5,193 research outputs found
Is Complexity a Source of Incompleteness?
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems
of a finitely-specified theory cannot be significantly more complex than the
theory itself}, for an appropriate measure of complexity. We show that the
measure is invariant under the change of the G\"odel numbering. For this
measure, the theorems of a finitely-specified, sound, consistent theory strong
enough to formalize arithmetic which is arithmetically sound (like
Zermelo-Fraenkel set theory with choice or Peano Arithmetic) have bounded
complexity, hence every sentence of the theory which is significantly more
complex than the theory is unprovable. Previous results showing that
incompleteness is not accidental, but ubiquitous are here reinforced in
probabilistic terms: the probability that a true sentence of length is
provable in the theory tends to zero when tends to infinity, while the
probability that a sentence of length is true is strictly positive.Comment: 15 pages, improved versio
Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes
We survey diverse approaches to the notion of information: from Shannon
entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov
complexity are presented: randomness and classification. The survey is divided
in two parts in the same volume. Part I is dedicated to information theory and
the mathematical formalization of randomness based on Kolmogorov complexity.
This last application goes back to the 60's and 70's with the work of
Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last
years.Comment: 40 page
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness
Planning with Incomplete Information
Planning is a natural domain of application for frameworks of reasoning about
actions and change. In this paper we study how one such framework, the Language
E, can form the basis for planning under (possibly) incomplete information. We
define two types of plans: weak and safe plans, and propose a planner, called
the E-Planner, which is often able to extend an initial weak plan into a safe
plan even though the (explicit) information available is incomplete, e.g. for
cases where the initial state is not completely known. The E-Planner is based
upon a reformulation of the Language E in argumentation terms and a natural
proof theory resulting from the reformulation. It uses an extension of this
proof theory by means of abduction for the generation of plans and adopts
argumentation-based techniques for extending weak plans into safe plans. We
provide representative examples illustrating the behaviour of the E-Planner, in
particular for cases where the status of fluents is incompletely known.Comment: Proceedings of the 8th International Workshop on Non-Monotonic
Reasoning, April 9-11, 2000, Breckenridge, Colorad
Counterfactual Logic and the Necessity of Mathematics
This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious program by Yli-Vakkuri and Hawthorne (2018), who seek to establish that mathematics is committed to its own necessity. I claim that their argument fails to establish this result for two reasons. First, their assumptions force our hand on a controversial debate within counterfactual logic. In particular, they license counterfactual strengthening— the inference from ‘If A were true then C would be true’ to ‘If A and B were true then C would be true’—which many reject. Second, the system they develop is provably equivalent to appending Deduction Theorem to a T modal logic. It is unsurprising that the combination of Deduction Theorem with T results in necessitation; indeed, it is precisely for this reason that many logicians reject Deduction Theorem in modal contexts. If Deduction Theorem is unacceptable for modal logic, it cannot be assumed to derive the necessity of mathematic
What Do Paraconsistent, Undecidable, Random, Computable and Incomplete mean? A Review of Godel's Way: Exploits into an undecidable world by Gregory Chaitin, Francisco A Doria, Newton C.A. da Costa 160p (2012) (review revised 2019)
In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking at how we actually use words in particular contexts. When we get clear about which language game we are playing, these topics are seen to be ordinary scientific and mathematical questions like any others. Wittgenstein’s insights have seldom been equaled and never surpassed and are as pertinent today as they were 80 years ago when he dictated the Blue and Brown Books. In spite of its failings—really a series of notes rather than a finished book—this is a unique source of the work of these three famous scholars who have been working at the bleeding edges of physics, math and philosophy for over half a century. Da Costa and Doria are cited by Wolpert (see below or my articles on Wolpert and my review of Yanofsky’s ‘The Outer Limits of Reason’) since they wrote on universal computation, and among his many accomplishments, Da Costa is a pioneer in paraconsistency.
Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019
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