if $vDash alpha Rightarrow vDash beta$ then $vDash alpha to beta$?
$begingroup$
if $vDash alpha Rightarrow vDash beta$ then $vDash alpha to beta$ ?
Is this proposition true? And what about converse? I also wonder about the difference between $Rightarrow$ and $to$ in this proposition.
logic propositional-calculus
edited Dec 18 '18 at 15:53
Mauro ALLEGRANZA
66.4k449115
asked Dec 18 '18 at 14:15
amoogaeamoogae
487
$endgroup$
add a comment |
$begingroup$
if $vDash alpha Rightarrow vDash beta$ then $vDash alpha to beta$ ?
Is this proposition true? And what about converse? I also wonder about the difference between $Rightarrow$ and $to$ in this proposition.
logic propositional-calculus
edited Dec 18 '18 at 15:53
Mauro ALLEGRANZA
66.4k449115
asked Dec 18 '18 at 14:15
amoogaeamoogae
487
$endgroup$
add a comment |
$begingroup$
if $vDash alpha Rightarrow vDash beta$ then $vDash alpha to beta$ ?
Is this proposition true? And what about converse? I also wonder about the difference between $Rightarrow$ and $to$ in this proposition.
logic propositional-calculus
edited Dec 18 '18 at 15:53
Mauro ALLEGRANZA
66.4k449115
asked Dec 18 '18 at 14:15
amoogaeamoogae
487
$endgroup$
if $vDash alpha Rightarrow vDash beta$ then $vDash alpha to beta$ ?
Is this proposition true? And what about converse? I also wonder about the difference between $Rightarrow$ and $to$ in this proposition.
logic propositional-calculus
logic propositional-calculus
edited Dec 18 '18 at 15:53
Mauro ALLEGRANZA
66.4k449115
asked Dec 18 '18 at 14:15
amoogaeamoogae
487
edited Dec 18 '18 at 15:53
Mauro ALLEGRANZA
66.4k449115
asked Dec 18 '18 at 14:15
amoogaeamoogae
487
edited Dec 18 '18 at 15:53
Mauro ALLEGRANZA
66.4k449115
edited Dec 18 '18 at 15:53
Mauro ALLEGRANZA
66.4k449115
edited Dec 18 '18 at 15:53
Mauro ALLEGRANZA
66.4k449115
66.4k449115
asked Dec 18 '18 at 14:15
amoogaeamoogae
487
asked Dec 18 '18 at 14:15
amoogaeamoogae
487
asked Dec 18 '18 at 14:15
amoogaeamoogae
487
487
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$to$ is the propositional connective "if..., then...". It is part of the language of propositional logic and it is used (with other connectives) to symbolize the formulas of the formal system.
The symbol $vDash alpha$ is not part of the language of the system but is part of the meta-language : it asserts the fact that formula $alpha$ is a tautology (or logically valid).
Thus, $Rightarrow$ is used in te meta-language to abbreviate "if..., then..." when expressing properties of the formal system.
Regarding the question :
if $⊨ α ⇒ ⊨ β$, then $⊨ α → β$ ?
the answer is no.
Consider the atom $p$ as $alpha$ and the atom $q$ as $beta$.
We have that $vDash p$ is $text {False}$.
Thus :
$vDash p Rightarrow vDash q$ holds,
because $text {False } Rightarrow text { False}$ is $text {True}$.
But $nvDash (p to q)$, because $p to q$ is not a tautology.
About the converse : assume that $vDash alpha to beta$ and $vDash alpha$.
We reason by contradiction, i.e. assume that $nvDash beta$. This means that for some truth assignment $v$ we have $v(beta)=$ f.
But from $vDash alpha$, we have that $v(alpha)=$ t.
Thus :
$v(alpha to beta)=$ f,
contradicting $vDash alpha to beta$.
answered Dec 18 '18 at 14:24
Mauro ALLEGRANZAMauro ALLEGRANZA
66.4k449115
$endgroup$
$begingroup$
Thank you very much! What about "if $ vDash alpha to beta $ then $ vDash alpha Rightarrow vDash beta $?" I think this is true, but I do not know how to prove it.
$endgroup$
– amoogae
Dec 18 '18 at 14:48
$begingroup$
Aha! Thank you! But I have another question. The meaning of $ Rightarrow $ is the same as the $ to $ in propositional logic?? For example, if $alpha$ is true and $beta$ is false then $alpha Rightarrow beta $ is false, if $alpha$ is false and $beta$ is false then $alpha Rightarrow beta $ is false ... and so on?
$endgroup$
– amoogae
Dec 18 '18 at 15:25
$begingroup$
@amoogae - The meaning of ⇒ is the same as the → in propositional logic? YES, but then if α is false and β is false then α⇒β is true.
