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# 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|MATH3306 SOME EXAMPLES

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## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|SOME EXAMPLES

Before we begin a closer discussion of the syntax of FOL, we give a few examples of vocabularies (alphabets) which can be used for describing some known structures.
Example 8.14 [Stacks]
Using the alphabet of stacks from Example 8.2, we may now set up the following (non-logical) axioms $\Gamma_{\text {Stack }}$ for the theory of stacks $(x, s \in \mathcal{V})$ :
(1) $\operatorname{St}($ empty)
(2) $\forall x, s: E l(x) \wedge S t(s) \rightarrow S t(p u s h(s, x))$
(3) $\forall s: S t(s) \rightarrow S t(p o p(s))$
(4) $\forall s: S t(s) \rightarrow E l(t o p(s))$.

These axioms describe merely the profiles of the functions and relate the extensions of the predicates. According to (1) empty is a stack, while (2) says that if $x$ is an element and $s$ is a stack then also the result of push $(s, x)$ is stack. (Usually, one uses some abbreviated notation to capture this information. In typed programming languages, for instance, it is taken care of by the typing system.) Further (non-logical) axioms, determining more specific properties of stacks, could then be:
(5) $\operatorname{pop}($ empty $) \equiv$ empty
(6) $\forall x, s: E l(x) \wedge S t(s) \rightarrow \operatorname{pop}(p u s h(s, x)) \equiv s$
(7) $\forall x, s: \operatorname{El}(x) \wedge S t(s) \rightarrow \operatorname{top}(p u s h(s, x)) \equiv x$.

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

In a given term $t$ or formula $A$, we may substitute a term for the free occurrences of any variable. This can be done simultaneously for several variables. A substitution is a function from some set of variables $X$ to terms, $\sigma: X \rightarrow \mathcal{T}_{\Sigma}$, and the result of applying it to a term/formula, $\bar{\sigma}(t) / \bar{\sigma}(A)$, is defined inductively.

Definition $8.18$ Application of a substitution $\sigma$ to terms, $\bar{\sigma}: \mathcal{T}{\Sigma} \rightarrow \mathcal{T}{\Sigma}$, is defined inductively on the structure of terms:
$$\begin{gathered} x \in \mathcal{V}:: \bar{\sigma}(x) \stackrel{\text { def }}{=} \begin{cases}\sigma(x) & \text { if } x \in \operatorname{dom}(\sigma) \ x & \text { if } x \notin \operatorname{dom}(\sigma)\end{cases} \ f\left(t_1, \ldots, t_k\right):: \bar{\sigma}\left(f\left(t_1, \ldots, t_k\right)\right) \stackrel{\text { def }}{=} f\left(\bar{\sigma}\left(t_1\right), \ldots, \bar{\sigma}\left(t_k\right)\right) \end{gathered}$$
This determines the result $\bar{\sigma}(t)$ of substitution $\sigma$ into any term $t$. Building on this, the application of substitution $\sigma$ to formulae, $\bar{\sigma}: \mathrm{WFF}^{\Sigma} \rightarrow \mathrm{WFF}^{\Sigma}$, is defined by induction on the complexity of formulae:
\begin{aligned} & \text { Atomic :: } \bar{\sigma}\left(P\left(t_1, \ldots, t_k\right)\right) \stackrel{\text { def }}{=} P\left(\bar{\sigma}\left(t_1\right), \ldots, \bar{\sigma}\left(t_k\right)\right) \ & \neg B:: \quad \bar{\sigma}(\neg B) \stackrel{\text { def }}{=} \neg \bar{\sigma}(B) \ & B \rightarrow C:: \quad \bar{\sigma}(B \rightarrow C) \stackrel{\text { def }}{=} \bar{\sigma}(B) \rightarrow \bar{\sigma}(C) \ & \exists x A:: \quad \bar{\sigma}(\exists x A) \stackrel{\text { def }}{=} \exists x \bar{\sigma}^{\prime}(A) \text { where } \sigma^{\prime}=\sigma \backslash\left({x} \times \mathcal{T}_{\Sigma}\right) \text {. } \ & \end{aligned}

## 数学代写|数理逻辑入门代写Introduction To Mathematical logic代考|SOME EXAMPLES

(1) $\mathrm{St}$ (空)
(2) $\forall x, s: E l(x) \wedge S t(s) \rightarrow S t(p u s h(s, x))$
(3) $\forall s: S t(s) \rightarrow S t(p o p(s))$
(4) $\forall s: S t(s) \rightarrow E l(\operatorname{top}(s))$.

(5)pop(空的) 三空的
(6) $\forall x, s: E l(x) \wedge S t(s) \rightarrow \operatorname{pop}($ push $(s, x)) \equiv s$
(7) $\forall x, s: \operatorname{El}(x) \wedge S t(s) \rightarrow \operatorname{top}($ push $(s, x)) \equiv x$.

## 数学代写数理逻辑入门代写Introduction To Mathematical logic代考|SUBSTITUTION

$$x \in \mathcal{V}:: \bar{\sigma}(x) \stackrel{\text { def }}{=}\left{\sigma(x) \quad \text { if } x \in \operatorname{dom}(\sigma) x \quad \text { if } x \notin \operatorname{dom}(\sigma) f\left(t_1, \ldots, t_k\right):: \bar{\sigma}\left(f\left(t_1, \ldots, t_k\right)\right) \stackrel{\text { def }}{=} f\left(\bar{\sigma}\left(t_1\right), \ldots, \bar{\sigma}\left(t_k\right)\right)\right.$$

$$\text { Atomic :: } \bar{\sigma}\left(P\left(t_1, \ldots, t_k\right)\right) \stackrel{\text { def }}{=} P\left(\bar{\sigma}\left(t_1\right), \ldots, \bar{\sigma}\left(t_k\right)\right) \quad \neg B:: \quad \bar{\sigma}(\neg B) \stackrel{\text { def }}{=} \neg \bar{\sigma}(B) B \rightarrow C:: \quad \bar{\sigma}(B \rightarrow C) \stackrel{\text { def }}{=} \bar{\sigma}(B) \rightarrow \bar{\sigma}(C) \quad \exists x A:: \quad \bar{\sigma}(\exists x A) \stackrel{\text { def }}{=}$$

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