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Prove the following statement on negation of quantifiers:

Statement: To negate a statement of the form $$ Q_1x_1 Q_2x_2 \ldots Q_nx_n\; P(x_1,x_2,\ldots,x_n), $$ where $Q_i$ is $\forall$ or $\exists$ for $1 \leq i \leq n$, we do the following:

(i) Change every $\forall$ to $\exists$ and every $\exists$ into $\forall$.

(ii) Replace $P$ by its negation.

Background: This problem appears in the book How to Think Like a Mathematician by Kevin Houston, where it comes up in a chapter with an introduction to induction. I remember spending considerable time on this problem and not being able to come up with anything satisfactory. It seemed like you had to go into some deep logic to try to come up with an answer (something the author did not intend). I communicated with the author and noted the exercise, but he never got back to me. I took this as a sign that the exercise was flawed (or at least was very inappropriate for where it was placed in the text). I am still curious as to how this problem may be solved, if at all possible. Or is it something more axiomatic in nature and not something you can really deduce per se?

Daniel W. Farlow
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  • You need the two "basic" equivalences : $\lnot \forall$ and $\exists \lnot$, and $\lnot \exists$ and $\forall \lnot$ to start the induction. – Mauro ALLEGRANZA Jan 07 '15 at 18:24
  • @MauroALLEGRANZA I know. The issue crops up when you actually try to make the induction argument. Try it out and you will see what I am talking about. – Daniel W. Farlow Jan 07 '15 at 18:29
  • @induktio Show us your work and where you are stuck, so that we know where to direct you. – DanielV Jan 07 '15 at 18:34

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Here's the argument spelt out in my Gödel book -- $\varphi$ is the predicate for which we aim to show by induction that $\forall n\varphi(n)$

From Peter Smith's Intro to Gödel Theorems

Peter Smith
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  • Interesting. What book is this from? I can mostly follow it, but I do not see at the end how $\varphi(n+1)$ has been established on the $\varphi(n)$ assumption, which is really the key inductive part. Would you not have to introduce $Q_{n+1}$ somewhere? – Daniel W. Farlow Jan 07 '15 at 19:39
  • We've looked at the two cases, where $Q_{n+1}$ are respectively $\exists v$ and then $\forall v$ -- it has to be one or the other! (The book, as I said, is my Gödel book -- I copied a chunk to avoid retyping!) – Peter Smith Jan 07 '15 at 23:51
  • Which Godel book? Surely he wrote more than just one book. – Daniel W. Farlow Jan 07 '15 at 23:56
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    You misunderstand: my book! :-) Peter Smith, An Introduction to Gödel's Theorems (CUP, 2nd edition 2013). – Peter Smith Jan 07 '15 at 23:58
  • Oh wow! My bad. Thanks :) – Daniel W. Farlow Jan 08 '15 at 00:00