$endgroup$
– Mauro ALLEGRANZA
Dec 18 '18 at 15:53
$begingroup$
Oh my mistake. Thank you!
$endgroup$
– amoogae
Dec 18 '18 at 16:06
$begingroup$
By the way, this story sounds to me as saying "the meta language of propositional logic is propositional logic". Can I understand it this way?
$endgroup$
– amoogae
Dec 19 '18 at 8:12
|
show 2 more comments
Your Answer
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1 Answer
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1 Answer
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$begingroup$
$to$ is the propositional connective "if..., then...". It is part of the language of propositional logic and it is used (with other connectives) to symbolize the formulas of the formal system.
The symbol $vDash alpha$ is not part of the language of the system but is part of the meta-language : it asserts the fact that formula $alpha$ is a tautology (or logically valid).
Thus, $Rightarrow$ is used in te meta-language to abbreviate "if..., then..." when expressing properties of the formal system.
Regarding the question :
if $⊨ α ⇒ ⊨ β$, then $⊨ α → β$ ?
the answer is no.
Consider the atom $p$ as $alpha$ and the atom $q$ as $beta$.
We have that $vDash p$ is $text {False}$.
Thus :
$vDash p Rightarrow vDash q$ holds,
because $text {False } Rightarrow text { False}$ is $text {True}$.
But $nvDash (p to q)$, because $p to q$ is not a tautology.
About the converse : assume that $vDash alpha to beta$ and $vDash alpha$.
We reason by contradiction, i.e. assume that $nvDash beta$. This means that for some truth assignment $v$ we have $v(beta)=$ f.
But from $vDash alpha$, we have that $v(alpha)=$ t.
Thus :
$v(alpha to beta)=$ f,
contradicting $vDash alpha to beta$.
answered Dec 18 '18 at 14:24
Mauro ALLEGRANZAMauro ALLEGRANZA
66.4k449115
$endgroup$
$begingroup$
Thank you very much! What about "if $ vDash alpha to beta $ then $ vDash alpha Rightarrow vDash beta $?" I think this is true, but I do not know how to prove it.
$endgroup$
– amoogae
Dec 18 '18 at 14:48
$begingroup$
Aha! Thank you! But I have another question. The meaning of $ Rightarrow $ is the same as the $ to $ in propositional logic?? For example, if $alpha$ is true and $beta$ is false then $alpha Rightarrow beta $ is false, if $alpha$ is false and $beta$ is false then $alpha Rightarrow beta $ is false ... and so on?
$endgroup$
– amoogae
Dec 18 '18 at 15:25
$begingroup$
@amoogae - The meaning of ⇒ is the same as the → in propositional logic? YES, but then if α is false and β is false then α⇒β is true.
$endgroup$
– Mauro ALLEGRANZA
Dec 18 '18 at 15:53
$begingroup$
Oh my mistake. Thank you!
$endgroup$
– amoogae
Dec 18 '18 at 16:06
$begingroup$
By the way, this story sounds to me as saying "the meta language of propositional logic is propositional logic". Can I understand it this way?
$endgroup$
– amoogae
Dec 19 '18 at 8:12
|
show 2 more comments
$begingroup$
$to$ is the propositional connective "if..., then...". It is part of the language of propositional logic and it is used (with other connectives) to symbolize the formulas of the formal system.
The symbol $vDash alpha$ is not part of the language of the system but is part of the meta-language : it asserts the fact that formula $alpha$ is a tautology (or logically valid).
Thus, $Rightarrow$ is used in te meta-language to abbreviate "if..., then..." when expressing properties of the formal system.
Regarding the question :
if $⊨ α ⇒ ⊨ β$, then $⊨ α → β$ ?
the answer is no.
Consider the atom $p$ as $alpha$ and the atom $q$ as $beta$.
We have that $vDash p$ is $text {False}$.
Thus :
$vDash p Rightarrow vDash q$ holds,
because $text {False } Rightarrow text { False}$ is $text {True}$.
But $nvDash (p to q)$, because $p to q$ is not a tautology.
About the converse : assume that $vDash alpha to beta$ and $vDash alpha$.
We reason by contradiction, i.e. assume that $nvDash beta$. This means that for some truth assignment $v$ we have $v(beta)=$ f.
But from $vDash alpha$, we have that $v(alpha)=$ t.
Thus :
$v(alpha to beta)=$ f,
contradicting $vDash alpha to beta$.
answered Dec 18 '18 at 14:24
Mauro ALLEGRANZAMauro ALLEGRANZA
66.4k449115
$endgroup$
$begingroup$
Thank you very much! What about "if $ vDash alpha to beta $ then $ vDash alpha Rightarrow vDash beta $?" I think this is true, but I do not know how to prove it.
