The meaning of $mathcal M vDash (forall y_1)…(forall y_n) phi(a_1,…a_m)$
$begingroup$
This is a question about the meaning of a mathematical sentence.
Let $mathcal M$ be a structure with universe $M$. Let $a_1,...,a_m in M$. Let $phi(x_1,...,x_m,y_1,...,y_n)$ be a quantifier-free formula. What does this mean:
$mathcal M vDash (forall y_1)...(forall y_n) phi(a_1,...a_m)$ $(*)$
Here is the definition of satisfaction,
$ mathcal M vDash psi$ if every sequence $(a_1,a_2...)$ from $M$ satisfy $psi$.
Also
$ mathcal M vDash forall(x_i)psi$ if every sequence that differs from $(a_n)$ in at most the $i^{th}$ component satisfy $psi$.
I have two question actually.
1) Are $(a_1,...,a_m)$ the free variables in $phi$? What if $phi$ is a sentence?
2) Can you give a definition to $(*)$? What I mean by definition is, "$(*)$ happens if this and this and this happens."
first-order-logic
asked Dec 22 '18 at 20:28
offretoffret
15111
$endgroup$
add a comment |
$begingroup$
This is a question about the meaning of a mathematical sentence.
Let $mathcal M$ be a structure with universe $M$. Let $a_1,...,a_m in M$. Let $phi(x_1,...,x_m,y_1,...,y_n)$ be a quantifier-free formula. What does this mean:
$mathcal M vDash (forall y_1)...(forall y_n) phi(a_1,...a_m)$ $(*)$
Here is the definition of satisfaction,
$ mathcal M vDash psi$ if every sequence $(a_1,a_2...)$ from $M$ satisfy $psi$.
Also
$ mathcal M vDash forall(x_i)psi$ if every sequence that differs from $(a_n)$ in at most the $i^{th}$ component satisfy $psi$.
I have two question actually.
1) Are $(a_1,...,a_m)$ the free variables in $phi$? What if $phi$ is a sentence?
2) Can you give a definition to $(*)$? What I mean by definition is, "$(*)$ happens if this and this and this happens."
first-order-logic
asked Dec 22 '18 at 20:28
offretoffret
15111
$endgroup$
$begingroup$
You wrote "$a_1,...,a_m in M$", so $(a_1,...,a_m)$ are not variable at all
$endgroup$
– Holo
Dec 22 '18 at 20:35
$begingroup$
The definition of (*) is that for any interpretation, $σ$, for $cal M$, we have $σ((forall y_1)...(forall y_n) phi(a_1,...a_m))$
$endgroup$
– Holo
Dec 22 '18 at 20:36
$begingroup$
See D.Marker, Model Theory, page 11 for the definition.
$endgroup$
– Mauro ALLEGRANZA
Dec 23 '18 at 11:03
add a comment |
$begingroup$
This is a question about the meaning of a mathematical sentence.
Let $mathcal M$ be a structure with universe $M$. Let $a_1,...,a_m in M$. Let $phi(x_1,...,x_m,y_1,...,y_n)$ be a quantifier-free formula. What does this mean:
$mathcal M vDash (forall y_1)...(forall y_n) phi(a_1,...a_m)$ $(*)$
Here is the definition of satisfaction,
$ mathcal M vDash psi$ if every sequence $(a_1,a_2...)$ from $M$ satisfy $psi$.
Also
$ mathcal M vDash forall(x_i)psi$ if every sequence that differs from $(a_n)$ in at most the $i^{th}$ component satisfy $psi$.
I have two question actually.
1) Are $(a_1,...,a_m)$ the free variables in $phi$? What if $phi$ is a sentence?
2) Can you give a definition to $(*)$? What I mean by definition is, "$(*)$ happens if this and this and this happens."
first-order-logic
asked Dec 22 '18 at 20:28
offretoffret
15111
$endgroup$
This is a question about the meaning of a mathematical sentence.
