109 research outputs found
Mathematical Logic: Proof theory, Constructive Mathematics
The workshop âMathematical Logic: Proof Theory, Constructive Mathematicsâ was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit
Computability and analysis: the legacy of Alan Turing
We discuss the legacy of Alan Turing and his impact on computability and
analysis.Comment: 49 page
Mathematical Logic: Proof Theory, Constructive Mathematics
[no abstract available
Hilbert's Program Then and Now
Hilbert's program was an ambitious and wide-ranging project in the philosophy
and foundations of mathematics. In order to "dispose of the foundational
questions in mathematics once and for all, "Hilbert proposed a two-pronged
approach in 1921: first, classical mathematics should be formalized in
axiomatic systems; second, using only restricted, "finitary" means, one should
give proofs of the consistency of these axiomatic systems. Although Godel's
incompleteness theorems show that the program as originally conceived cannot be
carried out, it had many partial successes, and generated important advances in
logical theory and meta-theory, both at the time and since. The article
discusses the historical background and development of Hilbert's program, its
philosophical underpinnings and consequences, and its subsequent development
and influences since the 1930s.Comment: 43 page
Intuitionistic Completeness of First-Order Logic
We establish completeness for intuitionistic first-order logic, iFOL, showing that is a formula is provable if and only if it is uniformly valid under the Brouwer Heyting Kolmogorov (BHK) semantics, the intended semantics of iFOL. Our proof is intuitionistic and provides an effective procedure Prf that converts uniform evidence into a formal first-order proof. We have implemented Prf . Uniform validity is defined using the intersection operator as a universal quantifier over the domain of discourse and atomic predicates. Formulas of iFOL that are uniformly valid are also intuitionistically valid, but not conversely. Our strongest result requires the Fan Theorem; it can also be proved classically by showing that Prf terminates using K¨onigâs Theorem.
The fundamental idea behind our completeness theorem is that a single evidence term evd witnesses the uniform validity of a minimal logic formula F. Finding even one uniform realizer guarantees validity because Prf (F, evd) builds a first-order proof of F, establishing its uniform validity and providing a purely logical normalized realizer.
We establish completeness for iFOL as follows. Friedman showed that iFOL can be embedded in minimal logic (mFOL). By his transformation, mapping formula A to F r(A). If A is uniformly valid, then so is F r(A), and by our Basic Completeness result, we can find a proof of F r(A) in minimal logic. Then we prove A from F r(A) in intuitionistic logic by a proof procedure fixed in advance. Our result resolves an open question posed by Beth in 1947
Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)
The Workshop "Mathematical Logic: Proof Theory,
Constructive Mathematics" focused on
proofs both as formal derivations in deductive systems as well as on
the extraction of explicit computational content from
given proofs in core areas of ordinary mathematics using proof-theoretic
methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory
Algebraizable Weak Logics
We extend the standard framework of abstract algebraic logic to the setting
of logics which are not closed under uniform substitution. We introduce the
notion of weak logics as consequence relations closed under limited forms of
substitutions and we give a modified definition of algebraizability that
preserves the uniqueness of the equivalent algebraic semantics of algebraizable
logics. We provide several results for this novel framework, in particular a
connection between the algebraizability of a weak logic and the standard
algebraizability of its schematic fragment. We apply this framework to the
context of logics defined over team semantics and we show that the classical
version of inquisitive and dependence logic is algebraizable, while their
intuitionistic versions are not
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