Finite axiomatizability of S_2 or T_2 is an open problem, and modal epistemologists following Williamson hold that possibility claims require more than the absence of known refutations.

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Reasons For

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Reason for

1.Williamson's safety condition requires that true beliefs couldn't easily have been false, demanding positive justification beyond mere absence of refutation.
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2.S_2 and T_2's open axiomatizability status suggests their logical structure resists complete formalization, requiring epistemic caution about possibility claims.
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3.Coherence without known contradiction is insufficient for metaphysical possibility; modal claims need constructive evidence, not just negative absence of counterexample.
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Reasons Against

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Reason against

1.Requiring positive proof of possibility beyond refutation-absence sets an unreasonably high bar that makes most ordinary modal reasoning epistemically unjustified.
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2.Finite axiomatizability status of formal systems doesn't determine whether natural-language modal claims require extra epistemic support beyond standard justification.
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3.Williamson's safety principle, while plausible for knowledge, may overconstrain modal epistemology and conflate metaphysical possibility with justified assertability.
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Key Terms

Absence of known refutations(as used in epistemology): A situation where we haven't found any proof that something is false, but that doesn't necessarily mean it's actually possible—it just means we haven't disproven it yet.
Epistemology / Modal epistemologists(as used in philosophy): Epistemology is the study of knowledge and how we know things. Modal epistemologists specifically study how we know about possibilities and what's possible versus impossible.
Finite axiomatizability(in logic and mathematics): A theory can be completely described using a limited, fixed list of basic starting assumptions (called axioms) rather than needing an infinite number of them.
Possibility claims(as used in modal logic and metaphysics): Statements about what could exist or could be true, rather than what definitely is true.
S_2 or T_2(as used in modal logic): Names for specific formal logical systems that philosophers use to study knowledge and possibility. Think of them as rule-based games with different sets of allowed moves.
Williamson: # Williamson Williamson most commonly refers to Timothy Williamson, a prominent British philosopher known for his work on knowledge, logic, and language. He's influential in contemporary philosophy for arguing that knowledge is more fundamental than belief and that traditional definitions of knowledge may be too restrictive. His ideas have shaped how philosophers think about what it means to know something and how language relates to reality.
modal(in logic and metaphysics): Dealing with possibility and necessity—questions about what could be true, what must be true, and what's merely contingent (could go either way).

Connections

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Truth & Knowledge1 linked Modality & Possibility1 linked

Coherence without known contradiction is insufficient for metaphysical possibili...Finite axiomatizability status of formal systems doesn't determine whether natur...

Details

Type: claim
Perspectives: 2 (1 for, 1 against)
Edits: 1 edit