Ironical trait is unfortunately inconsistent with logic (15)

I believe the answer is:

irrationalistic

I'm a little stuck... Click here to teach me more about this clue!

'inconsistent with logic' is the definition.
I can't tell whether this defines the answer.

'ironical trait is unfortunately' is the wordplay.
'unfortunately' indicates an anagram (a bad or unfortunate spelling).
'ironical'+'trait'+'is'='ironicaltraitis'
'ironicaltraitis' anagrammed gives 'IRRATIONALISTIC'.

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I've seen this clue in The Times.

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