I'm studying a book on logic, and I'm having difficulties with the book's rules of natural deduction. ##P_1,P_2,\dots## are variables that represent propositions like ##x>0##. All variables and the symbols ##\top## and ##\bot## are formulas. For all formulas ##\varphi,\psi##, the following strings of text are formulas too:(adsbygoogle = window.adsbygoogle || []).push({});

\begin{align*}

&\varphi\land\psi\\

&\varphi\lor\psi\\

&\varphi\to\psi

\end{align*}Edit:After taking a peak at Enderton, I realize that this isn't quite right. ##\top## and ##\bot## are not formulas. They are expressions, i.e. finite sequences of symbols, but not formulas. Also, it makes more sense to say that ##(\varphi\land\psi)## is a formula than to say that ##\varphi\land\psi## is a formula, because then we don't have to insert parentheses when we combine formulas. Instead, we can introduce a convention for when we can choosenotto write out all the parentheses.

Negation and equivalence are defined as abbreviations:

\begin{align*}

&\lnot\varphi=\varphi\to\bot\\

&\varphi\leftrightarrow\psi = (\varphi\to\psi)\land(\psi\to\varphi)

\end{align*} The book defines a bunch of rules. The simplest is the conjunction introduction rule:

$$\frac{\varphi\qquad\psi}{\varphi\land\psi}$$ The complete set of rules can be found on page 40 in the English edition. Link. They are explained (partially) on the pages before that. One of my problems is that the book does a few things that seem to violate the rules. The simplest example is this:

$$\frac{\frac{{}}{\top}}{P_1\to\top}$$ How is this not a violation of the implication introduction rule? It says explicitly that we can conclude that ##\varphi\to\psi## when (and only when) we have a derivation that starts with ##\varphi## and ends with ##\psi##?

I thought about viewing the above as an abbreviated notation for

$$\frac{\frac{P_1}{\top}}{P_1\to\top}$$ but that doesn't entirely make sense either, since there's no formula on top of the ##\top## in the ##\top## introduction rule. Maybe I should interpret that one similarly, i.e. as really saying that for all formulas ##\varphi##, we have

$$\frac{\varphi}{\top}$$ A bigger problem for me is that I don't understand the book's derivation of ##\top\leftrightarrow\lnot\bot## on page 41. A part of it goes like this:

$$\frac{\frac{[\bot]}{\lnot\bot}}{\top\to\lnot\bot}$$ Where did that ##\top## come from? The book claims that that step is using the implication introduction rule, but if I apply that, I would end up with ##\bot\to\lnot\bot##, not ##\top\to\lnot\bot##.

Maybe I'm missing something obvious, but I feel like the book didn't explain the rules well enough. For example, there's nothing in the statement of the implication introduction rule that says that assumptions other than the one that's being discharged when the rule is applied can be part of the derivation, and yet we see an example of that on page 41 (example 5.4.3, the ##\to I_1## step). That example helped me understand something about that rule that I couldn't have picked up from the statement of the rule. Maybe I have missed something similar about the other rules.

By the way, is there a better way to LaTeX these derivations than to use \frac?

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# Natural deduction: True iff not false

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