Negation on the Australian Plan

Abstract

We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. We defuse a number of objections to this Plan, raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan.

Truthmaker Semantics for Relevant Logic

“…We can expand the specific state in which Socrates drinks hemlock to encompass the state of Athens, which (as noted above) is not a truthmaker for 'Socrates drank hemlock'. So, on our analysis, the state Athens is not a truthmaker for: (4) If Socrates drinks hemlock then Socrates drinks hemlock Likewise, the states that I am currently typing and that wombats waddle are not truthmakers for (4). Very few states are.…”

“…This however leads to a definition of negation in terms of incompatibility, which risks conceptual circularity. ((Berto and Restall [4] argue otherwise). It is preferable within truthmaker semantics to analyse negation in terms of falsitymaking, with matehood acting as a restriction on (and not a definition of) truthmaker-falsitymaker pairs.…”

“…A better approach is to consider compatibility relations that hold between states [4,23]. Intuitively, states s and u are compatible when what one makes true the other doesz not make false and vice versa.…”

Putting the stars in the their places

“…Finally, we've given a semantics that, while still using the Routley star, uses it in a pretty innocuous way. In particular, we don't need to appeal to notions of incompatibility or “perp” (a la Berto & Restall, 2019 or Dunn, 1993) to explain the role the stars are playing. Again, this should be welcome in the community.…”

Another plan for negation

“…My point of departure is a recent defence of the "Australian Plan" for a paraconsistent negation by Berto and Restall in [1] against a recent attack by De and Omori in [4], defending the "American Plan". The heart of the controversy is whether to define paraconsistent negation model-theoretically by means of four-valued matrices (with gluts and gaps) or via frames of "points" (going also by the names 'situations', 'states', or 'cases' in general).…”