# The Reasoning Operators ⇒ and ⇔

rArr and hArr are not propositional operators

• different syntax:
(2x-6 = 0 harr 2x = 6) harr x = 3 is the same as
2x-6 = 0 harr 2x = 6 harr x = 3
(2x-6 = 0 hArr 2x = 6) hArr x = 3 is a syntax error
• so negation cannot be applied to rArr and hArr

• different meaning of chains:
2x-6 = 0 hArr 2x = 6 hArr x = 3 is valid for every x in RR
2x-6 = 0 harr 2x = 6 harr x = 3 yields F when x ne 3

• different reaction to context information
case x < 0: … x^2 = 9 hArr x = -3
We must — and can — and do — have UhArrF
• sqrt(x) = 2 hArr x=4 yields it when x = -1
F U T     F U rArr hArr ✓ ✓ ✓ ✓ ✓ – ✓ ✓ ✓ ✓ ✓ – – – ✓ – – ✓