Zach Weber

A theorem from Archimedes on the area of a circle is proved in a setting where some inconsistency is permissible, by using paraconsistent reasoning. The new proof emphasizes that the famous method of exhaustion gives approximations of areas closer than any consistent quantity. This is equivalent to the classical theorem in a classical context, but not in a context where it is possible that there are inconsistent innitesimals. The area of the circle is taken 'up to inconsistency'. The fact that the core of Archimedes's proof still works in a weaker logic is evidence that the integral calculus and analysis more generally are still practicable even in the event of inconsistency.

Even if you were the last person on Earth, you should not cut down all the trees—or so goes the Last Man thought experiment, which has been taken to show that nature has intrinsic value. But ‘Last Man’ is caught on a dilemma. If Last Man is too far inside the anthropocentric circle, so to speak, his actions cannot be indicative of intrinsic value. If Last Man is cast too far outside the anthropocentric circle, though, then value terms lose their cogency. The experiment must satisfy conditions in a seemingly impossible ‘goldilocks’ zone. To this end I propose a new version, the Ultramodal Last Man, which appeals to Routley's work in metaphysics and non-classical logic. With this ‘Last Last Man’, I argue that the Local/Global dilemma is resolved: impossible equations balance in ultramodal space. For defenders and critics alike, this helps to clarify the demands of intrinsic value, and renews a role for non-standard logics in value theory.

The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads to triviality.

In his ‘On t and u and what they can do’, Greg Restall presents an apparent problem for a handful of well-known non-classical solutions to paradoxes like the liar. In this article, we argue that there is a problem only if classical logic – or classical-enough logic – is presupposed. 1. Background Many have thought that invoking non-classical logic – in particular, a paracomplete or paraconsistent logic – is the correct response to the liar and related paradoxes. At the most basic level, the target non-classical idea is that some expressions, like ‘all and only the true propositions’, do not behave as we would expect from classical logic. Non-classical theorists argue that the class of all and only the truths is either incomplete or inconsistent: when you truly speak of all and only truths , you're either leaving some truths out, or you're letting some untruths in. Truth, in a slogan, is either gappy or glutty. Non-classicality is not a glib or easy-way-out response to the paradoxes. Innocuous-seeming notions can turn out to be philosophically substantial. Moreover, apparently correct forms of reasoning can turn out to be incorrect. To take a example, a glut theorist must hold that the following argument form is not in general valid. Either p is true or q is true. But p is untrue. So q is true. This argument form may strike our ears as acceptable. But if p is a truth value glut, then the inference fails to preserve truth. According to glut theorists, the inference breaks down in inconsistent contexts: if the subject matter involves gluts, then the inference is to be rejected. And the liar paradox, according to such theorists, shows that truth is exactly the kind of subject matter that yields inconsistency. In the …