Week 5

Taking stock

This week we will introduce more Boolean connectives, but no new methods. We will only introduce new examples of things you have already met. This will give you an opportunity to consolidate the logic topics we have introduced so far.

In week 1 we introduced predicates and universes. In weeks 2 & 3 we introduced syllogisms. These bring a host of new ideas, Aristotelian propositions, negation, contraposition at two different levels, rules of inference. The key ideas include predicates represented as boolean-valued functions on a universe of individuals, propositions that express relations between predicates, and rules that allow us to infer some propositions from others. You should know what it means for a proposition to be valid in a given universe, What it means for a rule to be sound, and how to provide a counter-example to a non-sound rule. Finally, we found various ways to transform sound rules to produce new sound rules.

In week 4 we introduced Gentzen's sequents.

You should make sure you understand when a sequent is valid, and what it means to provide a counter-example to a sequent -- a universe in which the sequent is not valid.

You should be able to reduce a sequent using the connectives ⋀ ⋁ ¬ to (the conjunction of) as set of simple sequents &madash;these are sequents Γ ⊨ Δ with no connectives (Γ Δ include only bare propostitional letters), such that Γ ⋂ Δ = ∅ (the intersection is empty), and to use this to determine whether a sequent is valid, and to produce a counter-example if it is not.

For examples of of reduction see:

Last Thursday's Q&A recording (26:56)

Open the tab below to see the pencil-and-paper notes created during the video.

Screenshots

Very few videos this week. I will add more in response to questions on Piazza, if there are specific topics or exercises you want me to cover.