$endgroup$
– amoogae
Dec 18 '18 at 14:48
$begingroup$
Aha! Thank you! But I have another question. The meaning of $ Rightarrow $ is the same as the $ to $ in propositional logic?? For example, if $alpha$ is true and $beta$ is false then $alpha Rightarrow beta $ is false, if $alpha$ is false and $beta$ is false then $alpha Rightarrow beta $ is false ... and so on?
$endgroup$
– amoogae
Dec 18 '18 at 15:25
$begingroup$
@amoogae - The meaning of ⇒ is the same as the → in propositional logic? YES, but then if α is false and β is false then α⇒β is true.
$endgroup$
– Mauro ALLEGRANZA
Dec 18 '18 at 15:53
$begingroup$
Oh my mistake. Thank you!
$endgroup$
– amoogae
Dec 18 '18 at 16:06
$begingroup$
By the way, this story sounds to me as saying "the meta language of propositional logic is propositional logic". Can I understand it this way?
$endgroup$
– amoogae
Dec 19 '18 at 8:12
|
show 2 more comments
$begingroup$
$to$ is the propositional connective "if..., then...". It is part of the language of propositional logic and it is used (with other connectives) to symbolize the formulas of the formal system.
The symbol $vDash alpha$ is not part of the language of the system but is part of the meta-language : it asserts the fact that formula $alpha$ is a tautology (or logically valid).
Thus, $Rightarrow$ is used in te meta-language to abbreviate "if..., then..." when expressing properties of the formal system.
Regarding the question :
if $⊨ α ⇒ ⊨ β$, then $⊨ α → β$ ?
the answer is no.
Consider the atom $p$ as $alpha$ and the atom $q$ as $beta$.
We have that $vDash p$ is $text {False}$.
Thus :
$vDash p Rightarrow vDash q$ holds,
because $text {False } Rightarrow text { False}$ is $text {True}$.
But $nvDash (p to q)$, because $p to q$ is not a tautology.
About the converse : assume that $vDash alpha to beta$ and $vDash alpha$.
We reason by contradiction, i.e. assume that $nvDash beta$. This means that for some truth assignment $v$ we have $v(beta)=$ f.
But from $vDash alpha$, we have that $v(alpha)=$ t.
Thus :
$v(alpha to beta)=$ f,
contradicting $vDash alpha to beta$.
answered Dec 18 '18 at 14:24
Mauro ALLEGRANZAMauro ALLEGRANZA
66.4k449115
$endgroup$
$to$ is the propositional connective "if..., then...". It is part of the language of propositional logic and it is used (with other connectives) to symbolize the formulas of the formal system.
The symbol $vDash alpha$ is not part of the language of the system but is part of the meta-language : it asserts the fact that formula $alpha$ is a tautology (or logically valid).
Thus, $Rightarrow$ is used in te meta-language to abbreviate "if..., then..." when expressing properties of the formal system.
Regarding the question :
if $⊨ α ⇒ ⊨ β$, then $⊨ α → β$ ?
the answer is no.
Consider the atom $p$ as $alpha$ and the atom $q$ as $beta$.
We have that $vDash p$ is $text {False}$.
Thus :
$vDash p Rightarrow vDash q$ holds,
because $text {False } Rightarrow text { False}$ is $text {True}$.
But $nvDash (p to q)$, because $p to q$ is not a tautology.
About the converse : assume that $vDash alpha to beta$ and $vDash alpha$.
We reason by contradiction, i.e. assume that $nvDash beta$. This means that for some truth assignment $v$ we have $v(beta)=$ f.
But from $vDash alpha$, we have that $v(alpha)=$ t.
Thus :
$v(alpha to beta)=$ f,
contradicting $vDash alpha to beta$.
answered Dec 18 '18 at 14:24
Mauro ALLEGRANZAMauro ALLEGRANZA
66.4k449115
edited Dec 18 '18 at 15:08
answered Dec 18 '18 at 14:24
Mauro ALLEGRANZAMauro ALLEGRANZA
66.4k449115
answered Dec 18 '18 at 14:24
Mauro ALLEGRANZAMauro ALLEGRANZA
66.4k449115
answered Dec 18 '18 at 14:24
Mauro ALLEGRANZAMauro ALLEGRANZA
66.4k449115
66.4k449115
$begingroup$
Thank you very much! What about "if $ vDash alpha to beta $ then $ vDash alpha Rightarrow vDash beta $?" I think this is true, but I do not know how to prove it.
