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# 数学代写|数理逻辑代考Mathematical logic代写|PHIL3007 Standardization of Connectives

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## 数学代写|数理逻辑代考Mathematical logic代写|Standardization of Connectives

When we define the notion of satisfaction in the next section we shall refer to the meaning of the connectives “not”, “and”, “or”, “if-then”, and “if and only if”. In ordinary language their meanings vary. For example, “or” is sometimes used in an inclusive sense and at other times in the exclusive sense “either-or”. However, for our purposes it is useful to fix a standard meaning: We shall always use “or” in the inclusive sense, that is, a compound proposition whose constituents are connected by “or” is true (has the truth-value $T$ ) iff at least one of the constituents is true; it is false (has the truth-value $F$ ) iff both constituents are false. For example, we specify in Definition $3.2$ below that a formula $(\varphi \vee \psi)$ is assigned the truth-value $T$ under an interpretation $\mathfrak{I}$ if and only if $\varphi$ is assigned the truth-value $T$ under $\mathfrak{I}$ or $\psi$ is assigned the truth-value $T$ under $\mathfrak{I}$. Because of our fixed standard meaning we have that $(\varphi \vee \psi)$ is assigned the truth-value $T$ under $\mathfrak{I}$ if and only if at least one of the formulas $\varphi, \psi$ is assigned $T$ under $\mathfrak{I}$.

According to our convention, the truth-value of a proposition compounded by “or” depends only on the truth-value of its constituents. Thus we can use a function
$$\dot{\vee}:{T, F} \times{T, F} \rightarrow{T, F}$$
to capture the meaning of “or”; the table of values (“truth-table”) is as follows:
\begin{tabular}{cc|c}
& & $\dot{\vee}$ \
\hline$T$ & $T$ & $T$ \
$T$ & $F$ & $T$ \
$F$ & $T$ & $T$ \
$F$ & $F$ & $F$
\end{tabular}
We proceed in a similar way with the connectives “and”, “if-then”, “if and only if”, and “not”. The truth-tables for the functions $\dot{\wedge}, \dot{\rightarrow}, \dot{\leftrightarrow}$, and $\dot{\neg}$ are:
\begin{tabular}{ll|l|l|l}
& & $\wedge$ & $\dot{\rightarrow}$ & $\dot{\leftrightarrow}$ \
\hline$T$ & $T$ & $T$ & $T$ & $T$ \
$T$ & $F$ & $F$ & $F$ & $F$ \
$F$ & $T$ & $F$ & $T$ & $F$ \
$F$ & $F$ & $F$ & $T$ & $T$
\end{tabular}
\begin{tabular}{l|l}
& $\dot{\dot{a}}$ \
\hline$T$ & $F$ \
$F$ & $T$
\end{tabular}

## 数学代写|数理逻辑代考Mathematical logic代写|The Satisfaction Relation

The satisfaction relation makes precise the notion of a formula being true under an interpretation. Again we fix a symbol set $S$. By “term”, “formula”, or “interpretation” we always mean “S-term”, “S-formula”, or “S-interpretation”. As a preliminary step we associate with every interpretation $\mathfrak{I}=(\mathfrak{A}, \beta)$ and every term $t$ an element $\mathfrak{I}(t)$ from the domain $A$. We define $\mathfrak{I}(t)$ by induction on terms.
3.1 Definition. (a) For a variable $x$ let $\mathfrak{I}(x):=\beta(x)$.
(b) For a constant $c \in S$ let $\mathfrak{I}(c):=c^{\mathfrak{A}}$.
(c) For an $n$-ary function symbol $f \in S$ and terms $t_{1}, \ldots, t_{n}$ let
$$\mathfrak{I}\left(f t_{1} \ldots t_{n}\right):=f^{\mathfrak{A}}\left(\mathfrak{I}\left(t_{1}\right), \ldots, \mathfrak{I}\left(t_{n}\right)\right) .$$
As an illustration, if $S=S_{\mathrm{gr}}$ and $\mathfrak{I}=(\mathfrak{A}, \beta)$ with $\mathfrak{A}=(\mathbb{R},+, 0)$ and $\beta\left(v_{0}\right)=2$, $\beta\left(v_{2}\right)=6$, then $\mathfrak{I}\left(v_{0} \circ\left(e \circ v_{2}\right)\right)=\mathfrak{I}\left(v_{0}\right)+\mathfrak{I}\left(e \circ v_{2}\right)=2+(0+6)=8$.

Now, using induction on formulas $\varphi$, we give a definition of the relation $\mathfrak{I}$ is a model of $\varphi$, where $\mathfrak{I}$ is an arbitrary interpretation. If $\mathfrak{I}$ is a model of $\varphi$, we also say that $\mathfrak{I}$ satisfies $\varphi$ or that $\varphi$ holds in $\mathfrak{I}$, and we write $\mathfrak{I}=\varphi$.

# 数理逻辑代写

## 数学代写|数理逻辑代考Mathematical logic代写|Standardization of Connectives

$$\dot{\vee}: T, F \times T, F \rightarrow T, F$$

## 数学代写|数理逻辑代考Mathematical logic代写|The Satisfaction Relation

$3.1$ 定义。(a) 对于变量 $x$ 让 $\mathfrak{J}(x):=\beta(x)$.
(b) 对于常数 $c \in S$ 让 $\mathfrak{J}(c):=c^{\mathfrak{A}}$.
(c) 对于一个n-ary 函数符号 $f \in S$ 和条款 $t_{1}, \ldots, t_{n}$ 让
$$\mathfrak{I}\left(f t_{1} \ldots t_{n}\right):=f^{\mathfrak{A}}\left(\mathfrak{I}\left(t_{1}\right), \ldots, \mathfrak{I}\left(t_{n}\right)\right) .$$

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