Are there some logical systems strong enough to contain theorems of first/second order Peano Arithmetic but constructed in such way that principle of explosion does not hold for them?
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You can see Inconsistent Mathematics. – Mauro ALLEGRANZA Mar 03 '15 at 13:47
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1Related: Can the principle of explosion be removed from constructive logic? – MJD Mar 05 '15 at 14:18