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Single Idea 13833
[filed under theme 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
]
Full Idea
'Dilution' (or 'Thinning') provides an essential contrast between deductive and inductive reasoning; for the introduction of new premises may spoil an inductive inference.
Gist of Idea
'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction
Source
Ian Hacking (What is Logic? [1979], §06.2)
Book Ref
'A Philosophical Companion to First-Order Logic', ed/tr. Hughes,R.I.G. [Hackett 1993], p.233
A Reaction
That is, inductive logic (if there is such a thing) is clearly non-monotonic, whereas classical inductive logic is monotonic.
Related Idea
Idea 13351
'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock]
The
18 ideas
with the same theme
[very useful sequents provable in propositional logic]:
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9522
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'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q
[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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9527
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The Distributive Laws can rearrange a pair of conjunctions or disjunctions
[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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9524
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We can change conditionals into disjunctions with P→Q -||- ¬P ∨ 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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9521
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'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- 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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