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New logical language

0

$begingroup$

We define a new language with the following values $1,0,-1$ and the following connectors ∨,∧,¬ :

$neg a = −a $

$𝑎 land 𝑏 = min𝑎, 𝑏$

$𝑎 vee 𝑏 = max𝑎, 𝑏$

$Gamma vdash^cdot phi$ iff $forall v,(v(phi) = 1~textor~exists gammainGamma (v(gamma) in −1,0))$

I managed to prove/disprove all questions but this one:

prove/refute: For every $T$ and $phi$, $Tvdash_CPLphi$ if and only if $Tvdash^cdotϕ$

Cheers!

logic propositional-calculus

share|cite|improve this question

edited Apr 9 at 6:08

Taroccoesbrocco

5,86471840

asked Apr 4 at 8:45

Kamal KhalailyKamal Khalaily

336

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  as defined in the question T ̇⊣ a is equal to Γ ⊢̇ 𝜑, which means Γ ⊢̇ 𝜑 iff ∀v(𝑣(𝜑) = 1 or ∃γ∈Γ (𝑣(𝛾) ∈ −1,0))...I think this happened due to translation issues from Hebrew to English.
  $endgroup$
  – Kamal Khalaily
  Apr 4 at 9:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  ⊢CPL is equal to ⊨; since T⊢CPL a means there exists a v, in which for every b in T, v(b)=v(a)=t.
  $endgroup$
  – Kamal Khalaily
  Apr 4 at 9:15
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  But when you say "T ̇⊣ a is equal to Γ ⊢̇ 𝜑", what is the relation between T and $Gamma$ and between 'a' and '𝜑'? Please, provide all the necessary definitions.
  $endgroup$
  – frabala
  Apr 4 at 9:23
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This means that T and Γ are sets while 𝜑 and a are statements (one element).
  $endgroup$
  – Kamal Khalaily
  Apr 4 at 9:24
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm not sure if I understand correctly. Do you mean to say "for every $T$ and $phi$, $Tvdash_CPLphi$ if and only if $Tvdash^cdotphi$"?
  $endgroup$
  – frabala
  Apr 4 at 9:30
  
  

  
  
  
  

 | 
show 2 more comments

0

$begingroup$

We define a new language with the following values $1,0,-1$ and the following connectors ∨,∧,¬ :

$neg a = −a $

$𝑎 land 𝑏 = min𝑎, 𝑏$

$𝑎 vee 𝑏 = max𝑎, 𝑏$

$Gamma vdash^cdot phi$ iff $forall v,(v(phi) = 1~textor~exists gammainGamma (v(gamma) in −1,0))$

I managed to prove/disprove all questions but this one:

prove/refute: For every $T$ and $phi$, $Tvdash_CPLphi$ if and only if $Tvdash^cdotϕ$

Cheers!

logic propositional-calculus

share|cite|improve this question

edited Apr 9 at 6:08

Taroccoesbrocco

5,86471840

asked Apr 4 at 8:45

Kamal KhalailyKamal Khalaily

336

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  as defined in the question T ̇⊣ a is equal to Γ ⊢̇ 𝜑, which means Γ ⊢̇ 𝜑 iff ∀v(𝑣(𝜑) = 1 or ∃γ∈Γ (𝑣(𝛾) ∈ −1,0))...I think this happened due to translation issues from Hebrew to English.
  $endgroup$
  – Kamal Khalaily
  Apr 4 at 9:13
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  ⊢CPL is equal to ⊨; since T⊢CPL a means there exists a v, in which for every b in T, v(b)=v(a)=t.
  $endgroup$
  – Kamal Khalaily
  Apr 4 at 9:15
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  But when you say "T ̇⊣ a is equal to Γ ⊢̇ 𝜑", what is the relation between T and $Gamma$ and between 'a' and '𝜑'? Please, provide all the necessary definitions.
  $endgroup$
  – frabala
  Apr 4 at 9:23
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This means that T and Γ are sets while 𝜑 and a are statements (one element).
  $endgroup$
  – Kamal Khalaily
  Apr 4 at 9:24
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I'm not sure if I understand correctly. Do you mean to say "for every $T$ and $phi$, $Tvdash_CPLphi$ if and only if $Tvdash^cdotphi$"?
  $endgroup$
  – frabala
  Apr 4 at 9:30
  
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

We define a new language with the following values $1,0,-1$ and the following connectors ∨,∧,¬ :

$neg a = −a $

$𝑎 land 𝑏 = min𝑎, 𝑏$

$𝑎 vee 𝑏 = max𝑎, 𝑏$

$Gamma vdash^cdot phi$ iff $forall v,(v(phi) = 1~textor~exists gammainGamma (v(gamma) in −1,0))$

I managed to prove/disprove all questions but this one:

prove/refute: For every $T$ and $phi$, $Tvdash_CPLphi$ if and only if $Tvdash^cdotϕ$

Cheers!

logic propositional-calculus

share|cite|improve this question

edited Apr 9 at 6:08

Taroccoesbrocco

5,86471840

asked Apr 4 at 8:45

Kamal KhalailyKamal Khalaily

336

$endgroup$

share|cite|improve this question

edited Apr 9 at 6:08

Taroccoesbrocco

5,86471840

asked Apr 4 at 8:45

Kamal KhalailyKamal Khalaily

336

share|cite|improve this question

edited Apr 9 at 6:08

Taroccoesbrocco

5,86471840

asked Apr 4 at 8:45

Kamal KhalailyKamal Khalaily

336

edited Apr 9 at 6:08

Taroccoesbrocco

5,86471840

edited Apr 9 at 6:08

Taroccoesbrocco

5,86471840

asked Apr 4 at 8:45

Kamal KhalailyKamal Khalaily

336

asked Apr 4 at 8:45

Kamal KhalailyKamal Khalaily

336