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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL

[very useful sequents provable in propositional logic]

18 ideas
We can change conditionals into negated conjunctions with P→Q -||- (P ∧ Q) [Lemmon]
We can change conjunctions into negated conditionals with P→Q -||- (P → Q) [Lemmon]
We can change conditionals into disjunctions with P→Q -||- P ∨ Q [Lemmon]
De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions [Lemmon]
The Distributive Laws can rearrange a pair of conjunctions or disjunctions [Lemmon]
'Modus ponendo tollens' (MPT) says P, (P ∧ Q) |- Q [Lemmon]
'Modus tollendo ponens' (MTP) says P, P ∨ Q |- Q [Lemmon]
The Law of Transposition says (P→Q) → (Q→P) [Hughes/Cresswell]
'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction [Hacking]
Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking]
Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with [Hacking]
'Assumptions' says that a formula entails itself (φ|=φ) [Bostock]
'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock]
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock]
'Negation' says that Γ,φ|= iff Γ|=φ [Bostock]
'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock]
'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock]
The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock]