p is true, then q must also be true".

2. How does the proof for this statement work?

The proof for "p=>r" using "p=>q" and "q=>r" follows the transitive property of implication. If we know that "p=>q" and "q=>r" are both true, then we can conclude that "p=>r" must also be true.

3. Why does the proof assume p is true?

The proof assumes that p is true because that is the given information or premise. In order to prove that "p=>r", we must start with the assumption that p is true and use the given implications to logically show that r must also be true.

4. Can this statement be proven without assuming p?

No, this statement requires the assumption of p in order to prove that "p=>r". It is a basic rule of implication that in order to prove "p=>q", we must assume that p is true.

5. What are some real-world examples of this statement?

An example of this statement in real-world scenarios could be: "If I study for my exam (p), then I will pass (q). If I pass (q), then I will graduate (r). Therefore, if I study (p), then I will graduate (r)."

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