13764
|
Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? [Edgington]
|
|
Full Idea:
Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? Are they non-truth-functional, like 'because' or 'before'? Do the values of A and B, in some cases, leave open the value of 'If A,B'?
|
|
From:
Dorothy Edgington (Conditionals [2001], 17.1)
|
|
A reaction:
I would say they are not truth-functional, because the 'if' asserts some further dependency relation that goes beyond the truth or falsity of A and B. Logical ifs, causal ifs, psychological ifs... The material conditional ⊃ is truth-functional.
|
13765
|
'If A,B' must entail ¬(A & ¬B); otherwise we could have A true, B false, and If A,B true, invalidating modus ponens [Edgington]
|
|
Full Idea:
If it were possible to have A true, B false, and If A,B true, it would be unsafe to infer B from A and If A,B: modus ponens would thus be invalid. Hence 'If A,B' must entail ¬(A & ¬B).
|
|
From:
Dorothy Edgington (Conditionals [2001], 17.1)
|
|
A reaction:
This is a firm defence of part of the truth-functional view of conditionals, and seems unassailable. The other parts of the truth table are open to question, though, if A is false, or they are both true.
|
13165
|
Geometrical proofs do not show causes, as when we prove a triangle contains two right angles [Proclus]
|
|
Full Idea:
Geometry does not ask 'why?' ..When from the exterior angle equalling two opposite interior angles it is shown that the interior angles make two right angles, this is not a causal demonstration. With no exterior angle they still equal two right angles.
|
|
From:
Proclus (Commentary on Euclid's 'Elements' [c.452], p.161-2), quoted by Paolo Mancosu - Explanation in Mathematics §5
|
|
A reaction:
A very nice example. It is hard to imagine how one might demonstrate the cause of the angles making two right angles. If you walk, turn left x°, then turn left y°, then turn left z°, and x+y+z=180°, you end up going in the original direction.
|
9569
|
The origin of geometry started in sensation, then moved to calculation, and then to reason [Proclus]
|
|
Full Idea:
It is unsurprising that geometry was discovered in the necessity of Nile land measurement, since everything in the world of generation goes from imperfection to perfection. They would naturally pass from sense-perception to calculation, and so to reason.
|
|
From:
Proclus (Commentary on Euclid's 'Elements' [c.452]), quoted by Charles Chihara - A Structural Account of Mathematics 9.12 n55
|
|
A reaction:
The last sentence is the core of my view on abstraction, that it proceeds by moving through levels of abstraction, approaching more and more general truths.
|