$endgroup$
– amoogae
Dec 18 '18 at 14:48
$begingroup$
Aha! Thank you! But I have another question. The meaning of $ Rightarrow $ is the same as the $ to $ in propositional logic?? For example, if $alpha$ is true and $beta$ is false then $alpha Rightarrow beta $ is false, if $alpha$ is false and $beta$ is false then $alpha Rightarrow beta $ is false ... and so on?
$endgroup$
– amoogae
Dec 18 '18 at 15:25
$begingroup$
@amoogae - The meaning of ⇒ is the same as the → in propositional logic? YES, but then if α is false and β is false then α⇒β is true.
$endgroup$
– Mauro ALLEGRANZA
Dec 18 '18 at 15:53
$begingroup$
Oh my mistake. Thank you!
$endgroup$
– amoogae
Dec 18 '18 at 16:06
$begingroup$
By the way, this story sounds to me as saying "the meta language of propositional logic is propositional logic". Can I understand it this way?
$endgroup$
– amoogae
Dec 19 '18 at 8:12
|
show 2 more comments
$begingroup$
Thank you very much! What about "if $ vDash alpha to beta $ then $ vDash alpha Rightarrow vDash beta $?" I think this is true, but I do not know how to prove it.
$endgroup$
– amoogae
Dec 18 '18 at 14:48
$begingroup$
Aha! Thank you! But I have another question. The meaning of $ Rightarrow $ is the same as the $ to $ in propositional logic?? For example, if $alpha$ is true and $beta$ is false then $alpha Rightarrow beta $ is false, if $alpha$ is false and $beta$ is false then $alpha Rightarrow beta $ is false ... and so on?
$endgroup$
– amoogae
Dec 18 '18 at 15:25
$begingroup$
@amoogae - The meaning of ⇒ is the same as the → in propositional logic? YES, but then if α is false and β is false then α⇒β is true.
$endgroup$
– Mauro ALLEGRANZA
Dec 18 '18 at 15:53
$begingroup$
Oh my mistake. Thank you!
$endgroup$
– amoogae
Dec 18 '18 at 16:06
$begingroup$
By the way, this story sounds to me as saying "the meta language of propositional logic is propositional logic". Can I understand it this way?
$endgroup$
– amoogae
Dec 19 '18 at 8:12
$begingroup$
Thank you very much! What about "if $ vDash alpha to beta $ then $ vDash alpha Rightarrow vDash beta $?" I think this is true, but I do not know how to prove it.
$endgroup$
– amoogae
Dec 18 '18 at 14:48
$begingroup$
Thank you very much! What about "if $ vDash alpha to beta $ then $ vDash alpha Rightarrow vDash beta $?" I think this is true, but I do not know how to prove it.
$endgroup$
– amoogae
Dec 18 '18 at 14:48
$begingroup$
Aha! Thank you! But I have another question. The meaning of $ Rightarrow $ is the same as the $ to $ in propositional logic?? For example, if $alpha$ is true and $beta$ is false then $alpha Rightarrow beta $ is false, if $alpha$ is false and $beta$ is false then $alpha Rightarrow beta $ is false ... and so on?
$endgroup$
– amoogae
Dec 18 '18 at 15:25
$begingroup$
Aha! Thank you! But I have another question. The meaning of $ Rightarrow $ is the same as the $ to $ in propositional logic?? For example, if $alpha$ is true and $beta$ is false then $alpha Rightarrow beta $ is false, if $alpha$ is false and $beta$ is false then $alpha Rightarrow beta $ is false ... and so on?
$endgroup$
– amoogae
Dec 18 '18 at 15:25
$begingroup$
@amoogae - The meaning of ⇒ is the same as the → in propositional logic? YES, but then if α is false and β is false then α⇒β is true.
$endgroup$
– Mauro ALLEGRANZA
Dec 18 '18 at 15:53
$begingroup$
@amoogae - The meaning of ⇒ is the same as the → in propositional logic? YES, but then if α is false and β is false then α⇒β is true.
$endgroup$
– Mauro ALLEGRANZA
Dec 18 '18 at 15:53
$begingroup$
Oh my mistake. Thank you!
$endgroup$
– amoogae
Dec 18 '18 at 16:06
$begingroup$
Oh my mistake. Thank you!
$endgroup$
– amoogae
Dec 18 '18 at 16:06
$begingroup$
By the way, this story sounds to me as saying "the meta language of propositional logic is propositional logic". Can I understand it this way?
$endgroup$
– amoogae
Dec 19 '18 at 8:12
$begingroup$
By the way, this story sounds to me as saying "the meta language of propositional logic is propositional logic". Can I understand it this way?
$endgroup$
– amoogae
Dec 19 '18 at 8:12
|
show 2 more comments
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