数学代写|数理逻辑代写Mathematical logic代考|MATH318

2023年1月6日

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数学代写|数理逻辑代写Mathematical logic代考|Structural Rules and Connective Rules

We divide the rules of the sequent calculus $\mathfrak{S}$ into the following categories: structural rules $(2.1,2.2)$, connective rules $(2.3,2.4,2.5,2.6)$, quantifier rules $(4.1,4.2)$, and equality rules $(4.3,4.4)$. We start with the two structural rules.
2.1 Antecedent Rule (Ant).
$\frac{\Gamma \varphi}{\Gamma^{\prime} \varphi}$ if every member of $\Gamma$ is also a member of $\Gamma^{\prime}$ (briefly: if $\Gamma \subseteq \Gamma^{\prime}$ ).
Note that a formula which occurs more than once in $\Gamma$ need only occur once in $\Gamma^{\prime}$.
2.2 Assumption Rule (Assm).
$\overline{\Gamma \varphi}$ if $\varphi$ is a member of $\Gamma$.
Correctness. (Ant): If a sequent $\Gamma \varphi$ is correct and $\Gamma \subseteq \Gamma^{\prime}$, then since $\Gamma \models \varphi$, also $\Gamma^{\prime} \models \varphi$
(Assm) is correct since $\Phi \models \varphi$ always holds for $\varphi \in \Phi$.
(Assm) reflects the trivial fact that one can conclude $\varphi$ from a set of assumptions which includes $\varphi$. (Ant) expresses the fact that one can re-order or add to assumptions.

Now we state the connective rules. (Remember that we restricted ourselves to the connectives $\neg$ and $\vee$; cf. (1) on page 33.) The first rule is concerned with negation and incorporates the commonly used method of proof by cases. In order to conclude $\varphi$ from $\Gamma$ one first considers the case where a condition $\psi$ holds and then treats the case where $\neg \psi$ holds. That is, one first has $\psi$ and then $\neg \psi$ as an additional assumption. We can translate this argument into a rule for sequents as follows.

数学代写|数理逻辑代写Mathematical logic代考|Henkin’s Theorem

Let $\Phi$ be a consistent set of formulas. In order to find an interpretation $\mathfrak{I}=(\mathfrak{A}, \beta)$ satisfying $\Phi$, we have at our disposal only the “syntactical” information given by the consistency of $\Phi$. Hence, we shall try to obtain a model using syntactical objects as far as possible. A first idea is to take as domain $A$ the set $T^S$ of all $S$-terms, to define $\beta$ by
$$\beta\left(v_i\right):=v_i \text { for } i \in \mathbb{N}$$
and to interpret, for instance, a unary function symbol $f$ by
$$f^{\mathfrak{A}}(t):=f t \text { for } t \in A$$ and a unary relation symbol $R$ by
$$R^{\mathfrak{A}}:={t \in A \mid \Phi \vdash R t} .$$
Then, for a variable $x$ we have $\mathfrak{I}(f x)=f^{\mathfrak{A}}(\beta(x))=f x$. Here a first difficulty arises concerning the equality symbol: If $y$ is a variable different from $x$, then $f x \neq f y$, hence $\mathfrak{I}(f x) \neq \mathfrak{I}(f y)$. If we choose $\Phi$ such that $\Phi \vdash f x \equiv f y($ e.g., $\Phi={f x \equiv f y}$ ), then $\mathfrak{I}$ is not a model of $\Phi$. Namely, by the Correctness Theorem IV.6.2 it follows that $\Phi \models f x \equiv f y$, and with $\mathfrak{I} \models \Phi$ we would have $\mathfrak{I}(f x)=\mathfrak{I}(f y)$.

We overcome this difficulty by defining an equivalence relation on terms and then using the equivalence classes rather than the individual terms as elements of the domain of $\mathfrak{I}$.

Let $\Phi$ be a set of formulas. We define an interpretation $\mathcal{J}^{\Phi}=\left(\mathfrak{T}^{\Phi}, \beta^{\Phi}\right)$. For this purpose we first introduce a binary relation $\sim$ on the set $T^S$ of $S$-terms by
1.1. $t_1 \sim t_2 \quad$ :iff $\quad \Phi \vdash t_1 \equiv t_2$.

数理逻辑代写

数学代写|数理逻辑代写Mathematical logic代考|Structural Rules and Connective Rules

$(4.1,4.2)$ ， 和平等规则 $(4.3,4.4)$. 我们从两个结构规则 开始。
$2.1$ 先行规则 (Ant)。
$\frac{\Gamma \varphi}{\Gamma^{\prime} \varphi}$ 如果每个成员 $\Gamma$ 也是成员 $\Gamma^{\prime}$ （简而言之: 如果 $\left.\Gamma \subseteq \Gamma^{\prime}\right)$.

$2.2$ 假设规则（Assm）。
$\overline{\Gamma \varphi}$ 如果 $\varphi$ 是的成员 $\Gamma$.

(Ass) 是正确的，因为 $\Phi \models \varphi$ 总是适用于 $\varphi \in \Phi$.
(Assm) 反映了一个可以得出结论的微不足道的事实 $\varphi$ 来

数学代写|数理逻辑代写Mathematical logic代考|Henkin’s Theorem

$\beta\left(v_i\right):=v_i$ for $i \in \mathbb{N}$

$$f^{\mathfrak{A}}(t):=f t \text { for } t \in A$$

$$R^{\mathfrak{A}}:=t \in A \mid \Phi \vdash R t .$$

$\mathfrak{I}(f x)=f^{\mathfrak{A}}(\beta(x))=f x$. 这里出现了关于等号的第

$\Phi \vdash f x \equiv f y$ (例如。， $\Phi=f x \equiv f y$ )，然后 $\mathfrak{I}$ 不 是的模型 $\Phi$. 即，根据正确性定理 IV.6.2，它遵循 $\Phi \models f x \equiv f y$ ，与 $\mathfrak{I} \models \Phi$ 我们会有 $\mathfrak{I}(f x)=\mathfrak{I}(f y)$.

1.1。 $t_1 \sim t_2 \quad$ :iff $\quad \Phi \vdash t_1 \equiv t_2$

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