Abstract
What does ‘Smith knows that it might be raining’ mean? Expressivism here faces a challenge, as its basic forms entail a pernicious type of transparency, according to which ‘Smith knows that it might be raining’ is equivalent to ‘it is consistent with everything that Smith knows that it is raining’ or ‘Smith doesn’t know that it isn’t raining’. Pernicious transparency has direct counterexamples and undermines vanilla principles of epistemic logic, such as that knowledge entails true belief and that something can be true without one knowing it might be. I re-frame the challenge in precise terms and propose a novel expressivist formal semantics that meets it by exploiting (i) the topic-sensitivity and fragmentation of knowledge and belief states and (ii) the apparent context-sensitivity of epistemic modality. The resulting form of assertibility semantics advances the state of the art for state-based bilateral semantics by combining attitude reports with context-sensitive modal claims, while evading various objectionable features. In appendices, I compare the proposed system to Beddor and Goldstein’s ‘safety semantics’ and discuss its analysis of a modal Gettier case due to Moss.
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Acknowledgements
Thanks to my reviewers for suggestions that improved the paper. This paper was presented at the Lingnan philosophy department’s WIP seminar in Hong Kong SAR (February 20, 2023) and at an APA committee session on Expressivism, at the 2023 Pacific Division APA Meeting at the Westin St. Francis in San Francisco, CA (April 5-8, 2023). Thanks to Nikolaj Pedersen for organizing the latter. Special thanks to Melissa Fusco, Ben Lennertz, and Dan Marshall for detailed commentary. Thanks also to the audiences for helpful comments, especially Ethan Brauer, Rafael De Clercq, and John MacFarlane. This publication is part of the project ‘The Semantics of Knowledge and Ignorance’ (ECS project number 23603221), funded by the Early Career Scheme of the University Grants Council (UGC) of Hong Kong.
Funding
Open Access Publishing Support Fund provided by Lingnan University. This work was supported by the University Grants Committee of Hong Kong via Early Career Scheme (ECS) grant no. 23603221.
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Appendices
Appendices
A Safety Semantics Minus Factivity
Beddor and Goldstein [4] exploit the traditional idea that knowledge is a composite: specifically, belief plus truth plus a condition that renders the belief ‘safe enough’ (cf. [79]). Their system validates the general factivity of knowledge ascriptions, virtually by stipulation. Compared to our own proposal, a question of motivation is immediately pertinent: if dropping Holism (with well-motivated context-sensitivity) gives room for a sensible theory of knowledge ascription, why deploy controversial tools like safety merely to defuse pernicious transparency? Indeed, theories that take knowledge to be a conjunction of truth, belief, and further conditions are inevitably controversial, given a history of difficulties (cf. [68, 79]). If truth is independent of the other conditions, generalized Gettierization looms [87]; if not, there is misleading redundancy. Our own account sidesteps such worries.
Beddor and Goldstein’s chief rationale for including a ‘truth condition’ is to assure Modal Factivity (in contrast, safely believing descriptive p entails p is true). Section 6.1 argued that Modal Factivity is disputable. We thus consider a variant of Beddor and Goldstein’s account that drops the truth condition, judged as a direct competitor to FaTE for meeting the challenge in Section 4.3.
We work with language
, including atoms, boolean connectives, ‘might’ operator \(\diamond \), objective possibility operator
, and belief operator B. Read
as ‘it could easily have been that \(\varphi \)’. A safety model \(\mathcal {S}\) is a quadruple \(\langle W, @, {\textbf {b}}, {\textbf {i}} \rangle \). W is the set of all possible worlds, including @. Each world w assigns a truth value w(p) (0 or 1) to each atomic sentence. Functions \({\textbf {b}}\) and \({\textbf {i}}\) map a possible world to an intension: \({\textbf {b}}(w)\) (we write \({\textbf {b}}^w\)) is the agent’s doxastic state at w (understood as a set of doxastic alternatives), while \({\textbf {i}}(w)\) (we write \({\textbf {i}}^w\)) is the worldly information at w: a set of worlds that intuitively are sufficiently ‘nearby’ w. We stipulate that \({\textbf {i}}^w\) is veridical at w, i.e., \(w \in {\textbf {i}}^w\).
