First-Order Logic and Satisfiability of Sets
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If $phi in Phi$ is a formula, and $c$ is a constant symbol not appearing in $Phi$, show that if $Phi$ is satisfiable, then $Phi, phi[c/x]$ is satisfiable.
Assume that $Phi$ is satisfiable, and let $phi in Phi$ and let $mathbf{c}$ be a constant symbol that does not appear in $Phi$. Then there exists an $mathcal{L}$-valuation $(mathcal{A},alpha)$ such that $mathcal{A}models Phi[alpha]$. Then $mathcal{A} models phi[alpha]$ for all assignments $alpha$. Then $mathcal{A}models phi[alpha[mathbf{c}^{mathcal{A}}[alpha]/x]]$. Then $mathcal{A}models phi[mathbf{c}/x]$. Then $mathcal{A}models Phi, phi[mathbf{c}/x]$.
proof-verification logic first-order-logic
asked Nov 17 at 5:27
numericalorange
1,639311
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Can someone see whether my solution is okay?
If $phi in Phi$ is a formula, and $c$ is a constant symbol not appearing in $Phi$, show that if $Phi$ is satisfiable, then $Phi, phi[c/x]$ is satisfiable.
Assume that $Phi$ is satisfiable, and let $phi in Phi$ and let $mathbf{c}$ be a constant symbol that does not appear in $Phi$. Then there exists an $mathcal{L}$-valuation $(mathcal{A},alpha)$ such that $mathcal{A}models Phi[alpha]$. Then $mathcal{A} models phi[alpha]$ for all assignments $alpha$. Then $mathcal{A}models phi[alpha[mathbf{c}^{mathcal{A}}[alpha]/x]]$. Then $mathcal{A}models phi[mathbf{c}/x]$. Then $mathcal{A}models Phi, phi[mathbf{c}/x]$.
proof-verification logic first-order-logic
asked Nov 17 at 5:27
numericalorange
1,639311
What does it mean $mathbf{c}^mathcal{A}[alpha]$? The assignment $alpha$ should concern only the interpretation of variables, shouldn't it?
– Taroccoesbrocco
Nov 18 at 6:57
add a comment |
up vote
0
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up vote
0
down vote
favorite
Can someone see whether my solution is okay?
If $phi in Phi$ is a formula, and $c$ is a constant symbol not appearing in $Phi$, show that if $Phi$ is satisfiable, then $Phi, phi[c/x]$ is satisfiable.
Assume that $Phi$ is satisfiable, and let $phi in Phi$ and let $mathbf{c}$ be a constant symbol that does not appear in $Phi$. Then there exists an $mathcal{L}$-valuation $(mathcal{A},alpha)$ such that $mathcal{A}models Phi[alpha]$. Then $mathcal{A} models phi[alpha]$ for all assignments $alpha$. Then $mathcal{A}models phi[alpha[mathbf{c}^{mathcal{A}}[alpha]/x]]$. Then $mathcal{A}models phi[mathbf{c}/x]$. Then $mathcal{A}models Phi, phi[mathbf{c}/x]$.
proof-verification logic first-order-logic
asked Nov 17 at 5:27
numericalorange
1,639311
Can someone see whether my solution is okay?
If $phi in Phi$ is a formula, and $c$ is a constant symbol not appearing in $Phi$, show that if $Phi$ is satisfiable, then $Phi, phi[c/x]$ is satisfiable.
Assume that $Phi$ is satisfiable, and let $phi in Phi$ and let $mathbf{c}$ be a constant symbol that does not appear in $Phi$. Then there exists an $mathcal{L}$-valuation $(mathcal{A},alpha)$ such that $mathcal{A}models Phi[alpha]$. Then $mathcal{A} models phi[alpha]$ for all assignments $alpha$. Then $mathcal{A}models phi[alpha[mathbf{c}^{mathcal{A}}[alpha]/x]]$. Then $mathcal{A}models phi[mathbf{c}/x]$. Then $mathcal{A}models Phi, phi[mathbf{c}/x]$.
proof-verification logic first-order-logic
proof-verification logic first-order-logic
asked Nov 17 at 5:27
numericalorange
1,639311
asked Nov 17 at 5:27
numericalorange
1,639311
asked Nov 17 at 5:27
numericalorange
1,639311
asked Nov 17 at 5:27
numericalorange
1,639311
asked Nov 17 at 5:27
numericalorange
1,639311
1,639311
What does it mean $mathbf{c}^mathcal{A}[alpha]$? The assignment $alpha$ should concern only the interpretation of variables, shouldn't it?
– Taroccoesbrocco
Nov 18 at 6:57
add a comment |
What does it mean $mathbf{c}^mathcal{A}[alpha]$? The assignment $alpha$ should concern only the interpretation of variables, shouldn't it?
– Taroccoesbrocco
Nov 18 at 6:57
What does it mean $mathbf{c}^mathcal{A}[alpha]$? The assignment $alpha$ should concern only the interpretation of variables, shouldn't it?
– Taroccoesbrocco
Nov 18 at 6:57
What does it mean $mathbf{c}^mathcal{A}[alpha]$? The assignment $alpha$ should concern only the interpretation of variables, shouldn't it?
– Taroccoesbrocco
Nov 18 at 6:57
add a comment |
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What does it mean $mathbf{c}^mathcal{A}[alpha]$? The assignment $alpha$ should concern only the interpretation of variables, shouldn't it?
– Taroccoesbrocco
Nov 18 at 6:57