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classical logic - rules for quantifiers

0

$begingroup$

I have these formulas of CL:

(a) ∀xP(x,x)

(b) ∀x∀y∀z(P(x,y)∧P(y,z) → P(x,z))

(c) ∀x∀y(P(x,y) → ¬P(y,x)

and I have been trying to prove weather (a),(b) ⊨ (c). First I would use ∀l and then my question is, can I apply this rule for every single ∀x (y or z) and then just have many eigenvariables or is there another way?

If anyone knows of exercises similar to this for practice that would be very helpful

logic proof-theory sequent-calculus

share|cite|improve this question

edited Mar 20 at 14:10

Graham Kemp

87.6k43578

asked Mar 20 at 11:25

rumetalmIrumetalmI

263

$endgroup$

• 
  
  
  

  
  7
  

  
  
  
  
  
  

  
  $begingroup$
  Are you sure that you have to prove that $forall x P(x,x), forall x forall y forall z (P(x,y) land P(y,z) to P(x,z)) vdash forall x forall y (P(x,y) to lnot P(y,x)$ ? If you interpret $P$ as the equality, you can see that such a sequent is not provable.
  $endgroup$
  – Taroccoesbrocco
  Mar 20 at 11:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In general, do you already have the completeness theorem for first-order logic at hand? Then it may be useful for any kind of derivability question to consider the corresponding version using the semantic entailment $models$.
  $endgroup$
  – blub
  Mar 20 at 11:40
  
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  (a), (b) $vDash$ (c) means that there is no interpretation that satisfies (a) and (b) but falsifies (c). You can check with a simple domain $D = a,b $ such that $P(a,a), P(a,b), P(b,a)$ and $P(b,b)$ all hold. In this case, both premises are TRUE and the conclusion is FALSE.
  $endgroup$
  – Mauro ALLEGRANZA
  Mar 20 at 12:58
  
  

  
  
  
  

add a comment | 

0

$begingroup$

I have these formulas of CL:

(a) ∀xP(x,x)

(b) ∀x∀y∀z(P(x,y)∧P(y,z) → P(x,z))

(c) ∀x∀y(P(x,y) → ¬P(y,x)

and I have been trying to prove weather (a),(b) ⊨ (c). First I would use ∀l and then my question is, can I apply this rule for every single ∀x (y or z) and then just have many eigenvariables or is there another way?

If anyone knows of exercises similar to this for practice that would be very helpful

logic proof-theory sequent-calculus

share|cite|improve this question

edited Mar 20 at 14:10

Graham Kemp

87.6k43578

asked Mar 20 at 11:25

rumetalmIrumetalmI

263

$endgroup$

• 
  
  
  

  
  7
  

  
  
  
  
  
  

  
  $begingroup$
  Are you sure that you have to prove that $forall x P(x,x), forall x forall y forall z (P(x,y) land P(y,z) to P(x,z)) vdash forall x forall y (P(x,y) to lnot P(y,x)$ ? If you interpret $P$ as the equality, you can see that such a sequent is not provable.
  $endgroup$
  – Taroccoesbrocco
  Mar 20 at 11:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  In general, do you already have the completeness theorem for first-order logic at hand? Then it may be useful for any kind of derivability question to consider the corresponding version using the semantic entailment $models$.
  $endgroup$
  – blub
  Mar 20 at 11:40
  
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  (a), (b) $vDash$ (c) means that there is no interpretation that satisfies (a) and (b) but falsifies (c). You can check with a simple domain $D = a,b $ such that $P(a,a), P(a,b), P(b,a)$ and $P(b,b)$ all hold. In this case, both premises are TRUE and the conclusion is FALSE.
  $endgroup$
  – Mauro ALLEGRANZA
  Mar 20 at 12:58
  
  

  
  
  
  

add a comment | 

$begingroup$

I have these formulas of CL:

(a) ∀xP(x,x)

(b) ∀x∀y∀z(P(x,y)∧P(y,z) → P(x,z))

(c) ∀x∀y(P(x,y) → ¬P(y,x)

and I have been trying to prove weather (a),(b) ⊨ (c). First I would use ∀l and then my question is, can I apply this rule for every single ∀x (y or z) and then just have many eigenvariables or is there another way?

If anyone knows of exercises similar to this for practice that would be very helpful

logic proof-theory sequent-calculus

share|cite|improve this question

edited Mar 20 at 14:10

Graham Kemp

87.6k43578

asked Mar 20 at 11:25

rumetalmIrumetalmI

263

$endgroup$

share|cite|improve this question

edited Mar 20 at 14:10

Graham Kemp

87.6k43578

asked Mar 20 at 11:25

rumetalmIrumetalmI

263

share|cite|improve this question

edited Mar 20 at 14:10

Graham Kemp

87.6k43578

asked Mar 20 at 11:25

rumetalmIrumetalmI

263

edited Mar 20 at 14:10

Graham Kemp

87.6k43578

edited Mar 20 at 14:10

Graham Kemp

87.6k43578

asked Mar 20 at 11:25

rumetalmIrumetalmI

263

asked Mar 20 at 11:25

rumetalmIrumetalmI

263