Definition 10
(Safety Semantics) Given safety model \(\mathcal {S}\), \(w \in W\) and \({\textbf {s}} \subseteq W\):
\(\llbracket p \rrbracket ^{w, {\textbf {s}}} = 1\qquad \quad \,\,\,\,\) iff \(\quad w(p)=1\)
\(\llbracket \lnot \varphi \rrbracket ^{w, {\textbf {s}}} = 1\qquad \quad \) iff \(\quad \llbracket \varphi \rrbracket ^{w, {\textbf {s}}} = 0\)
\(\llbracket \varphi \wedge \psi \rrbracket ^{w, {\textbf {s}}} = 1\quad \,\,\) iff \(\quad \llbracket \varphi \rrbracket ^{w, {\textbf {s}}} = 1\) and \(\llbracket \psi \rrbracket ^{w, {\textbf {s}}} = 1\)
\(\llbracket \diamond \varphi \rrbracket ^{w, {\textbf {s}}} = 1\qquad \quad \,\) iff \(\quad \exists v \in {\textbf {s}}\): \(\llbracket \varphi \rrbracket ^{v, {\textbf {s}}} = 1\)
\(\llbracket B\varphi \rrbracket ^{w, {\textbf {s}}}=1\qquad \quad \) iff \(\quad \llbracket \varphi \rrbracket ^{{\textbf {b}}^w} = 1\)
\(\llbracket \)
\(\varphi \rrbracket ^{w,{\textbf {s}}} = 1\) iff \(\exists v\in {\textbf {i}}^w\): \(\llbracket \varphi \rrbracket ^{v,{\textbf {i}}^v}=1\)
where \(\llbracket \varphi \rrbracket ^{\textbf {s}} = 1\) iff \(\forall w \in {\textbf {s}}\): \(\llbracket \varphi \rrbracket ^{w,{\textbf {s}}} = 1\).
So
is true at \(\langle w,{\textbf {s}} \rangle \) when there is a world v compatible with the worldly information at w (intuitively, v is ‘nearby’ w) such that \(\varphi \) is true at \(\langle v,{\textbf {i}}^v \rangle \). To capture assertibility, we take \({\textbf {s}} \Vdash \varphi \) to mean \(\llbracket \varphi \rrbracket ^{\textbf {s}} = 1\). Then, \(\varphi |{\hspace{-0.7pt}}|{\hspace{-5.0pt}}= \psi \) holds when: for every safety model \(\mathcal {S}\) and intension \({\textbf {s}}\), if \(\llbracket \varphi \rrbracket ^{\textbf {s}} = 1\) and \(@ \in {\textbf {s}}\) then \(\llbracket \psi \rrbracket ^{\textbf {s}} = 1\).
We define
and \(K \varphi := B\varphi \wedge \blacksquare (B\varphi \supset \varphi )\). Thus, \(\blacksquare (B\varphi \supset \varphi )\) operates as the ‘safety condition’. Routine proofs establish that safety semantics then validates KB and Descriptive Factivity, without validating K-Transparency, Modal Omniscience, or Modal Factivity. However, our safety theory also has some problematic features.
Proposition 23
Per safety semantics, Attitude Inheritance fails: \(K(p \wedge \diamond q) \nVDash K\diamond (p \wedge q)\).
