Logical Systems
Specific criticisms of the logical systems devised by Frege and Russell are now more of historical interest than in arguing against Logicism as a contemporary doctrine.
Putting aside the defects of these particular systems the most telling argument which remains against logicism is probably that the ontological premises necessary to carry through the development of mathematics cannot be shown to be principles of logic.


Ontology
Modern mathematics is conducted in a rich ontological framework (such as is found in classical first order set theory), yet no convincing arguments have ever been presented to show that the existence of this elaborate hierarchy of sets (or any other adequate ontology) is logically necessary.

Consistency of Finite Universe
It is consistent to add to first order logic an axiom asserting that there is only one individual, and it cannot therefore be logically necessary that there are more than one.


Gödel's results
Gödel's incompleteness results have also impacted on the credibility of Logicism.
Gödel's first incompleteness theorem shows that, however well patched up, neither Russell's Theory of Types nor any other formal logical system is capable of proving all true facts of elementary arithmetic.

