Feasible Operations on Proofs: The Logic of Proofs for Bounded Arithmetic

This paper presents a semantics for the logic of proofs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{LP}$\end{document} in which all the operations on proofs are realized by feasibly computable functions. More precisely, we will show that the completeness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{LP}$\end{document} for the semantics of proofs of Peano Arithmetic extends to the semantics of proofs in Buss’ bounded arithmetic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{S}^{1}_{2}$\end{document} . In view of applications in epistemology of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathsf{LP}$\end{document} in particular and justification logics in general this result shows that explicit knowledge in the propositional framework can be made computationally feasible.