Proof
Let \(\mathcal {S}\) be a safety model: (i) \(W = \{w_1, w_2\}\) with \(@ = w_1\), (ii) \(w_1(p) = w_1(q)= 1\), (iii) \(w_2(p) = w_2(q)= 0\), (iv) \({\textbf {b}}^{w_1} = \{ w_1\}\), (v) \({\textbf {b}}^{w_2} = W\), (vi) \({\textbf {i}}^{w_1} = W\), and (vii) \({\textbf {i}}^{w_2} = \{ w_2\}\). Then \(\llbracket B(p \wedge \diamond q) \rrbracket ^{\{w_1\}}=1 \), as there are only \(p \wedge q\)-worlds in \({\textbf {b}}^{w_1}\). Further, \(\llbracket \blacksquare (B(p \wedge \diamond q) \supset (p \wedge \diamond q)) \rrbracket ^{\{w_1\}}=1 \), as for every \(v \in {\textbf {i}}^{w_1}\), if there is a q-world and only p-worlds in \({\textbf {b}}^v\) (note that \(w_1\) meets this condition, but not \(w_2\)), then v is a p-world and there is a q-world in \({\textbf {i}}^v\). Altogether: \(\llbracket K(p \wedge \diamond q) \rrbracket ^{\{w_1\}}=1 \). However, there exists \(v \in {\textbf {i}}^{w_1}\) (namely, \(w_2\)) such that: there is a \(p \wedge q\)-world in \({\textbf {b}}^v\) (namely, \(w_1\)) but no \(p \wedge q\)-world in \({\textbf {i}}^v\) (as only \(w_2\) is in \({\textbf {i}}^{w_2}\)). Thus, \(\llbracket \blacksquare (B \diamond (p \wedge q) \supset \diamond (p \wedge q)) \rrbracket ^{\{w_1\}}=0\). Thus, \(\llbracket K\diamond (p \wedge q) \rrbracket ^{\{w_1\}}=0\). \(\square \)
Thus, our safety theory misses a key advantage of expressivist frameworks like FaTE (cf. Section 4.1): by itself, it lacks resources to answer an important element of the precisified challenge from transparency (cf. Section 4.3).
This isn’t the end of its problematic logical features.
Proposition 24
Per safety semantics, \(K\diamond (p \wedge q) \nVDash K\diamond p \).
Proof
Let \(\mathcal {S}\) be a safety model: (i) \(W = \{w_1, w_2, w_3\}\) with \(w_1 = @\), (ii) \(w_1(p)=w_3(p)= w_1(q)=w_2(q)= 1\), (iii) \(w_2(p)=w_3(q)= 0\), (iv) \({\textbf {b}}^{w_1}= \{ w_1\}\), (v) \({\textbf {b}}^{w_2}= {\textbf {b}}^{w_3}= \{ w_3\}\), (vi) \({\textbf {i}}^{w_1} = \{w_1, w_2\}\), and (vii) \({\textbf {i}}^{w_2} = \{w_2\}\). As there is a \(p \wedge q\)-world in \({\textbf {b}}^{w_1}\), we have \(\llbracket B\diamond (p \wedge q) \rrbracket ^{\{w_1\}}=1\). Further, \(\forall w \in {\textbf {i}}^{w_1}\), if there is a \(p\wedge q\)-world in \({\textbf {b}}^w\) then there’s one in \({\textbf {i}}^w\). So, \(\llbracket \blacksquare (B\diamond (p \wedge q) \supset \diamond (p \wedge q))\rrbracket ^{\{w_1\}}=1\). Thus, \(\llbracket K\diamond (p \wedge q) \rrbracket ^{\{w_1\}}=1\). However, by (v) and (vii), there is a p-world in \({\textbf {b}}^{w_2}\) but not in \({\textbf {i}}^{w_2}\). So, \(\exists w \in {\textbf {i}}^{w_1}\) s.t. there’s a p-world in \({\textbf {b}}^w\) but not in \({\textbf {i}}^w\). So, \(\llbracket \blacksquare (B\diamond p \supset \diamond p)\rrbracket ^{\{w_1\}}\ne 1\). So, \(\llbracket K\diamond p \rrbracket ^{\{w_1\}} \ne 1\). \(\square \)
Thus, unlike FaTE, the current theory erroneously predicts that ‘Smith knows that it might be cloudy and damp’ does not entail ‘Smith knows that it might be cloudy’.
