For natural deduction, I have used a package called bussproofs. imported by putting

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1 General tips This tutorial assumes that you know a decent amount about using LaTeX, and will only cover how to typeset proofs themselves. There are two packages I have used to typeset proof trees in LaTeX, bussproofs and prooftrees. For further details on these packages, and customisation they offer, the documentation for the packages can be found at pdf and pdf. Another website to look at for tips is latex-for-logicians/ Natural Deduction For natural deduction, I have used a package called bussproofs. imported by putting It is \usepackage{bussproofs} in your preamble. Importing this package gives you access to the environment prooftree. To typeset natural deduction proofs you have four basic commands, namely AxiomC, UnaryInfC, BinaryInfC and TernaryInfC. The AxiomC-command lets you introduce assumptions, and the other commands allow you to apply rules to make deductions from those assumptions, naming how many assumptions are in the given rule. An example is probably the best way to learn here. The code \AxiomC{$p \land q$} \RightLabel{\scriptsize{$\land E$}} \UnaryInfC{$p$} \AxiomC{$p \to (q \to r)$} \RightLabel{\scriptsize{$\to E$}} \BinaryInfC{$q \to r$} \AxiomC{$p \land q$} \RightLabel{\scriptsize{$\land E$}} \UnaryInfC{$q$} \RightLabel{\scriptsize{$\neg E$}} 1

2 \BinaryInfC{$r$} \RightLabel{\scriptsize{$\to I_1$}} \UnaryInfC{$p \land q \to r$} \RightLabel{\scriptsize{$\to I_2$}} \UnaryInfC{$(p \to (q \to r)) \to (p \land q \to r)$} produces the proof tree p q E p p (q r) q r E p q E q E r I 1 p q r I 2 (p (q r)) (p q r) Note that the formulas are in squiggly brackets. This is required, as far as I can tell. Here I have included comments using the command RightLabel. This is not required, but make the proofs easier to read. Sequent Calculus The sequent calculus is similar, using the same package and environment. However, here the commands are Axiom, UnaryInf, BinaryInf (the corresponding command TernaryInf is not necessary for the sequent calculus). Another thing which is important is the command fcenter, which needs to be included in each formula. This is the symbol used in the sequent calculus, typically = (implies from the package amsmath) or (vdash). Also, a definition of fcenter (using def) needs to be included somewhere in the document, typically in the preamble. The package typesets proofs in the order in which you write them down, so write the tree down from top to bottom. Example code \def\fcenter{\mbox{\ $\vdash$\ }} \Axiom$p \fcenter p,q$ \RightLabel{\scriptsize{$R \neg$}} 2

3 \UnaryInf$\fCenter \neg p, p, q$ \Axiom$q \fcenter p,q$ \RightLabel{\scriptsize{$R \neg$}} \UnaryInf$\fCenter \neg q, p, q$ \RightLabel{\scriptsize{$R \land$}} \BinaryInf$\fCenter p, q, \neg p \land \neg q$ \RightLabel{\scriptsize{$R \lor$}} \UnaryInf$\fCenter p \lor q, \neg p \land \neg q$ \RightLabel{\scriptsize{$L \neg$}} \UnaryInf$\neg (p \lor q) \fcenter \neg p \land \neg q$ \RightLabel{\scriptsize{$R \to$}} \UnaryInf$\fCenter \neg (p \lor q) \to (\neg p \land \neg q)$ with output Resolution p p, q q p, q R R p, p, q q, p, q R p, q, p q R p q, p q L (p q) p q (p q) ( p q) The way I have typeset resolution is identical to the way I did natural deduction. Here labels are only really useful to indicate which substitutions, if any, have been made. \AxiomC{$\{\neg p, \neg q, r\}, \{p\}, \{q\}, \{\neg r\}, \{\neg q, r\}, \{r\}, \emptyset$} \UnaryInfC{$\{\neg p, \neg q, r\}, \{p\}, \{q\}, {\color{red}\{\neg r\}}, \{\neg q, r\}, {\color{red}\{r\}}$} \UnaryInfC{$\{\neg p, \neg q, r\}, \{p\}, {\color{red}\{q\}}, \{\neg r\}, {\color{red}\{\neg q, r\}}$} \UnaryInfC{${\color{red}\{\neg p, \neg q, r\}}, {\color{red}\{p\}}, \{q\}, \{\neg r\}$} 3 R

4 Tableau calculus { p, q, r}, {p}, {q}, { r}, { q, r}, {r}, { p, q, r}, {p}, {q}, { r}, { q, r}, {r} { p, q, r}, {p}, {q}, { r}, { q, r} { p, q, r}, {p}, {q}, { r} Here things start to get a bit more complicated. The package I ve used is called prooftrees, and sets downward-branching proof trees. If you import the package, you will create a command called prooftree, which is the same as the bussproofs package. In order to avoid this, if you import the package with the option tableaux, i.e. have \usepackage[tableaux]{prooftrees} in your preamble, you should create the same environment, but with the name tableau instead. On a slight technical note: For this to work, you need an updated version of these packages. If you get some error here, it might be due to an outdated version of prooftrees and/or forest (which prooftrees depends on). Updated versions can be downloaded from the CTAN website. The syntax is a bit more complicated. There is a type of preamble to each proof, with some settings for the proof. Then follows the proof using a type of nested bracket syntax to represent the tree. A very simple example is \begin{tableau} {} [a[b[c][d]]] \end{tableau} which produces 1. a 2. b 3. c d Note that the line numbers are automatically included. Here is a more complicated example \begin{tableau} { 4

5 to prove={\forall x (q \to p(x)) \to (q \to \forall x p(x)}, close with=\ensuremath{\times} } [(\forall x (q \to p(x)) \to (q \to \forall x p(x))^0 [\forall x (q \to p(x))^1, just={$1, \to^0$} [(q \to \forall x p(x))^0, just={$1, \to^0$} [q^1, just={$3, \to^0$} [\forall x p(x)^0, just={$3, \to^0$} [p(a)^0, just={$5, \forall^0$} [(q \to p(a))^1, just={$2, \forall^1$} [q^0, close={4,8}, just={$7, \to^1$}] [p(a)^1, close={6,8}]]]]]]]] \end{tableau} with output x(q p(x)) (q xp(x) ( x(q p(x)) (q xp(x)) 0 x(q p(x)) 1 (q xp(x)) 0 q 1 xp(x) 0 p(a) 0 (q p(a)) 1 1, 0 1, 0 3, 0 3, 0 5, 0 2, 1 8. q 0 4,8 p(a) 1 6,8 7, 1 showing how you can add notation for closed branches and showing which rule you are using and where the formula(s) used in the rule is/are coming from. 5