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# 数学代写|离散数学代写Discrete Mathematics代考|CSC226 A Gentzen-Style System for Natural Deduction

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## 数学代写|离散数学代写Discrete Mathematics代考|A Gentzen-Style System for Natural Deduction

The process of discharging premises when constructing a deduction is admittedly a bit confusing. Part of the problem is that a deduction tree really represents the last of a sequence of stages (corresponding to the application of inference rules) during which the current set of “active” premises, that is, those premises that have not yet been discharged (closed, cancelled) evolves (in fact, shrinks). Some mechanism is needed to keep track of which premises are no longer active and this is what this business of labeling premises with variables achieves. Historically, this is the first mechanism that was invented. However, Gentzen (in the 1930s) came up with an alternative solution that is mathematically easier to handle. Moreover, it turns out that this notation is also better suited to computer implementations, if one wishes to implement an automated theorem prover.

The point is to keep a record of all undischarged assumptions at every stage of the deduction. Thus, a deduction is now a tree whose nodes are labeled with pairs of the form $\langle\Gamma, P\rangle$, where $P$ is a proposition, and $\Gamma$ is a record of all undischarged assumptions at the stage of the deduction associated with this node.

Instead of using the notation $\langle\Gamma, P\rangle$, which is a bit cumbersome, Gentzen used expressions of the form $\Gamma \rightarrow P$, called sequents

It should be noted that the symbol $\rightarrow$ is used as a separator between the left-hand side $\Gamma$, called the antecedent, and the right-hand side $P$, called the conclusion (or succedent ) and any other symbol could be used. Of course $\rightarrow$ is reminiscent of implication but we should not identify $\rightarrow$ and $\Rightarrow$. Still, it turns out that a sequent $\Gamma \rightarrow P$ is provable if and only if $\left(P_1 \wedge \cdots \wedge P_m\right) \Rightarrow P$ is provable, where $\Gamma=\left(P_1, \ldots, P_m\right)$.
During the construction of a deduction tree, it is necessary to discharge packets of assumptions consisting of one or more occurrences of the same proposition. To this effect, it is convenient to tag packets of assumptions with labels, in order to discharge the propositions in these packets in a single step. We use variables for the labels, and a packet labeled with $x$ consisting of occurrences of the proposition $P$ is written as $x: P$.

## 数学代写|离散数学代写Discrete Mathematics代考|Clearing Up Differences Among :-Introduction,-Elimination, and RAA

The differences between the rules, $\neg$-introduction, $\perp$-elimination, and the proof-bycontradiction rule (RAA) are often unclear to the uninitiated reader and this tends to cause confusion. In this section we try to clear up some common misconceptions about these rules.
Confusion 1. Why is RAA not a special case of $\neg$-introduction?
The only apparent difference between $\neg$-introduction (on the left) and RAA (on the right) is that in RAA, the premise $P$ is negated but the conclusion is not, whereas in $\neg$-introduction the premise $P$ is not negated but the conclusion is.

The important difference is that the conclusion of RAA is not negated. If we had applied $\neg$-introduction instead of RAA on the right, we would have obtained
\begin{aligned} \Gamma, \neg P^x \ \mathscr{D} \ \frac{\perp}{\neg \neg P} \quad x(\neg \text {-intro }) \end{aligned}
where the conclusion would have been $\neg \neg P$ as opposed to $P$. However, as we already said earlier, $\neg \neg P \Rightarrow P$ is not provable intuitionistically. Consequently, RAA is not a special case of $\neg$-introduction. On the other hand, one may view $\neg$ introduction as a “constructive” version of RAA applying to negated propositions (propositions of the form $\neg P$ ).

## 数学代写|离散数学代可Discrete Mathematics代考|Clearing Up Differences Among :-Introduction,-Elimination, and RAA

$$\Gamma, \neg P^x \mathscr{D} \frac{\perp}{\neg \neg P} \quad x(\neg \text {-intro })$$

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