Takeaways for Week One
Judgements and evidence
- KNOW: A judgement is a statement (proposition, utterance, enunciation). “claim”
- KNOW: An evident judgement is a statement for which I have evidence (proof)
Rules
- BE ABLE TO: express what constitutes as evidence of a judgement in the form of rules
- BE ABLE TO: assemble rules into derivations that prove judgements
- KNOW: Judgements above the line of a rule are called premises
- KNOW: Judgements below the line of a rule are called conclusions
- KNOW: Rules with no premises are called axioms.
Simultaneous generation of judgements
- KNOW: judgements can be mutually defined using rules
- BE ABLE TO: define rules that mutually define judgements and use them to assemble derivations
Derivable and admissible rules
- KNOW: A rule is admissible if whenever we have a derivation of the premises, then we know we can construct a derivation of the conclusion.
- BE ABLE TO: prove that a rule is admissible
- KNOW: A rule is derivable if we can use a derivation of its premise as a building block in deriving its conclusion
- BE ABLE TO: prove that a rule is admissible
Induction
- UNDERSTAND: the principle of induction
- BE ABLE TO: state the associated induction principle given a set of rules
- BE ABLE TO: prove something by induction!
Simultaneous Induction
- KNOW: induction can also be used for mutually defined judgements
- BE ABLE TO: state the associated induction principle given a set of mutually defined rules
- BE ABLE TO: prove something by simultaneous induction
Key:
KNOW = a fact that you have memorised / written down in a place you can find it again
UNDERSTAND = something you can explain to another, where you don’t just know what, but why and how
BE ABLE TO = something you can replicate again on your own and without guidance