Let $mathcal M$ be a structure with universe $M$. Let $a_1,...,a_m in M$. Let $phi(x_1,...,x_m,y_1,...,y_n)$ be a quantifier-free formula. What does this mean:
$mathcal M vDash (forall y_1)...(forall y_n) phi(a_1,...a_m)$ $(*)$
Here is the definition of satisfaction,
$ mathcal M vDash psi$ if every sequence $(a_1,a_2...)$ from $M$ satisfy $psi$.
Also
$ mathcal M vDash forall(x_i)psi$ if every sequence that differs from $(a_n)$ in at most the $i^{th}$ component satisfy $psi$.
I have two question actually.
1) Are $(a_1,...,a_m)$ the free variables in $phi$? What if $phi$ is a sentence?
2) Can you give a definition to $(*)$? What I mean by definition is, "$(*)$ happens if this and this and this happens."
first-order-logic
first-order-logic
asked Dec 22 '18 at 20:28
offretoffret
15111
asked Dec 22 '18 at 20:28
offretoffret
15111
asked Dec 22 '18 at 20:28
offretoffret
15111
asked Dec 22 '18 at 20:28
offretoffret
15111
asked Dec 22 '18 at 20:28
offretoffret
15111
15111
$begingroup$
You wrote "$a_1,...,a_m in M$", so $(a_1,...,a_m)$ are not variable at all
$endgroup$
– Holo
Dec 22 '18 at 20:35
$begingroup$
The definition of (*) is that for any interpretation, $σ$, for $cal M$, we have $σ((forall y_1)...(forall y_n) phi(a_1,...a_m))$
$endgroup$
– Holo
Dec 22 '18 at 20:36
$begingroup$
See D.Marker, Model Theory, page 11 for the definition.
$endgroup$
– Mauro ALLEGRANZA
Dec 23 '18 at 11:03
add a comment |
$begingroup$
You wrote "$a_1,...,a_m in M$", so $(a_1,...,a_m)$ are not variable at all
$endgroup$
– Holo
Dec 22 '18 at 20:35
$begingroup$
The definition of (*) is that for any interpretation, $σ$, for $cal M$, we have $σ((forall y_1)...(forall y_n) phi(a_1,...a_m))$
$endgroup$
– Holo
Dec 22 '18 at 20:36
$begingroup$
See D.Marker, Model Theory, page 11 for the definition.
$endgroup$
– Mauro ALLEGRANZA
Dec 23 '18 at 11:03
$begingroup$
You wrote "$a_1,...,a_m in M$", so $(a_1,...,a_m)$ are not variable at all
$endgroup$
– Holo
Dec 22 '18 at 20:35
$begingroup$
You wrote "$a_1,...,a_m in M$", so $(a_1,...,a_m)$ are not variable at all
$endgroup$
– Holo
Dec 22 '18 at 20:35
$begingroup$
The definition of (*) is that for any interpretation, $σ$, for $cal M$, we have $σ((forall y_1)...(forall y_n) phi(a_1,...a_m))$
$endgroup$
– Holo
Dec 22 '18 at 20:36
$begingroup$
The definition of (*) is that for any interpretation, $σ$, for $cal M$, we have $σ((forall y_1)...(forall y_n) phi(a_1,...a_m))$
$endgroup$
– Holo
Dec 22 '18 at 20:36
$begingroup$
See D.Marker, Model Theory, page 11 for the definition.
$endgroup$
– Mauro ALLEGRANZA
Dec 23 '18 at 11:03
$begingroup$
See D.Marker, Model Theory, page 11 for the definition.
$endgroup$
– Mauro ALLEGRANZA
Dec 23 '18 at 11:03
add a comment |
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$begingroup$
You wrote "$a_1,...,a_m in M$", so $(a_1,...,a_m)$ are not variable at all
$endgroup$
– Holo
Dec 22 '18 at 20:35
$begingroup$
The definition of (*) is that for any interpretation, $σ$, for $cal M$, we have $σ((forall y_1)...(forall y_n) phi(a_1,...a_m))$
$endgroup$
– Holo
Dec 22 '18 at 20:36
$begingroup$
See D.Marker, Model Theory, page 11 for the definition.
$endgroup$
– Mauro ALLEGRANZA
Dec 23 '18 at 11:03