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Truth-table generator for classical propositional logic


Hatt is a command-line program which prints truth tables for expressions in classical propositional logic, and a library allowing its parser, evaluator, truth table generator and other functionality to be used in other programs.


Hatt is available from Hackage. To install it with cabal-install, update your list of known packages and then install Hatt.

$ cabal update
$ cabal install hatt

To build it from source, cd into the directory containing the Hatt source files, including hatt.cabal, and run cabal install.

Valid Hatt expressions

The following are all valid expression forms which can be parsed by Hatt, where ϕ and ψ are metalinguistic variables standing in for any valid expression. Hatt accepts the standard Unicode symbols for logical connectives. This is useful when pasting example formulae from the web or using Hatt in --pretty mode.

  • Variables: P, Q, a, b etc.---basically anything in the character class [a-zA-Z]
  • Negation: or ¬ϕ
  • Conjunction: (ϕ & ψ) or (ϕ ∧ ψ)
  • Disjunction: (ϕ | ψ) or (ϕ ∨ ψ)
  • Conditional: (ϕ -> ψ) or (ϕ → ψ)
  • Biconditional: (ϕ <-> ψ) or (ϕ ↔ ψ)

Parentheses are not required around top-level formulae, regardless of whether the primary connective is binary. For example, the expression a | b is valid and will be parsed correctly, as would p <-> (q & ~r), although the parenthesised versions of both these expressions ((a | b) and (p <-> (q & ~r))) are also fine.

Standard operator precedence for logical operators is supported. All the binary connectives associate to the right, so a -> b -> c is interpreted as a -> (b -> c). To override these rules, use parentheses as normal. Note that right-associativity only matters for the material conditional: all the other logical operators are associative, so a * (b * c) is equivalent to (a * b) * c where * is conjunction, disjunction or the biconditional.

Using the hatt command-line program

The default mode is interactive: you start the program, enter expressions at the prompt, and their truth tables are printed. Here's an example session.

$ hatt
Entering interactive mode. Type `help` if you don't know what to do!
> A | B
A B | (A | B)
T T | T
T F | T
F T | T
F F | F
> p -> (q & ~r)
p q r | (p -> (q & ~r))
T T T | F
T T F | T
T F T | F
T F F | F
F T T | T
F T F | T
F F T | T
F F F | T
> e <-> f
e f | (e <-> f)
T T | T
T F | F
F T | F
F F | T
> exit

The --evaluate flag lets you pass a single expression to be evaluated directly.

$ hatt --evaluate="P -> (Q | ~R)"
P Q R | (P -> (Q | ~R))
T T T | F
T T F | F
T F T | F
T F F | F
F T T | F
F T F | F
F F T | T
F F F | F

By default, hatt will print ASCII representations of expressions. If you have a Unicode-capable terminal, try passing the --pretty option to pretty-print expressions using the more common logical symbols.

$ hatt --evaluate="P -> (Q | ~R)" --pretty
P Q R | (P → (Q ∨ ¬R))
T T T | F
T T F | F
T F T | F
T F F | F
F T T | F
F T F | F
F F T | T
F F F | F

You can enable pretty-printing while in interactive mode by using the pretty command.

If you pass the --coloured flag, hatt will colour the truth values in the tables which it prints: green for true, red for false. You can enable colouring during interactive mode by using the colour command.

You can print out the normal forms of expressions too, by prefixing an expression with nnf, dnf or cnf.

$ hatt --pretty
> nnf ~(P -> (Q & R))
(P ∧ (¬Q ∨ ¬R))

The three supported normal forms are negation normal form, conjunctive normal form and disjunctive normal form.

Using Hatt in other programs

Hatt exposes the Data.Logic.Propositional module, which provides a simple API for parsing, evaluating, and printing truth tables. It also includes the Data.Logic.Propositional.NormalForms module which exports functions to convert logical expressions into normal forms. Details of the foregoing can be found in the library documentation.

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