The Journal of Symbolic Logic

Research Article

The liar paradox and fuzzy logic

Petr Hájeka1, Jeff Parisa2 and John Shepherdsona3

a1 Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodarenskou Vezi 2, 182 07 Prague 8, Czech Republic, E-mail:

a2 Department of Mathematics, University of Manchester, Manchester M139PL, U.K., E-mail:

a3 Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, U.K., E-mail:


Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying “x is true” and satisfying the “dequotation schema” for all sentences φ? This problem is investigated in the frame of Łukasiewicz infinitely valued logic.

(Received August 01 1997)

(Revised August 02 1998)