47 research outputs found
A note on strictly positive logics and word rewriting systems
We establish a natural translation from word rewriting systems to strictly
positive polymodal logics. Thereby, the latter can be considered as a
generalization of the former. As a corollary we obtain examples of undecidable
strictly positive normal modal logics. The translation has its counterpart on
the level of proofs: we formulate a natural deep inference proof system for
strictly positive logics generalizing derivations in word rewriting systems. We
also formulate some open questions related to the theory of modal companions of
superintuitionistic logics that was initiated by L.L. Maximova and V.V.
Rybakov.Comment: 9 page
Reflection calculus and conservativity spectra
Strictly positive logics recently attracted attention both in the description
logic and in the provability logic communities for their combination of
efficiency and sufficient expressivity. The language of Reflection Calculus RC
consists of implications between formulas built up from propositional variables
and constant `true' using only conjunction and diamond modalities which are
interpreted in Peano arithmetic as restricted uniform reflection principles.
We extend the language of RC by another series of modalities representing the
operators associating with a given arithmetical theory T its fragment
axiomatized by all theorems of T of arithmetical complexity , for all
n>0. We note that such operators, in a precise sense, cannot be represented in
the full language of modal logic.
We formulate a formal system RC extending RC that is sound and, as
we conjecture, complete under this interpretation. We show that in this system
one is able to express iterations of reflection principles up to any ordinal
. On the other hand, we provide normal forms for its variable-free
fragment. Thereby, the variable-free fragment is shown to be decidable and
complete w.r.t. its natural arithmetical semantics.
Whereas the normal forms for the variable-free formulas of RC correspond in a
unique way to ordinals below , the normal forms of RC are
more general. It turns out that they are related in a canonical way to the
collections of proof-theoretic ordinals of (bounded) arithmetical theories for
each complexity level .
Finally, we present an algebraic universal model for the variable-free
fragment of RC based on Ignatiev's Kripke frame. Our main theorem
states the isomorphism of several natural representations of this algebra
Topological interpretations of provability logic
Provability logic concerns the study of modality as provability in
formal systems such as Peano arithmetic. Natural, albeit quite surprising,
topological interpretation of provability logic has been found in the 1970's by
Harold Simmons and Leo Esakia. They have observed that the dual
modality, corresponding to consistency in the context of formal arithmetic, has
all the basic properties of the topological derivative operator acting on a
scattered space. The topic has become a long-term project for the Georgian
school of logic led by Esakia, with occasional contributions from elsewhere.
More recently, a new impetus came from the study of polymodal provability
logic GLP that was known to be Kripke incomplete and, in general, to have a
more complicated behavior than its unimodal counterpart. Topological semantics
provided a better alternative to Kripke models in the sense that GLP was shown
to be topologically complete. At the same time, new fascinating connections
with set theory and large cardinals have emerged.
We give a survey of the results on topological semantics of provability logic
starting from first contributions by Esakia. However, a special emphasis is put
on the recent work on topological models of polymodal provability logic. We
also included a few results that have not been published so far, most notably
the results of Section 6 (due the second author) and Sections 10, 11 (due to
the first author).Comment: 36 page
Some abstract versions of G\"odel's second incompleteness theorem based on non-classical logics
We study abstract versions of G\"odel's second incompleteness theorem and
formulate generalizations of L\"ob's derivability conditions that work for
logics weaker than the classical one. We isolate the role of contraction rule
in G\"odel's theorem and give a (toy) example of a system based on modal logic
without contraction invalidating G\"odel's argument.Comment: 16 page
Positive provability logic for uniform reflection principles
We deal with the fragment of modal logic consisting of implications of
formulas built up from the variables and the constant `true' by conjunction and
diamonds only. The weaker language allows one to interpret the diamonds as the
uniform reflection schemata in arithmetic, possibly of unrestricted logical
complexity. We formulate an arithmetically complete calculus with modalities
labeled by natural numbers and \omega, where \omega corresponds to the full
uniform reflection schema, whereas n<\omega corresponds to its restriction to
arithmetical \Pi_{n+1}-formulas. This calculus is shown to be complete w.r.t. a
suitable class of finite Kripke models and to be decidable in polynomial time.Comment: 34 page
Topological completeness of the provability logic GLP
Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics.
