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# 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|MHF5306 Deduction Theorem in G and N

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|Deduction Theorem in G and N

Observe that Gentzen’s rules $(\vdash \vee)$ and $(\wedge \vdash)$, Section 8.4, indicate the semantics of sequents. $A_1 \ldots A_n \vdash_{\mathcal{G}} B_1 \ldots B_m$ corresponds by these rules to $A_1 \wedge \ldots \wedge A_n \vdash_g B_1 \vee \ldots \vee B_m$, and by rule $(\vdash \rightarrow)$ to $\vdash_q\left(A_1 \wedge \ldots \wedge A_n\right) \rightarrow$ $\left(B_1 \vee \ldots \vee B_m\right)$ which is a simple formula with the expected semantics corresponding to the semantics of the original sequent.

Now $\mathcal{G}$ for $\mathrm{FOL}$, unlike $\mathcal{N}$, is a truly natural deduction system. Rule $(\vdash \rightarrow)$ is the unrestricted Deduction Theorem built into $\mathcal{G}$. Recall that it was not so for $\mathcal{N}$ – Deduction Theorem $8.29$ allowed us to use a restricted version of the rule: $\frac{\Gamma . A \vdash_{\mathcal{N}} B}{\Gamma \vdash_{\mathcal{N}} A \rightarrow B}$ only if $A$ is closed! Without this restriction, the rule would be unsound, e.g.:

1. $A \vdash_{\mathcal{N}} A$
2. $A \vdash_{\mathcal{N}} \forall x A \quad$ L.8.24.(4)
3. $\vdash_{\mathcal{N}} A \rightarrow \forall x A \quad D T !$
4. $\vdash_{\mathcal{N}} \exists x A \rightarrow \forall x A \quad \exists \mathrm{I}$
The conclusion of this proof is obviously invalid (verify this) and we could derive it only using a wrong application of DT in line 3.

In $\mathcal{G}$, such a proof cannot proceed beyond step 1. Rule $(\vdash \forall)$ requires replacement of $x$ from $\forall x A$ by a fresh $x^{\prime}$, i.e., not occurring (freely) in the sequent. Attempting this proof in $\mathcal{G}$ would lead to the following:

1. $A\left(x^{\prime}\right) \vdash_{\mathfrak{g}} A(x)$
2. $A\left(x^{\prime}\right) \vdash_g \forall x A(x) \quad(\vdash \forall), x$ fresh $\left(x \neq x^{\prime}\right)$
3. $\exists x A(x) \vdash_{\mathcal{G}} \forall x A(x) \quad(\exists \vdash), x^{\prime}$ fresh
4. $\quad \vdash_{\mathcal{g}} \exists x A(x) \rightarrow \forall x A(x) \quad(\vdash \rightarrow)$

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|PRENEX NORMAL FORM

We have seen in Corollaries $6.7$ and $6.8$ that every PL formula can be written equivalently in DNF and CNF. A normal form which is particularly useful in the study of FOL is Prenex Normal Form.

Definition 10.1 [PNF] A formula $A$ is in Prenex Normal Form iff it has the form $Q_1 x_1 \ldots Q_n x_n B$, where $Q_i$ are quantifiers and $B$ contains no quantifiers.
The quantifier part $Q_1 x_1 \ldots Q_n x_n$ is called the prefix, and the quantifier-free part $B$ the matrix of $A$.

To show that each formula is equivalent to some formula in PNF we need the next lemma.

Proof. Exercise $5.15$ showed the version for PL. The proof is by induction on the complexity of $F[A]$, with a special case considered first: $F[A]$ is :
$A::$ This is a special case in which we have trivially $F[A]=$ $A \Leftrightarrow B=F[B]$.
In the rest of the proof, assume that we are not in the special case.
Aтоміс :: If $F[A]$ is atomic then either we have the special case, or no replacement is made, i.e., $F[A]=F[B]$, since $F$ has no subformula $A$.
$$\neg C[A]:: \text { By IH } C[A] \Leftrightarrow C[B] \text {. So } \neg C[A] \Leftrightarrow \neg C[B] \text {. }$$

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代 考|Deduction Theorem in $\mathbf{G}$ and $\mathbf{N}$

$A \vdash_{\mathcal{N}} A$

$A \vdash_{\mathcal{N}} \forall x A \quad$ L.8.24.(4)

$\vdash_{\mathcal{N}} A \rightarrow \forall x A \quad D T$ !

$\vdash_{\mathcal{N}} \exists x A \rightarrow \forall x A \quad \exists \mathrm{I}$

$A\left(x^{\prime}\right) \vdash_{\mathfrak{g}} A(x)$

$A\left(x^{\prime}\right) \vdash_g \forall x A(x) \quad(\vdash \forall), x$ 新鲜的 $\left(x \neq x^{\prime}\right)$

$\exists x A(x) \vdash \mathcal{G} \forall x A(x) \quad(\exists \vdash), x$ 新鮮的

$\vdash_g \exists x A(x) \rightarrow \forall x A(x) \quad(\vdash \rightarrow)$

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代 考|PRENEX NORMAL FORM

$A::$ :这是一个特殊情况，我们有微不足道的 $F[A]=A \Leftrightarrow B=F[B]$.

Атоміс :: 如果 $F[A]$ 是原子的，那么要么我们有特殊情况，要么不进行替换，即 $F[A]=F[B]$ ，自从 $F$ 你没有 子公式 $A$.
$$\neg C[A]:: \text { By IH } C[A] \Leftrightarrow C[B] \text {. So } \neg C[A] \Leftrightarrow \neg C[B] \text {. }$$

## MATLAB代写

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