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I am working through Mathematical Logic by Chiswell and Hodges and am stuck on pages 24-26 (Section 2.6 - Arguments using 'not'). This section starts off by stating that "If $\phi$ is a statement, we write $(\lnot \phi)$ for the statement expressing that $\phi$ is not true". A few sentences later, while explaining how $\lnot$ is used in arguments and introducing the new notion of $\bot$ , the text says, "In derivations we shall treat $(\lnot \phi)$ exactly as if it was written $(\phi \to \bot)$.

1.) Based on the above, I am taking $(\phi \to \bot)$ to be the definition of $(\lnot \phi)$. Is this correct? The authors do not seem to state this explicitly, instead choosing to only say that $(\lnot \phi)$ behaves like $(\phi \to \bot)$; this makes me think I might be missing something.

If my current understanding IS correct, then all is well (that is, the text goes on to introduce the natural deduction rules $(\lnot E)$ and $(\lnot I)$, which I think make sense) until I reach Example 2.6.2, which is a proof of the statement "There are infinitely many prime numbers" that is given in order to motivate the next natural deduction rule, reductio ad absurdum (RAA).

The proof starts out as follows:

Theorem: There are infinitely many prime numbers.

Proof: Assume not. Then there are only finitely many prime numbers $p_1,...,p_n$ (and then the proof goes on...)

2.) Again, taking $(\phi \to \bot)$ to be the definition of $\lnot \phi$, I am confused about how this proof can conclude from "Assume not" that "Then there are only finitely many prime numbers $p_1,...,p_n$". In the case of this proof, if $\phi$ is the statement "There are infinitely many prime numbers", then it seems like the proof is saying that $(\lnot \phi)$ is "there are only finitely many prime numbers", but I am not sure why that is the case; it would seem to me that $(\lnot \phi)$ would instead just have to be "if there are infinitely many prime numbers, then absurdity", and I don't know how the proof would then proceed.

Artyom Elessar
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  • For your first question, yes. $\lnot\phi$ is defined as $\phi\to\bot$. This way, negations like $\lnot\lnot\phi$ won't be written like $(\phi\to\bot)\to\bot$, which becomes longer. – soupless Sep 09 '23 at 20:25
  • Concerning your second question: 'if there are infinitely many prime numbers, then absurdity' is equivalent to 'there are only finitely many primes'. If the first sentence is true, its antecedent cannot be true and so the second sentence is true. – sequitur Sep 11 '23 at 00:19
  • You are right: the book does not define $\lnot$ in terms of $\bot$. The negation sign is primitive and then the authors introduce the sign $\bot$ (pronounced ‘absurdity’ or ‘bottom’ according to taste) for a statement which is definitely false, for example, ‘$0 = 1$’ . If so, we may say that $\lnot \phi$ and $\phi \to \bot$ behave the same. But you have to consider that, having not yet introduced the rules for managing $\lnot$, this assertion is quite vague. – Mauro ALLEGRANZA Sep 11 '23 at 07:34
  • Only if you are working in Classical Logic, having available truth table, you can compute the TT for $\phi \to \bot$ [the authors at this stage have already introduced $\to$] and verify that it behaves exactly as expected. – Mauro ALLEGRANZA Sep 11 '23 at 07:35
  • The proof about the prime numbers is using *reductio ad absurdum" which is the principle that $\lnot \phi \to \bot$ implies $\phi$, specifically with $\phi$ meaning that there are infinitely many prime numbers. The authors assume $\lnot\phi$ and derive $\bot$, which is a proof of $\lnot \phi \to \bot$, from which they conclude $\phi$. This principle is not accepted by some mathematicians and it is also unnecessary in this proof, which is constructive: it shows that given any finite set of primes we can find a prime that is not in the set and that means that the set of primes is infinite. – Rob Arthan Sep 11 '23 at 19:37
  • Thank you all for the comments – Artyom Elessar Sep 14 '23 at 00:02
  • I'm understanding now that $\lnot$ is primitive, and I am re-reading the first quote in my post to be the definition of $\lnot$. For the second question, I think I can then more clearly pinpoint my confusion: How do we know that "the statement expressing $\phi$ is not true" is unique? For the infinite primes example, I would take $\lnot \phi$ by definition to mean "It is not true that there are infinitely many primes". I think most would say this means the same thing as "there are only finitely many primes", however could there not be other possibilities such as "primes don't exist"? – Artyom Elessar Sep 14 '23 at 00:03
  • I suppose another way of asking the above is, how do we know ("It is not true that there are infinitely primes" iff "There are only finitely many prime numbers"), as opposed to ("It is not true that there are infinitely primes" if "There are only finitely many prime numbers")? – Artyom Elessar Sep 14 '23 at 00:14

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