Jeremy Goodman

Forcing is a fundamental set-theoretic technique, with which many independence results can be established. A famous example is the independence of the continuum hypothesis in ZFC set theory. Forcing is also well-known to be complex, and therefore difficult to master. Here, we provide a gentle introduction of forcing, by developing forcing for second-order logic. Second-order logic can be interpreted as a rudimentary kind of set theory. Although very limited as a theory of sets, second-order logic is rich enough to capture a form of the continuum hypothesis, as well as the generalized continuum hypothesis, which is slightly easier to state. We develop forcing for second-order logic in the form of possibility semantics, and use this to show that a second-order version of the generalized continuum hypothesis cannot be derived in a standard proof system for second-order logic. Mathematically, the results we obtain in this way are much weaker than standard independence results in set theory. However, as a consequence, we are able to avoid many technical complexities in the presentation of forcing. In this way, we hope that forcing for second-order logic can serve an expository function, of providing an accessible route towards understanding some of the central ideas behind forcing.

Central to certain versions of logical atomism are claims to the effect that every proposition is a truth-functional combination of elementary propositions. Assuming that propositions form a Boolean algebra, we consider a number of natural formal regimentations of informal claims in this vicinity, and show that they are equivalent. For a number of reasons, such as the need to accommodate quantifiers, logical atomists might consider only complete Boolean algebras, and take into account infinite truth-functional combinations. We show that in such a variant setting, some of the regimentations come apart, and explore how they relate to each other. We also discuss how they relate to the claim that propositions form a double powerset algebra, which has been proposed by a number of authors as a way of capturing the central logical atomist idea.

In their 2016 ‘Taking a Chance on KK’, Jeremy Goodman and Bernhard Salow offer an interesting epistemic model that validates KK. The springboard for my discussion will be a mistake in their presentation, one that points to a real problem with the model. My remarks will have a broad interest, since certain of the key examples point to intriguing questions concerning the epistemic contours of testimony and memory. I'll also raise instructive difficulties for a theoretically important weakening of the JTB treatment of knowledge.

It is very natural to think that if two people, x and y, are qualitatively identical and have committed qualitatively identical actions, then it cannot be the case that one has committed something wrong whereas the other did not. That is to say, if x and y differ in their moral status, then it must be because x and y are qualitatively different, and not simply because x is identical to x and not identical to y. In this fictional dialogue between Socrates and Cantor involving infinitely many qualitatively identical agents, this assumption is challenged.

It is widely accepted that knowledge is necessary for proper assertion. More controversial is the thesis that omega-knowledge – infinitely higher-order knowledge – is necessary for proper assertion. Proponents of the ‘KK’ principle take the former thesis to entail the latter, since they take knowledge to entail omega-knowledge. But in Iterated Knowledge, Simon Goldstein argues against the KK principle and in favor of the thesis that omega-knowledge is necessary for proper assertion. This paper argues that reflection on some independently plausible principles about the relationship between knowledge, assertion, and vagueness shows that both Goldstein and the proponents of KK are mistaken. First: if (as is widely accepted) vagueness is a barrier to knowledge, then one can't omega-know that it's vague what one omega-knows. But if omega-knowledge is plentiful, then it's clearly assertable (and thus knowable) that sometimes it's vague what one omega-knows. So if vagueness is a barrier to knowledge, then omega-knowledge can't be necessary for proper assertion, and knowledge can't entail omega-knowledge. Second: even if vagueness isn't a barrier to knowledge, it's a barrier to proper assertion. But this presents a dilemma for Goldstein: either being in a position to assert something entails being in a position to assert that one is in a position to assert it, or it doesn't. If it does, then one can't assert that it's sometimes vague what one is in a position to assert. But that looks like a problem, because it seems clear that one can assert that sometimes it's vague what one is in a position to assert. But if being in a position to assert something doesn't entail being in a position to assert that one is in a position to assert it, then the principle that assertion requires omega-knowledge fails to do the explanatory work Goldstein needs it to do. Either way, we should reject arguments for omega-knowledge anti-skepticism grounded in judgments about proper assertion.

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language. This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given.

(Goodsell, Journal of Philosophical Logic, 51(1), 127-150 2022) establishes the noncontingency of sentences of first-order arithmetic, in a plausible higher-order modal logic. Here, the same result is derived using significantly weaker assumptions. Most notably, the assumption of rigid comprehension—that every property is coextensive with a modally rigid one—is weakened to the assumption that the Boolean algebra of properties under necessitation is countably complete. The results are generalized to extensions of the language of arithmetic, and are applied to answer a question posed by Bacon and Dorr (2024).

