Ideas from 'Logical Necessity' by Ian Rumfitt [2010], by Theme Structure

[found in 'Modality' (ed/tr Hale,B/Hoffman,A) [OUP 2010,]].

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4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
The logic of metaphysical necessity is S5
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Soundness in argument varies with context, and may be achieved very informally indeed
There is a modal element in consequence, in assessing reasoning from suppositions
We reject deductions by bad consequence, so logical consequence can't be deduction
5. Theory of Logic / D. Assumptions for Logic / 3. Contradiction
Contradictions include 'This is red and not coloured', as well as the formal 'B and not-B'
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths)
10. Modality / A. Necessity / 3. Types of Necessity
A distinctive type of necessity is found in logical consequence [Hale/Hoffmann,A]
10. Modality / A. Necessity / 6. Logical Necessity
Logical necessity is when 'necessarily A' implies 'not-A is contradictory'
A logically necessary statement need not be a priori, as it could be unknowable
Narrow non-modal logical necessity may be metaphysical, but real logical necessity is not
10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
If a world is a fully determinate way things could have been, can anyone consider such a thing?