julmue / Clank

Clank: proof assistant for propositional logics classical PC, K3, L3, LP, RM

Clank - Theorem Prover (Paraconsistent Propositional Logics)

A paraconsistent logic is a logic where the consequence relation is not explosive, i. e. where the law {A, ¬A}⊨B does not hold. A very good introduction can be found on the Stanford Encyclopedia of Philosophie

An explosive consequence relation is acceptable for mathematics, but a nuisance in domains that have to operate on sets of possibly inconsistent premisses like medical assistance systems or applications in robotics; these are the domains where paraconsistent logics shine.

Supported Logics

Clank is a prover for various systems of paraconsistent propositional logic and also vanilla propositional logic:

• PC: Vanilla propositional calculus
• K3: System Kleen
• L3: System Lucasiewicz
• LP: System Priest
• RM: System Routly-Meyer

Grammar

The syntax of the propositional formulas:

Formula = Var 	-- all aplphanumeric strings can be used as identifiers
        | a && b	-- conjunction
        | a || b    -- disjunction	
        | a -> b    -- implication
        | a <-> b   -- equivalence

Usage

Clank has a command line interface. Arguments are devided in Mandatory Arguments and Options;
All mandatory arguments and at least one option have to be provided for a successful query.

Mandatory arguments are:

• -f, --formula: One or more formulas of propositional logic.

• -s, --semantics: One of the following:

  • pc: propositional calculus
  • k3: system Kleene
  • l3: system Lucasiewicz
  • lp: system Priest
  • rm: system Routly-Meyer

Options are:

• -c, --classification: classification of the formula into theorem, satisfiable formulas, and contradictions.
• -p, --property: property assertion; whether a given formula / set of formulas is sat, valid or unsat.
• -ms, --models: the set of models of a given formula (with respect to a given semantics).
• -nf, --normalform: not yet implemented
• --t, --entails: whether the first formula is the conclusion the remaining formulas are the set of premisses.
• -h, --help: the help file (the help is also shown in case of an invalid query)

The order of arguments doesn't matter. Several queries be issued at once.

Use clank --help to get a full overview of the available commands. Here are some simple examples (all exmaples use classical propositional calculus):

• Get the models of a ∧ b → a:

  clank -f 'a && b -> a' -s pc -ms

  "[[(a,T),(b,T)],[(a,T),(b,F)],[(a,F),(b,T)],[(a,F),(b,F)]]"

• Classify a ∧ b → b:

  clank -f 'p -> (p -> q) -> q' -s pc -c

  sat

• Check whether p → (p → q) → q is valid:

  clank -f 'p -> (p -> q) -> q' -s pc -p valid

  true

• Check whether q implies p → q:

  clank -f 'p -> q' 'q' -s pc -t

  True