substitution lemmas for first order logic
How can i prove $ text{$models$}_{Sigma} ((forall x varphi ) Leftrightarrow (forall y [varphi]_{y}^{x}))$.
Being $ Sigma $ a signature, $ varphi$ a formula, and $ [varphi]_{y}^{x} $ the subsistution of $ x$ by $ y $.
When $ y$ is not in the free variables of $ varphi$ and $y triangleright_{Sigma} x : varphi$, which means that $x$ is free for $y$ in $ varphi $
In my opinion i would have to use the substitution's lemma or the one of the hidden variables but i just can't seem to find a way to prove it.
first-order-logic satisfiability hilbert-calculus
asked Jun 2 '16 at 21:46
Bernardo Varanda
324
add a comment |
How can i prove $ text{$models$}_{Sigma} ((forall x varphi ) Leftrightarrow (forall y [varphi]_{y}^{x}))$.
Being $ Sigma $ a signature, $ varphi$ a formula, and $ [varphi]_{y}^{x} $ the subsistution of $ x$ by $ y $.
When $ y$ is not in the free variables of $ varphi$ and $y triangleright_{Sigma} x : varphi$, which means that $x$ is free for $y$ in $ varphi $
In my opinion i would have to use the substitution's lemma or the one of the hidden variables but i just can't seem to find a way to prove it.
first-order-logic satisfiability hilbert-calculus
asked Jun 2 '16 at 21:46
Bernardo Varanda
324
Couldn't you prove the theorem were true for each of the conditional conjuncts of the bi-conditional using the rules for substitution and induction on the relevant formulae, and then just concatenate the two conditionals?
– DMA
Nov 26 at 16:58
add a comment |
How can i prove $ text{$models$}_{Sigma} ((forall x varphi ) Leftrightarrow (forall y [varphi]_{y}^{x}))$.
Being $ Sigma $ a signature, $ varphi$ a formula, and $ [varphi]_{y}^{x} $ the subsistution of $ x$ by $ y $.
When $ y$ is not in the free variables of $ varphi$ and $y triangleright_{Sigma} x : varphi$, which means that $x$ is free for $y$ in $ varphi $
In my opinion i would have to use the substitution's lemma or the one of the hidden variables but i just can't seem to find a way to prove it.
first-order-logic satisfiability hilbert-calculus
asked Jun 2 '16 at 21:46
Bernardo Varanda
324
How can i prove $ text{$models$}_{Sigma} ((forall x varphi ) Leftrightarrow (forall y [varphi]_{y}^{x}))$.
Being $ Sigma $ a signature, $ varphi$ a formula, and $ [varphi]_{y}^{x} $ the subsistution of $ x$ by $ y $.
When $ y$ is not in the free variables of $ varphi$ and $y triangleright_{Sigma} x : varphi$, which means that $x$ is free for $y$ in $ varphi $
In my opinion i would have to use the substitution's lemma or the one of the hidden variables but i just can't seem to find a way to prove it.
first-order-logic satisfiability hilbert-calculus
first-order-logic satisfiability hilbert-calculus
asked Jun 2 '16 at 21:46
Bernardo Varanda
324
asked Jun 2 '16 at 21:46
Bernardo Varanda
324
edited Jun 2 '16 at 22:36
asked Jun 2 '16 at 21:46
Bernardo Varanda
324
asked Jun 2 '16 at 21:46
Bernardo Varanda
324
asked Jun 2 '16 at 21:46
Bernardo Varanda
324
324
Couldn't you prove the theorem were true for each of the conditional conjuncts of the bi-conditional using the rules for substitution and induction on the relevant formulae, and then just concatenate the two conditionals?
– DMA
Nov 26 at 16:58
add a comment |
Couldn't you prove the theorem were true for each of the conditional conjuncts of the bi-conditional using the rules for substitution and induction on the relevant formulae, and then just concatenate the two conditionals?
– DMA
Nov 26 at 16:58
Couldn't you prove the theorem were true for each of the conditional conjuncts of the bi-conditional using the rules for substitution and induction on the relevant formulae, and then just concatenate the two conditionals?
– DMA
Nov 26 at 16:58
Couldn't you prove the theorem were true for each of the conditional conjuncts of the bi-conditional using the rules for substitution and induction on the relevant formulae, and then just concatenate the two conditionals?
– DMA
Nov 26 at 16:58
add a comment |
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Couldn't you prove the theorem were true for each of the conditional conjuncts of the bi-conditional using the rules for substitution and induction on the relevant formulae, and then just concatenate the two conditionals?
– DMA
Nov 26 at 16:58