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It is not the case that Weak completeness (every valid formula is provable) is the standard notion of completeness in propositional modal logic, making 'only' misleading.
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Reasons For
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Reason for
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1.
Strong completeness is provable for K, T, S4, S5 with finite axiomatizations, so 'only weak' undersells what modal logic actually achieves.
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2.
The word 'only' reasonably flags that weak completeness omits finite consistency preservation, a philosophically significant limitation.
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3.
Pedagogically, distinguishing weak from strong completeness clarifies what's proven, even if weak is standard—'only' aids precision, not misleading.
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Reasons Against
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Reason against
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1.
Weak completeness (valid → provable) is achieved by major modal logics K, T, S4, S5, making it the standard in textbooks and research.
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2.
Strong completeness (Γ ⊨ φ → Γ ⊢ φ) fails for infinitary cases in modal logic, so weak completeness is the realistic standard.
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3.
Using 'only' suggests weak completeness is unusual or limited, when it's actually the primary completeness notion modal logicians establish.
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