Science & Logic

PHIL30043: The Power and Limits of Logic

PHIL30043: The Power and Limits of Logic is a University of Melbourne undergraduate subject. It covers the metatheory of classical first order predicate logic, beginning at the Soundness and Completeness Theorems (proved not once but twice, first for a tableaux proof system for predicate logic, then a Hilbert proof system), through the Deduction Theorem, Compactness, Cantor’s Theorem, the Downward Löwenheim–Skolem Theorem, Recursive Functions, Register Machines, Representability and ending up at Gödel’s Incompleteness Theorems and Löb’s Theorem. Kurt Gödel, seated The subject is taught to University of Melbourne undergraduate students (for Arts students as a part of the Philosophy major, for non-Arts students, as a breadth subject). Details for enrolment are here. I make use of video lectures I have made freely available on Vimeo. Outline The course is divided into four major sections and a short prelude. Here is a list of all of the videos, in case you’d like to follow along with the content. Prelude Logical Equivalence Disjunctive Normal Form Why DNF Works Prenex Normal Form Models for Predicate Logic Trees for Predicate Logic Completeness Introducing Soundness and Completeness Soundness for Tree Proofs Completeness for Tree Proofs Hilbert Proofs for Propositional Logic Conditional Proof Hilbert Proofs for Predicate Logic Theories Soundness and Completeness for Hilbert Proofs for Predicate Logic Compactness Counting Sets Diagonalisation Compactness Non-Standard Models Inexpressibility of Finitude Downward Löwenheim–Skolem Theorem Computability Functions Register Machines Recursive Functions Register Machine computable functions are Recursive The Uncomputable Undecidability and Incompleteness Deductively Defined Theories The Finite Model Property Completeness Introducing... -

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PY4601: Paradoxes

py4601: Paradoxes is an honours Philosophy module at the University of St Andrews. It’s coordinated by my colleague, Patrick Greenough, and I’m teaching a small slice at the end on the liar paradox. If you’d like to see what I am covering, you can see some slides and notes here. Here’s what we’re covering in the whole module: A paradox is a plausible argument for an absurd conclusion. Better still: a paradox is an apparently plausible argument, with apparently plausible premises, which leads to an apparently absurd conclusion using apparently valid reasoning. In this module, we are going to look at three groups of paradoxes (and the connections between them): Epistemological Paradoxes (e.g. various forms of Scepticism, Lottery Paradox, Dogmatism Paradox). Paradoxes of Vagueness and... -

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PY1012: Reasoning

py1012: Reasoning introduces the essential concepts and techniques of critical reasoning, formal propositional logic, and basic predicate logic. Among the central questions are these: what distinguishes an argument from a mere rhetorical ploy? What makes an argument a good one? How can we formally prove that a conclusion follows from... -

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Proofs for Relevant Consequence, with Star and Perp

Abstract: In this talk, I show how to incorporate insights from the model-theoretic semantics for negation (insights due to J. Michael Dunn, in his paper “Star and Perp: Two Treatments of Negation”), into a proof-first understanding of the semantics of negation. I then discuss how a logical pluralist may understand... -

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Proofs for Relevant Consequence, with Star and Perp Abstract: In this talk, I show how to incorporate insights from the model-theoretic semantics for negation (insights due to J. Michael Dunn, in his paper “Star and Perp: Two Treatments of Negation”), into a proof-first understanding of the semantics of negation. I then discuss how a logical pluralist may understand the underlying accounts of proofs and their significance. The result is a new perspective from which to view the connection between relevance and different notions... Consequently.org -

PY4601: Paradoxes py4601: Paradoxes is an honours Philosophy module at the University of St Andrews. It’s coordinated by my colleague, Patrick Greenough, and I’m teaching a small slice at the end on the liar paradox. If you’d like to see what I am covering, you can see some slides and notes here. Here’s what we’re covering in the whole module: A paradox is a plausible argument for an absurd conclusion. Better still: a paradox is an apparently... Consequently.org -

PY1012: Reasoning py1012: Reasoning introduces the essential concepts and techniques of critical reasoning, formal propositional logic, and basic predicate logic. Among the central questions are these: what distinguishes an argument from a mere rhetorical ploy? What makes an argument a good one? How can we formally prove that a conclusion follows from some premises? In addressing these questions, we will also cover topics such as ambiguity, argument forms and analyses, induction compared to deduction, counterexamples, truth-tables, natural... Consequently.org -

Erdős Number: 3 According to the AMS’s handy Mathematics Collaboration Distance calculator, my Erdős number is down to three, given the following path: Vedran Čačić, Pavel Pudlák, Greg Restall, Alasdair Urquhart, Albert Visser, “Decorated linear order types and the theory of concatenation,” Logic Colloquium 2007, p. 1–13, ed. F. Delon, U. Kohlenbach, P. Maddy and F. Stephan, Cambridge University Press, 2010. Noga Alon and Pavel Pudlák, “Equilateral Sets in $l^n_p$,” Geometric & Functional Analysis, 13 (2003), 467-482. Noga... Consequently.org -

Wombat, Conditional, or Inference? As my colleague and PY1012 Reasoning co-lecturer, Franz Berto knows, it’s never too early to introduce your students to wombats, or to the difference between a conditional and an inference. A slide from my first week’s PY1012 lecture. Yes, next semester’s classes are just about to start, and I’m in the depths of preparation. Consequently.org -