The Preface Paradox is often discussed for its implications for rational belief. Much less discussed is a variant of the Preface Paradox for knowledge. In this paper, I argue that the most plausible closure-friendly resolution to the Preface Paradox for Knowledge is to say that in any given context, we do not know much. I call this view “Socraticism”. I argue that Socraticism is the most plausible view on two accounts—(1) this view is compatible with the claim that most of our knowledge ascriptions are true, and (2) provided that (1) is true, the costs of accepting Socraticism are much less than the costs of accepting any other resolution to the Paradox. I argue for (1) in Section 2 by developing a question-sensitive contextualist model for knowledge that shows how Socraticism is compatible with the claim that most of our knowledge ascriptions are true. I also argue how this contextualist model can achieve this result where other contextualist models fail. I then consider other closure-friendly solutions to the paradox in Section 3 and show how accepting those solutions forces us to give up a number of plausible epistemic principles.

Contingentists—who hold that it is contingent what there is—are divided on the claim that having a property or standing in a relation requires being something. This claim can be formulated as a natural schematic principle of higher-order modal logic. On this formulation, I argue that contingentists who are also higher-order contingentists—and so hold that it is contingent what propositions, properties and relations there are—should reject the claim. Moreover, I argue that given higher-order contingentism, having a property or standing in a relation does not even require possibly being something.

The 'content view', in slogan form, is 'Perceptual experiences have representational content'. I explain why the content view should be reformulated to remove any reference to 'experiences'. I then argue, against Bill Brewer, Charles Travis and others, that the content view is true. One corollary of the discussion is that the content of perception is relatively thin (confined, in the visual case, to roughly the output of 'mid-level' vision). Finally, I argue (briefly) that the opponents of the content view are partially vindicated, because perceptual error is due to false belief.

We argue that all comparative expressions in natural language obey a principle that we call Comparability: if x and y are at least as F as themselves, then either x is at least as F as y or y is at least as F as x. This principle has been widely rejected among philosophers, especially by ethicists, and its falsity has been claimed to have important normative implications. We argue that Comparability is needed to explain the goodness of several patterns of inference that seem manifestly valid, that the purported failures of Comparability would have absurd consequences, and that the influential arguments against Comparability are less compelling than they may have initially seemed.

This article presents a new interpretation of the great Ming dynasty philosopher Wang Yangming’s celebrated doctrine of the “unity of knowledge and action”. Wang held that action was not unified with all knowledge, but only with an elevated form of knowledge, which he sometimes called “genuine knowledge”. I argue for a new interpretation of this notion, according to which genuine knowledge requires freedom from a form of doxastic conflict. I propose that, in Wang’s view, a person is free from this form of doxastic conflict if and only if they are acting virtuously.

Suppose you’d like to believe that p, whether or not it’s true. What can you do to help? A natural initial thought is that you could engage in Intentionally Biased Inquiry : you could look into whether p, but do so in a way that you expect to predominantly yield evidence in favour of p. This paper hopes to do two things. The first is to argue that this initial thought is mistaken: intentionally biased inquiry is impossible. The second is to show that reflections on intentionally biased inquiry strongly support a controversial ‘access’ principle which states that, for all p, if p is part of our evidence, then that p is part of our evidence is itself part of our evidence.

In explaining the notion of a fundamental property or relation, metaphysicians will often draw an analogy with languages. The fundamental properties and relations stand to reality as the primitive predicates and relations stand to a language: the smallest set of vocabulary God would need in order to write the “book of the world.” This paper attempts to make good on this metaphor. To that end, a modality is introduced that, put informally, stands to propositions as logical truth stands to sentences. The resulting theory, formulated in higher-order logic, also vindicates the Humean idea that fundamental properties and relations are freely recombinable and a variant of the structural idea that propositions can be decomposed into their fundamental constituents via logical operations. Indeed, it is seen that, although these ideas are seemingly distinct, they are not independent, and fall out of a natural and general theory about the granularity of reality.

