Church and Fitch have argued that from the verificationationist thesis “for every proposition, if this proposition is true, then it is possible to know it” we can derive that for every truth there is someone who knows that truth. Moreover, Humberstone has shown that from the latter proposition we can derive that someone knows every truth, hence that there is an omniscient being. In his article “Omnificence”, John Bigelow adapted these arguments in order to argue that from the assumption "every (...) contingent proposition is such that if it is true something brought it about that it is true" we can derive that there is an omnificent being: a being that brings it about that every true contingent proposition is true. In my reply to his article, I show that Bigelow’s argument is flawed because there is some formal property that the knowledge operator has but that the bringing about operator lacks. This is the property of distributing over conjunctions. I explain why what brings it about that some conjunctive proposition is true need not bring it about that its conjuncts are true. (shrink)

In virtue of Fitch-Church proof, also known as the knowability paradox, we are able to prove that if everything is knowable, then everything is known. I present two ‘onto-theological’ versions of the proof, one concerning collective omniscience and another concerning omnificence. I claim these arguments suggest new ways of exploring the intersection between logical and ontological givens that is a grounding theme of religious thought. What is more, they are good examples of what I call semi-paradoxes: apparently sound arguments whose (...) conclusion is not properly unacceptable, but simply arguable. (shrink)

Anthony Brueckner, in a recent article, proffers ‘a new way of thinking about Bostrom's Simulation Argument’ . His comments, however, misconstrue the argument; and some words of explanation are in order.The Simulation Argument purports to show, given some plausible assumptions, that at least one of three propositions is true . Roughly stated, these propositions are: almost all civilizations at our current level of development go extinct before reaching technological maturity; there is a strong convergence among technologically mature civilizations such that (...) almost all of them lose interest in creating ancestor-simulations; almost all people with our sorts of experiences live in computer simulations. I also argue that conditional on you should assign a very high credence to the proposition that you live in a computer simulation. However, pace Brueckner, I do not argue that we should believe that we are in simulation. 1 In fact, I believe that we are probably not simulated. The Simulation Argument purports to show only that, as well as , at least one of – is true; but it does not tell us which one.Brueckner also writes: " It is worth noting that one reason why Bostrom thinks that the number of Sims [computer-generated minds with experiences similar to those typical of normal, embodied humans living in a Sim-free early 21 st century world] will vastly outstrip the number of humans is that Sims ‘will run their … ". (shrink)

We illustrate, with three examples, the interaction between boolean and modal connectives by looking at the role of truth-functional reasoning in the provision of completeness proofs for normal modal logics. The first example (§ 1) is of a logic (more accurately: range of logics) which is incomplete in the sense of being determined by no class of Kripke frames, where the incompleteness is entirely due to the lack of boolean negation amongst the underlying non-modal connectives. The second example (§ 2) (...) focusses on the breakdown, in the absence of boolean disjunction, of the usual canonical model argument for the logic of dense Kripke frames, though a proof of incompleteness with respect to the Kripke semantics is not offered. An alternative semantic account is developed, in terms of which a completeness proof can be given, and this is used (§ 3) in the discussion of the third example, a bimodal logic which is, as with the first example, provably incomplete in terms of the Kripke semantics, the incompleteness being due to the lack of disjunction (as a primitive or defined boolean connective). (shrink)

Section 1 recalls a point noted by A. N. Prior forty years ago: that a certain formula in the language of a purely implicational intermediate logic investigated by R. A. Bull is unprovable in that logic but provable in the extension of the logic by the usual axioms for conjunction, once this connective is added to the language. Section 2 reminds us that every formula is interdeducible with (i.e. added to intuitionistic logic, yields the same intermediate logic as) some conjunction-free (...) formula. Thus it would seem that any detour going via formulas with conjunction can be avoided, which raises a puzzle: how is this consistent with the point from Section 1? Sections 3 and 4 raise and discuss this puzzle. In fact, the puzzle turns out on closer inspection not to be so puzzling after all, but it does serve as a convenient centrepiece around which to organize a discussion of the phenomenon illustrated by the Bull–Prior example. Section 5 notes that Prior's observation can be extended to the case of the result of adding disjunction to Bull's logic, while Section 6 includes some further remarks aimed at diagnosing one source of possible residual puzzlement. A subtext of our discussion—spanning several of the notes—is that this work by Bull and Prior has been overlooked, their results having to be rediscovered, by many algebraists and logicians in more recent years. (shrink)

Humberstone has shown that if some set of agents is collectively omniscient (every true proposition is known by at least one agent) then one of them alone must be omniscient. The result is paradoxical as it seems possible for a set of agents to partition resources whereby at the level of the whole community they enjoy eventual omniscience. The Humberstone paradox only requires the assumption that knowledge distributes over conjunction and as such can be viewed as a reductio against the (...) universal validity of that principle. A new route to this paradox is presented which does not require the distribution principle, building on earlier work of Jago and Williamson on Fitch’s paradox. The result relies on an axiom strictly weaker than one necessary for the Jago-Fitch variant. It is shown that the same reasoning behind the variant form of Humberstone’s paradox can recover Bigelow’s results in action theory in a way that is immune to an objection brought against it by Guigon and Humberstone. (shrink)

If what is known need not be closed under logical consequence, then a distinction arises between something's being known to be the case (by a specific agent) and its following from something known (to that subject). When each of these notions is represented by a sentence operator, we get a bimodal logic in which to explore the relations between the two notions.

Two proofs are given which show that if some set of truths fall under finitely many concepts (so-called Collectivity), then they all fall under at least one of them even if we do not know which one. Examples are given in which the result seems paradoxical. The first proof crucially involves Moorean propositions while the second is a reconstruction and generalization of a proof due to Humberstone free from any reference to such propositions. We survey a few solution routes including (...) Tennant-style restriction strategies. It is concluded that accepting Collectivity for some set of truths while also denying that any of the involved concepts in isolation capture all of them requires that one of these concepts cannot be closed under conjunction elimination. This is surprising since the paper surveys several applications in which Collectivity and the latter closure condition seemed jointly satisfiable for concepts of actual philosophical interest. (shrink)