1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Problem in Logic - Hilbert Systems

  1. Oct 12, 2012 #1


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    1. The problem statement, all variables and given/known data
    In a Hilbert System, prove:
    [tex]\phi[x|m] \rightarrow\forall y((y=m)\rightarrow\phi[x|y])[/tex]
    where [itex]\phi[/itex] is a formula, [itex]y, x[/itex] are variables and [itex]m[/itex] is a constant.
    [itex]\phi[a|b][/itex] denotes the formula obtained by substituting [itex]b[/itex] for [itex]a[/itex] in [itex]\phi[/itex]
    This problem crops up in my attempting to prove Godel's diagonal lemma (aka 'Fixed Point Theorem').

    2. Relevant equations
    The axioms of a Hilbert system are standard, as set out in http://en.wikipedia.org/wiki/Hilbert_system

    3. The attempt at a solution
    \phi[x|m]\ \ \ \ \ \ \text{1. Hypothesis}\\
    y=m\ \ \ \ \ \ \text{2. Hypothesis}\\
    (y=m)\rightarrow(\phi[x|m]\rightarrow\phi[x|y))\ \ \ \ \ \ \text{3. Axiom Schema 6 (substitution of equal variables)}\\
    \phi[x|m]\rightarrow\phi[x|y]))\ \ \ \ \ \ \text{4. Modus Ponens applied to 2 and 3}\\
    \phi[x|y]\ \ \ \ \ \ \text{5. Modus Ponens applied to 1 and 4}\\
    (y=m)\rightarrow\phi[x|y]\ \ \ \ \ \ \text{6. Deduction Theorem applied to 2 and 5)}\\
    \phi[x|m]\rightarrow((y=m)\rightarrow\phi[x|y])\ \ \ \ \ \ \text{7. Deduction Theorem applied to 1 and 6)}\\
    This is almost what's required, except it's missing the universal quantifier [itex]\forall y[/itex] and I can't use Axiom Schema 4 of Generalisation to add it because y is free in this formula.
    I tried a different approach that gave me the quantifier where I needed it, but it didn't have the right form.

    Any suggestions gratefully received.

    PS the equations are horribly aligned. Any suggestions about how to better align them in TeX would be appreciated too.
    Last edited: Oct 12, 2012
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted

Similar Threads - Problem Logic Hilbert Date
Logical Point in Topological Problem May 4, 2017
Rules of Inference problem Oct 3, 2016
Problem with rectangles May 15, 2015
Logic behind the Lifting Water problem Sep 13, 2014
CNF/DNF logic problem Jan 21, 2013