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53 lines
2.2 KiB
TeX
53 lines
2.2 KiB
TeX
\subsubsection{Natural Deduction}
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\paragraph{Parenthesis}
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This task (if it were to even ever appear in the exams) is simply applying precedences, as well as remembering associativity.
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The precedences are as follows:
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\[
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\neg > \land > \lor > \rightarrow
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\]
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The associativities are right for $\land$ and $\lor$, whereas $\rightarrow$ is left associative. This means that
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\[
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A \rightarrow B \rightarrow C \text{ is parenthesized as } A \rightarrow (B \rightarrow C)
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\]
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\paragraph{Derivation trees}
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The most important tip here is to write as little as possible, by defining variables (such as $\Gamma$, $\Gamma'$, etc) for each step,
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so you can simply write that instead of all the content.
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\paragraph{Defining rules}
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This is often the hardest type of task. First state the obvious cases, the basic introduction and elimination rules according to the given definition.
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Remember that the rules are defined top-down, so, for introducing $A \leftrightarrow B$, defined as $(A \rightarrow B) \land (B \rightarrow A)$, we define the rules as follows:
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\[
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\begin{prooftree}
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\hypo{\Gamma \vdash (A \rightarrow B) \land (B \rightarrow A)}
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\infer1[$\leftrightarrow$-intro]{\Gamma \vdash A \leftrightarrow B}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash A \leftrightarrow B}
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\infer1[$\leftrightarrow$-elim]{\Gamma \vdash (A \rightarrow B) \land (B \rightarrow A)}
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\end{prooftree}
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\]
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The hard part is coming up with useful extra rules. Here, and with all rules defined by an AND, we can also define it slightly differently:
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\[
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\begin{prooftree}
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\hypo{\Gamma \vdash A \rightarrow B}
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\hypo{\Gamma \vdash B \rightarrow A}
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\infer2[$\leftrightarrow$-I]{\Gamma \vdash A \leftrightarrow B}
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\end{prooftree}
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\]
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Similarly, we can also provide elimination rules for the left or right:
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\[
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\begin{prooftree}
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\hypo{\Gamma \vdash A \leftrightarrow B}
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\infer1[$\leftrightarrow$-EL]{\Gamma \vdash (A \rightarrow B)}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash A \leftrightarrow B}
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\infer1[$\leftrightarrow$-ER]{\Gamma \vdash (B \rightarrow A)}
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\end{prooftree}
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\]
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