If a viable alternative theory matches the platonist/fictionalist account's consistency with speaker-intention data, then P4's claim that it is the only such theory is demonstrably false.

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Consistency with speaker-intention data(as a measure of how well a theory explains human language use): Whether a theory matches up with evidence about what people actually mean and intend when they use certain words or statements.
P4(Relational principle linking knowledge to its object.): Knowledge is what it is in relation to what it is knowledge of — the nature of knowledge is constituted by its object.
Platonism (in philosophy of mathematics)(as a theory about what numbers and mathematical entities are): The view that abstract mathematical objects like numbers exist independently in their own realm, separate from the physical world and our minds.
Viable alternative theory(as a challenger to an existing theory): A competing explanation or framework that is realistic, workable, and could actually be true as a serious option.

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If a viable alternative theory matches the platonist/fictionalist account's consistency with speaker-intention data, then P4's claim that it is the *only* such theory is demonstrably false.