Thursday's Q&A recording
Open the tab below to see the pencil-and-paper notes created during the video.Screenshots
Over the past two weeks we've looked at Aristotle's Syllogisms.
We've formalised Aristotle's Categorical Propositions using four
forms: a ⊫ b, a ⊫ ¬b, a ⊭ b,
a ⊭ ¬b, and discovered that we can obtain all 15 sound
syllogisms from barbara, using substitutions and contraposition.
Our first video is a brief recap of Aristotle's rules.
Slides pp.2-5
Slides pp.6-14
In this video we suggest you think about venn diagrams on a sphere to visualise the inherent duality — the fact that there are always two ways of looking at things — in Boolean logic.
We recommend that you wait until after you have completed the Syllogisms test on Wednesday, before watching the videos below.
One important new ingredient this week, and next, is introduction of more operations on predicates. We already have negation; now we add conjunction and disjunction.Slides pp.15-18
Slides pp.17-21
The following videos introduce sequents, a far-reaching generalisation of the idea underlying Aristotle's propositions. We have already discussed the introduction of multiple antecedents (Multiple predicates coming before the turnstile). Now we introduce multiple succeedents (multiple predicates coming after the turnstile). Here, "multiple" includes any finite number of predicates, zero or more, including none.
This video introduces the idea of putting multiple predicates to the right of the turnstile.
Slides pp.1-3
We interpret additional predicates before the turnstile. These simply express validity in a sub-universe. This means that for any sound rule the corresponding rule with additional predicates is also sound.
Slides pp.3-10
Additional predicates after the turnstile behave similarly.
Slides pp.10-15
We can now give Gentzen's rules for ¬ ⋀ ⋁
Slides pp.15-18
We use the rules to reduce a sequent to a conjunction of simpler sequents. In this example we find that the expression asserted by the sequent is a tautology — it is equivalent to the empty conjunction, so is valid in every universe.
Slides pp.19-33
We use the rules to reduce a sequent to a conjunction of simple sequents, sequents that only mentions propositional letters, with no connectives, and no repetitions — in this example, we find a single simple sequent, equivalent to our starting sequent in the sense that both are valid in the same universes. This shows our starting sequent is not a tautology. It is simple to provide a counterexample to a simple sequent, and this provides a counter-example to the strtting sequent.
Slides pp.34-49
In this video we introduce the idea of valuations. From the perspective of our logic, two individuals with the same properties (i.e. satisfying the same predicates) are indistinguishable. We cannot count how many indistinguishable things there are of each kind, but we can check whether or not there are some of a given kind. Each kind is specified by giving a valuation saying which predicates are true, and which false, for individuals of this kind.
A valuation corresponds to a line in a truthtable. It is simply a function from predicates to Booleans.
Slides pp.50-57