B Modal Gettier Cases
Sarah Moss argues that an adequate theory of knowledge ascription should accommodate modal Gettier cases.
-
(65)
Fake Letters. Alice enters a psychology study with her friend Bert. As part of the study, each participant is given a detailed survey of romantic questions about their friend. After the study is over, each participant is informed of the probability that they find their friend attractive. Several disgruntled lab assistants have started mailing out fake letters, telling nearly every participant that they probably find their friend attractive. Alice happens to receive a letter from a diligent lab assistant. Her letter correctly reports that she probably does find Bert attractive. Alice reads the letter and comes to have high credence that she finds Bert attractive. [Accordingly, she comes to believe that she might find Bert attractive.] [55, pg.103, additional sentence appended]
Given Fake Letters, one reasonably judges that Alice might find Bert attractive and justifiably believes that she might find Bert attractive, but she fails to know that she might find Bert attractive, as she could easily have been misled. As Alice in fact finds Bert attractive, she also cannot know that she doesn’t find him attractive. So Fake Letters serves as an intuitive counterexample to K-Transparency [4, 53, 55].
FaTE has resources for modeling such cases. Consider a TF model \(\mathcal {T}\) and information state \({\textbf {s}}\) where: (i) \({\textbf {s}}\) is compatible with p (i.e., \({\textbf {s}}\) is partly about p and is consistent with p), (ii) at every world in \({\textbf {s}}\), the agent’s doxastic state at that world contains a fragment that is compatible with p, and (iii) at every world in \({\textbf {s}}\), the agent’s epistemic state at that world has no fragment compatible with p (in particular, no such fragment is about p, i.e., no such fragment has content whose subject matter includes that of p). It follows that \({\textbf {s}} |{\hspace{-0.7pt}}|{\hspace{-4.6pt}}- \diamond p \wedge B\diamond p \wedge \lnot K \diamond p \wedge \lnot K \lnot p\), as required.
To more pointedly model Fake Letters, we can extend FaTE to include justified belief operators. Let a TF model with justification be a TF model \(\mathcal {J}\) supplemented with a function \({\textbf {J}}\) that maps a world to fragments of justified belief: \({\textbf {J}}(w)\) is Smith’s total justified belief state at w. We assume that every justified belief fragment is a type of belief fragment (\({\textbf {J}}(w) \subseteq {\textbf {B}}(w)\), for all w) and every knowledge fragment is a type of justified belief fragment (\({\textbf {K}}(w) \subseteq {\textbf {J}}(w)\), for all w). Again, our semantic treatment remains silent on the epistemological question as to what makes a fragment justified. For sensible constraints on subject matter, we assume:
-
SM5.
\(\texttt {t}(B\varphi ) \subseteq \texttt {t}(J\varphi )\) and \(\texttt {t}(J\varphi ) \subseteq \texttt {t}(K\varphi )\).
-
SM6.
If \(\texttt {t}(\varphi )\subseteq \texttt {t}(\psi )\) then: \(\texttt {t}(J\varphi )\subseteq \texttt {t}(J\psi )\).