A natural topological semantics of GLP interprets modalities as derivative
operators of a polytopological space. Such spaces satisfying all the axioms of
GLP are called GLP-spaces. We develop some constructions to build nontrivial
GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces
On provability logics with linearly ordered modalities
We introduce the logics GLP(\Lambda), a generalization of Japaridze's
polymodal provability logic GLP(\omega) where \Lambda is any linearly ordered
set representing a hierarchy of provability operators of increasing strength.
We shall provide a reduction of these logics to GLP(\omega) yielding among
other things a finitary proof of the normal form theorem for the variable-free
fragment of GLP(\Lambda) and the decidability of GLP(\Lambda) for recursive
orderings \Lambda. Further, we give a restricted axiomatization of the
variable-free fragment of GLP(\Lambda)
Reflection algebras and conservation results for theories of iterated truth
We consider extensions of the language of Peano arithmetic by transfinitely
iterated truth definitions satisfying uniform Tarskian biconditionals. Without
further axioms, such theories are known to be conservative extensions of the
original system of arithmetic. Much stronger systems, however, are obtained by
adding either induction axioms or reflection axioms on top of them. Theories of
this kind can interpret some well-known predicatively reducible fragments of
second-order arithmetic such as iterated arithmetical comprehension.
We obtain sharp results on the proof-theoretic strength of these systems
using methods of provability logic. Reflection principles naturally define
unary operators acting on the semilattice of axiomatizable extensions of our
basic theory of iterated truth. The substructure generated by the top element
of this algebra provides a canonical ordinal notation system for the class of
theories under investigation.
Using these notations we obtain conservativity relationships for iterated
reflection principles of different logical complexity levels corresponding to
the levels of the hyperarithmetical hierarchy, i.e., the analogs of Schmerl's
formulas. These relationships, in turn, provide proof-theoretic analysis of our
systems and of some related predicatively reducible theories. In particular, we
uniformly calculate the ordinals characterizing the standard measures of their
proof-theoretic strength, such as provable well-orderings, classes of provably
recursive functions, and -ordinals.Comment: 49 page
A Many-Sorted Variant of Japaridze's Polymodal Provability Logic
We consider a many-sorted variant of Japaridze's polymodal provability logic
. In this variant, which is denoted ,
propositional variables are assigned sorts , where
variables of finite sort are interpreted as -sentences
of the arithmetical hierarchy, while those of sort range over
arbitrary ones. We prove that is arithmetically complete
with respect to this interpretation. Moreover, we relate to
its one-sorted counterpart and prove that the former inherits
some well-known properties of the latter, like Craig interpolation and PSpace
decidability. We also study a positive variant of which
allows for an even richer arithmetical interpretation---variables are permitted
to range over theories rather than single sentences. This interpretation in
turn allows the introduction of a modality that corresponds to the full uniform
reflection principle. We show that our positive variant of
is arithmetically complete.Comment: {A version of this article has been published in the Logic Journal of
the IGPL, 26(5): 505--538 (2018
Axiomatizing Origami planes
We provide a variant of an axiomatization of elementary geometry based on
logical axioms in the spirit of Huzita--Justin axioms for the Origami
constructions. We isolate the fragments corresponding to natural classes of
Origami constructions such as Pythagorean, Euclidean, and full Origami
constructions. The sets of Origami constructible points for each of the classes
of constructions provides the minimal model of the corresponding set of logical
axioms.
Our axiomatizations are based on Wu's axioms for orthogonal geometry and some
modifications of Huzita--Justin axioms. We work out bi-interpretations between
these logical theories and theories of fields as described in J.A. Makowsky
(2018). Using a theorem of M. Ziegler (1982) which implies that the first order
theory of Vieta fields is undecidable, we conclude that the first order theory
of our axiomatization of Origami is also undecidable.Comment: 25 page