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Single Idea 13352
[filed under theme 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
]
Full Idea
The Principle of Cutting is the general point that entailment is transitive, extending this to cover entailments with more than one premiss. Thus if γ |= φ and φ,Δ |= ψ then γ,Δ |= ψ. Here φ has been 'cut out'.
Gist of Idea
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z
Source
David Bostock (Intermediate Logic [1997], 2.5.C)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.31
A Reaction
It might be called the Principle of Shortcutting, since you can get straight to the last conclusion, eliminating the intermediate step.
The
18 ideas
with the same theme
[very useful sequents provable in propositional logic]:
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9521
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'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q
[Lemmon]
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9522
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'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q
[Lemmon]
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9525
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We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q)
[Lemmon]
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9524
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We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q
[Lemmon]
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9523
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De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions
[Lemmon]
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9527
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The Distributive Laws can rearrange a pair of conjunctions or disjunctions
[Lemmon]
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9526
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We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q)
[Lemmon]
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9541
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The Law of Transposition says (P→Q) → (¬Q→¬P)
[Hughes/Cresswell]
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13833
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'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction
[Hacking]
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13834
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Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C'
[Hacking]
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13835
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Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with
[Hacking]
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13350
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'Assumptions' says that a formula entails itself (φ|=φ)
[Bostock]
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13351
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'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference
[Bostock]
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13352
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'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z
[Bostock]
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13353
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'Negation' says that Γ,¬φ|= iff Γ|=φ
[Bostock]
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13354
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'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ
[Bostock]
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13355
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'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|=
[Bostock]
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13356
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The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ
[Bostock]
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