FaTE with Justification: We extend the semantics for FaTE with:
\({\textbf {s}} |{\hspace{-0.7pt}}|{\hspace{-4.6pt}}- J\varphi \qquad \) iff \(\qquad \texttt {t}(J\varphi ) \subseteq {\textbf {s}}\) and \(\forall w \in {\textbf {s}}\): \(\exists {\textbf {j}} \in {\textbf {J}}(w)\): \({\textbf {j}} |{\hspace{-0.7pt}}|{\hspace{-4.6pt}}- \varphi \)
\({\textbf {s}} -{\hspace{-4.6pt}}|{\hspace{-0.6pt}}| J\varphi \qquad \) iff \(\qquad \texttt {t}(J\varphi ) \subseteq {\textbf {s}}\) and \(\forall w \in {\textbf {s}}\): \(\forall {\textbf {j}} \in {\textbf {J}}(w)\): \({\textbf {j}} \nVdash \varphi \)
Fake Letters can then be modeled with a model \(\mathcal {J}\) and proposition \({\textbf {s}}\) with the following features: (i) \({\textbf {s}}\) is compatible with p (i.e., \({\textbf {s}}\) is partly about p and is consistent with p), (ii) at every world in \({\textbf {s}}\), the agent’s justified belief state at that world contains a fragment that is compatible with p, and (iii) at every world in \({\textbf {s}}\), the agent’s knowledge state contains no fragment compatible with p (in particular, no such fragment is about p, i.e., no such fragment has content whose subject matter includes that of p). It follows that \({\textbf {s}} |{\hspace{-0.7pt}}|{\hspace{-4.6pt}}- \diamond p \wedge B\diamond p \wedge J \diamond p \wedge \lnot K \diamond p \wedge \lnot K \lnot p\).
In short, FaTE can diagnose a modal Gettier case with respect to \(\diamond p\) as a situation where an agent’s cognitive system contains belief fragments about p’s subject matter, but no knowledge fragments about p’s subject matter. This doesn’t imply that the agent is unable to grasp p’s subject matter, enter into reasoning with content about that subject matter, or attend to the question as to whether p is true: intuitively, these functions could manifest via the agent’s belief fragments on p’s subject matter.
However, a deeper worry points again to FaTE’s limitations. If no fragment of her knowledge is about p’s subject matter, our agent has no knowledge at all on that subject matter. But, intuitively, modal Gettier cases exist where the agent in question has some knowledge about the subject matter of p. In Fake Letters, it would be odd to deny that Alice at least knows that either she finds Bert attractive or she doesn’t (\(p \vee \lnot p\)). By the lights of FaTE, Alice must have a knowledge fragment about p’s subject matter, grounding her knowledge that \(p \vee \lnot p\). More pointedly, recall (Section 6) that FaTE validates Restricted K-Transparency: \(K(p \vee \lnot p) \wedge \lnot K\lnot p {={\hspace{-5.0pt}}|{\hspace{-0.7pt}}|}{|{\hspace{-0.7pt}}|{\hspace{-5.0pt}}=} K\diamond p\). According to FaTE, if \(K(p \vee \lnot p)\) and \(\lnot K\lnot p\) hold, our agent cannot be in a modal Gettier case, contrary to our intuitions about Fake Letters.
There is an answer: shift to Contextualist FaTE, as one of its chief virtues is that it invalidates Restricted K-Transparency (Section 6.2). Hence, Contextualist FaTE offers improved tools for modeling modal Gettier cases, with nuanced explanatory options. By its lights, if ‘Alice knows \(p \vee \lnot p\)’ is true, but ‘Alice knows \(\lnot p\)’ and ‘Alice knows \(\diamond p\)’ are false, there must be a contextually salient distinction in play: there must be a relevant alternative c such that (i) Alice’s information fails to eliminate c and (ii) Alice would be positioned to deny p if she were to learn c holds. In Fake Letters, there is an obvious candidate for c: the possibility that Alice’s survey indicates that she doesn’t find Bert attractive. Just as ‘Smith knows the coin might land Tails’ is true in Coin 1 (Section 6.2) only if Smith knows that the coin isn’t double-headed, so ‘Alice knows she might find Bert attractive’ is true in Fake Letters only if Alice knows that her survey doesn’t indicate that she doesn’t find Bert attractive.
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Hawke, P. Modal Knowledge for Expressivists. J Philos Logic (2024). https://doi.org/10.1007/s10992-024-09759-2
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DOI: https://doi.org/10.1007/s10992-024-09759-2