Logical Methods Publication Day Today MIT Press releases our book, Logical Methods into the big wide world. It was an absolute delight to work on this long-term project with my co-author and friend, Shawn Standefer. A Stack of Copies of Logical Methods Consequently.org -

Logical Methods As the cover blurb says Logical Methods is an accessible introduction to philosophical logic, suitable for undergraduate courses and above. Rigorous yet accessible, Logical Methods introduces logical tools used in philosophy—including proofs, models, modal logics, meta-theory, two-dimensional logics, and quantification—for philosophy students at the undergraduate level and above. The approach developed by Shawn Standefer and I developed is distinct from other texts because it presents proof construction on equal footing with model building and emphasizes... Consequently.org -

True Contradictions in Theology? In The Contradictory Christ, Jc Beall argues that paraconsistent logic provides a way to show how the central claims of Christology can all be true, despite their paradoxical appearances. For Beall, claims such as “Christ is peccable” and “Christ is impeccable” are both true, with no change of subject matter or ambiguity of meaning of any term involved in each claim. Since to say that Christ is impeccable is to say that Christ is not... Consequently.org -

Natural Deduction with Alternatives: on structural rules, and identifying assumptions Abstract: In this talk, I will introduce natural deduction with alternatives, explaining how this framework provides a simple, well-behaved, single conclusion natural deduction system for a range of logical systems, including classical logic, (classical) linear logic, relevant logic and affine logic, in addition to the familar intuitionistic restrictions of these systems. Each of these proof systems have identical connective rules. As we expect in substructural logics, different logical systems are given by varying the structural... Consequently.org -

Collection Frames: What, How and Why? Abstract: In this talk, I give a breezy introduction to Collection Frames (joint work with Shawn Standefer), with an emphasis on how they are technically equivalent to, but conceptually simpler than Routley–Meyer ternary relational frames. The talk is an online presentation at the New Directions in Relevant Logic Online Workshop. The slides for the talk are available here. Consequently.org -

Structural Rules in Natural Deduction with Alternatives Natural deduction with alternatives extends Gentzen–Prawitz-style natural deduction with a single structural addition: negatively signed assumptions, called alternatives. It is a mildly bilateralist, single-conclusion natural deduction proof system in which the connective rules are unmodified from the usual Prawitz introduction and elimination rules–the extension is purely structural. This framework is general: it can be used for (1) classical logic, (2) relevant logic without distribution, (3) affine logic, and (4) linear logic, keeping the connective rules... Consequently.org -

Classical Logic and Intuitionistic Logic: looking both ways Abstract: We know a great many technical results concerning the relationship between classical logic and intuitionistic logic, whether in the propositional, first-order or higher-order languages. We also know quite a lot about the relationship between intuitionistic and classical theories. In this talk, I will explore some of what these results might mean, from the perspective of partisans of one side or other of the divide, and what kinds of pluralism might be tenable, in the... Consequently.org -

The Many Uses of Proofs: logic and philosophy, language and more Abstract: This talk is a free-wheeling introduction to my research, starting from work in substructural logics and logical pluralism, and ending at the many uses of proofs, including giving an account of how our modal vocabulary has the meaning that it does, and the connections between proof norms and the semantics and pragmatics of dialogue. The talk is a face-to-face presentation at the University of St-Andrews Computer Science Department’s Research in School day. The slides... Consequently.org -

True Contradictions? Why, and Why Not? Abstract: In this talk, I introduce the difference between paraconsistency (adopting a logic for which a contradiction need not entail everything) and dialetheism (the notion that there are true contradictions), and I explain some reasons why one might take there to be true contradictions. I focus on Jc Beall’s recent work on contradictory Christology as one such motivation, and discuss some attractions of the view, as well as some shortcomings to be further explored. The... Consequently.org -

Collection Frames for Distributive Substructural Logics We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalization of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on... Consequently.org -

Exploring Three-Valued Models for Identity Abstract: There is a very natural way to interpret the propositional connectives and quantifiers, relative to the algebra of three semantic values, {0, i, 1} where 0 and 1 are understood as the traditional values of falsity and truth, and the third value is some intermediate value. The evaluation clauses do not, by themselves, determine the logic, because for that, you need to determine how models are used to provide a counterexample to a sequent.... Consequently.org -

Proofs and Models in Philosophical Logic This is a short book, in the Cambridge Elements series in Philosophical Logic. This is a general introduction to recent work in proof theory and model theory of non-classical logics, with a focus on the application of non-classical logic to the semantic paradoxes and (to a lesser extent), the sorites paradox. After a short introduction motivating general notions of proof and of models, I introduce and motivate a simple natural deduction system, and present the... Consequently.org -

Questions, Justification Requests, Inference, and Definition In this paper, I examine connections between the speech acts of assertion, denial, polar questions and justification requests, and the common ground. When we pay attention to the structure of norms governing polar questions, we can clarify the distinction between strong and weak denial, together with the parallel distinction between strong and weak assertion, and the distinct way that these speech acts interact with the common ground. In addition, once we pay attention to the... Consequently.org -