David Corfield negation

What is Hegel’s determinate negation? Kant in ‘Attempt to introduce the concept of negative magnitudes into philosophy’ distinguishes ‘real’ and ‘logical’ negation.

Logical: ‘A is not B’ or ‘It is not the case that A is B’

Real: two predicates are opposed to each other but not through the law of contradiction. Development of simple term negation of Aristotle. Cancelling of opposites.

In Critique, three categories of quality – reality, negation, limitation, and corresponding judgements – affirmative, negative, infinite.

Three judgements of relation – categorical, hypothetical, disjunctive, and corresponding categories – substance, event, community.

Distinction between negations:

$A: Type$

$x: A \vdash B(x) : Prop$

$B(a) \to 0$ is logical negation.

Colour: Set

Colour of: Thing $\to$ Colour

‘The colour of this thing is red’: Prop is determinate because of the structure of Colour?

• This thing: Thing
• Colour of this thing: Colour
• Red: Colour
• $Id_{Colour}$(Colour of this thing, Red): Prop
• $x$: thing $\vdash$ $Id_{Colour}$(colour of $x$, Red): Prop.

So ‘is red’ is a predicate of thing (i.e., a dependent type ‘x is red’ for x: thing). But it is part of a family of contrary predicates ‘is y’ for each y: colour. So that each thing satisfies one of the family, assuming monochrome things.

The worry for Fregean logic is that such disjunctive judgement is not straightforwardly written in.

‘It is not the case that this thing is red’: Prop

This thing is not red.