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Going through a book on formal logic, I have encountered the following problem. Since I am somewhat new to formal logic, I am confused about how to approach it.

A certain formal theory has exactly two axioms:

(a) 2 + 2 = 4 -> (2 + 2 = 4 -> 2 + 3 = 6)
(b) 2 + 2 = 4

and has modus ponens, i.e., (P->Q, P) -> Q

Find all theorems of this theory.

I understand that the axioms themselves are theorems. How can I find the others?

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  • $\begingroup$ Hint: think about it like like a self-assembly kit. You have been given some basic parts (the axioms) and some tools for combining existing parts to make new ones (the inference rules). In your case you only have one inference rule (modus ponens) and a very small supply of axioms. What can you build? $\endgroup$
    – Rob Arthan
    Mar 28, 2017 at 21:17
  • $\begingroup$ What confuses me is the double implication in axiom (a). How does one understand it? $\endgroup$ Mar 28, 2017 at 23:45

1 Answer 1

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A theorem is any statement that can be derived from the axioms (in addition to the axioms themselves).

So, using your Modus Ponens, we can combine (a) and (b) to get:

2 + 2 = 4 -> 2 + 3 = 6

And there is one more statement we can infer ... do you see which one?

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  • $\begingroup$ I guess I am confused what statement (a) even means. Does it mean that (2+2=4) implies (2+3=6) when 2+2=4? $\endgroup$ Mar 28, 2017 at 23:41
  • $\begingroup$ Yes, the other implied statement is 2+3=6 because it follows from (b) after (a) is applied. However, the first question still stands :-) $\endgroup$ Mar 28, 2017 at 23:49
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    $\begingroup$ Yes, statement (a) means that 2+2=4 implies 2+3 =6 when 2+2=4. Why anyone would ever make such a bizarre claim is beyond me, but that's what it is. Indeed, don't look too much into the meaning of that statement ... It's just used for an example to demonstrate how you can repeatedly apply Modus Ponens and thus get more theorems. $\endgroup$
    – Bram28
    Mar 29, 2017 at 0:05

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