This three-part chapter explores a higher-order logic we call ‘Classicism’, which extends a minimal classical higher-order logic with further axioms which guarantee that provable coextensiveness is sufficient for identity. The first part presents several different ways of axiomatizing this theory and makes the case for its naturalness. The second part discusses two kinds of extensions of Classicism: some which take the view in the direction of coarseness of grain (whose endpoint is the maximally coarse-grained view that coextensiveness is sufficient for identity), and some which take the view in the direction of fineness of grain (whose endpoint is the maximally fine-grained theory containing all distinctness claims compatible with Classicism). The third part introduces some techniques for constructing models of Classicism, and uses them to prove the consistency of many of the extensions of Classicism introduced in the second part.

We develop a theory of necessity operators within a version of higher-order logic that is neutral about how fine-grained reality is. The theory is axiomatized in terms of the primitive of *being a necessity*, and we show how the central notions in the philosophy of modality can be recovered from it. Various questions are formulated and settled within the framework, including questions about the ordering of necessities under strength, the existence of broadest necessities satisfying various logical conditions, and questions about their logical behaviour. We also wield the framework to probe the conditions under which a logicist account of necessities is possible, in which the theory is completely reducible to logic.

Can conjunctive propositions be identical without their conjuncts being identical? Can universally quantified propositions be identical without their instances being identical? On a common conception of propositions, on which they inherit the logical structure of the sentences which express them, the answer is negative both times. Here, it will be shown that such a negative answer to both questions is inconsistent, assuming a standard type-theoretic formalization of theorizing about propositions. The result is not specific to conjunction and universal quantification, but applies to any binary operator and propositional quantifier. It is also shown that the result essentially arises out of giving a negative answer to both questions, as each negative answer is consistent by itself.

This paper presents a puzzle involving embedded attitude reports. We resolve the puzzle by arguing that attitude verbs take restricted readings: in some environments the denotation of attitude verbs can be restricted by a given proposition. For example, when these verbs are embedded in the consequent of a conditional, they can be restricted by the proposition expressed by the conditional’s antecedent. We formulate and motivate two conditions on the availability of verb restrictions: a constraint that ties the content of restrictions to the “dynamic effects” of sentential connectives and a constraint that limits the availability of restriction effects to present tense verbs with first-person subjects. However, we also present some cases that make trouble for these conditions, and outline some possible ways of modifying the view to account for the recalcitrant data. We conclude with a brief discussion of some of the connections between our semantics for attitude verbs and issues concerning epistemic modals and theories of knowledge.

This paper defends the view, put roughly, that to think that p is to guess that p is the answer to the question at hand, and that to think that p rationally is for one’s guess to that question to be in a certain sense non-arbitrary. Some theses that will be argued for along the way include: that thinking is question-sensitive and, correspondingly, that ‘thinks’ is context-sensitive; that it can be rational to think that p while having arbitrarily low credence that p; that, nonetheless, rational thinking is closed under entailment; that thinking does not supervene on credence; and that in many cases what one thinks on certain matters is, in a very literal sense, a choice. Finally, since there are strong reasons to believe that thinking just is believing, there are strong reasons to think that all this goes for belief as well.

Most philosophers have largely downplayed any relevance of multiple meanings of the folk concept of truth in the empirical domain. However, confusions about what truth is have surged in political and everyday discourse. In order to resolve these confusions, we argue that we need a more accurate picture of how the term ‘true’ is in fact used. Our experimental studies reveal that the use of ‘true’ shows substantial variance within the empirical domain, indicating that ‘true’ is ambiguous between a correspondence and a coherence reading. We then explore the consequences of these results for the project of re-engineering truth.

Current views of metaphysical ground suggest that a true conjunction is immediately grounded in its conjuncts, and only its conjuncts. Similar principles are suggested for disjunction and universal quantification. Here, it is shown that these principles are jointly inconsistent: They require that there is a distinct truth for any plurality of truths. By a variant of Cantor’s Theorem, such a fine-grained individuation of truths is inconsistent. This shows that the notion of grounding is either not in good standing, or that natural assumptions about it need to be revised.

Is the fact that our universe contains fine-tuned life evidence that we live in a multiverse? Ian Hacking and Roger White influentially argue that it is not. We approach this question through a systematic framework for self-locating epistemology. As it turns out, leading approaches to self-locating evidence agree that the fact that our own universe contains fine-tuned life indeed confirms the existence of a multiverse. This convergence is no accident: we present two theorems showing that, in this setting, any updating rule that satisfies a few reasonable conditions will have the same feature. The conclusion that fine-tuned life provides evidence for a multiverse is hard to escape.