from July 2019 to October 2019

Offered through the University of Melbourne

PHIL20030: Meaning, Possibility and Paradox is a University of Melbourne undergraduate subject introducing logic to philosophy students. It’s taught by Greg Restall and Shawn Standefer.

This year, we have completely revised our curriculum. Now the subject introduces the proof theory and model theory of propositional, modal and predicate logic–in that order. We’re writing an introductory text Logical Methods, which we’re trialling with this class, as well as producing explanatory videos to use along with the text.

Here’s the outline of the subject.

Preliminaries

• Introduction
• Arguments and Trees
• Sentences and Formulas

Propositional Logic

• Connectives: and & if
• Conjunction
• Conditional
• Biconditional
• More connectives: not & or
• Negation and falsum
• Disjunction
• Our System of Proofs
• Facts about proofs & provability
• Normalisation
• The Subformula Property
• Consequences of Normalisation
• Models & counterexamples
• Models and truth tables
• Counterexamples and validity
• Model-theoretic validity
• Soundness & completeness
• Soundness
• Completeness
• Proofs first or models first?
• Heyting algebras
• Necessity & possibility
• Possible worlds models
• Validity
• Strict conditionals and ambiguities
• Propositions
• Another notion of necessity
• Equivalence relations and epistemic logic
• Actuality & two-dimensional logic
• Actuality models and double indexing
• Validity
• Fixity and diagonal propositions
• Real world validity
• Natural deduction for modal logics
• Natural deduction for S4
• Natural deduction for S5
• Features of S5

Predicate Logic

• Quantifiers
• Syntax
• Natural deduction for CQ
• What is provable?
• Generality and eliminating detours
• Models for first-order logic
• Models and assignments of values
• Substitution
• Counterexamples and validity
• Compactness and what this means

One novelty in our approach to the subject is the balance between proof theory and model theory. We introduce propositional logic by way of Gentzen/Prawitz-style natural deduction—for intuitionistic logic—and along the way, each time we introduce the rules for a connective, we show that they are in harmony. So, it’s not too hard to show that proofs in the whole system can be normalised and we get the subformula property for normal proofs. (So, we can gesture in the direction of provability being analytic in a strong sense, since a normal proof literally analyses the premises and conclusion into components and connects them using the fundamental rules governing the concepts involved.)

Once that’s done, we then introduce Boolean valuations (and truth tables), and we can show that the proof system is sound but not complete for validity defined as the absence of a Boolean counterexample. Approaching things this way means we have an interesting discussion about soundness and completeness, and about intuitionistic and classical logic, and whether we should be happy with the gap between proofs and models or not, and if not, whether we should close that gap by adding to our proof system (that way lies classical natural deduction), or whether we should close the gap by enriching our class of models to serve as counterexamples (here we sketch Heyting algebras, as generalisations of Boolean valuations, but we point to Kripke models, too). There’s also scope for a discussion of whether we should understand logic in a proof-first way or a model-first way (or both, or neither), and how proofs and models relate to however it is that words and concepts get their meanings.

With that done, we’re halfway through the subject. Having arrived at Boolean valuations, it’s a short hop, skip and jump to Carnap’s models for modality, and their generalisation, universal models for the modal logic S5. So, we look at these models for possibility and necessity, and show how these possible worlds models can be used to analyse modality, strict conditionality, and similar notions.

Then with models like these we can be of service to our colleagues by introducing double-indexing and two-dimensional modal logic, and the analysis of fixedly diagonal propositions, and the relationship between analyticity, necessity and a priority.

With these model-theoretic considerations in hand, we turn to the question of what it might be to derive a modal claim, and we turn to the natural deduction rules for modals, which introduce constraints on assumptions. One way to prove that $$A$$ is necessary, after all, is to prove $$A$$ from claims of the form $$\Box B$$, for those claims hold not only here, but also in any alternate circumstances, too. So, we get natural deduction systems for S4 and S5 rather straightforwardly.

Proving something more general than $$A$$ by proving $$A$$ from premises satisfying certain conditions sounds familiar if you’ve dealt with quantifiers before. How to you show that everything is an $$F$$? By proving that $$Fa$$ when we have assumed nothing about $$a$$. Then our proof applies no matter what $$a$$ is. So, we can generalise the conditions for modal proof to proofs with quantifiers too. So, we introduce the logic of first-order quantifiers with natural deduction first, and once we’ve done that, we turn back to models at last.

So, the introduction to logic has a rhythm, taking us from proofs to models of propositional logic, through models and then proofs for modal logic, and then to proofs and models for predicate logic. Along the way we look at issues in the philosophy of logic and the applications of logic to different issues in philosophy.

Although this curriculum and the course material is all ours, we are indebted to our colleagues for many discussions concerning the pedagogy of logic. I’ll single out two here. Allen Hazen talked to GR for many years about the pedagogical virtues of introducing modal logic before predicate logic to philosophy students. And Dave Ripley has, for the last couple of years, introduced logic using intuitionistic natural deduction and classical truth tables, making a virtue out of the soundness and incompleteness of the pairing between the proof theory and the model theory. Neither Allen nor Dave would teach things how we have, but we’ve valued talking over the pedagogy with them over the years.

If you’d like to compare your mastery of logic, in comparison to what our students are learning, you can try your own hand at our in-class tasks for weeks 1 to 6.

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