Propositional Identity and Logical Necessity

In two early papers, Max Cresswell constructed two formal logics of propositional identity,  and , which he observed to be respectively deductively equivalent to modal logics 4 and 5. Cresswell argued informally that these equivalences respectively “give . . . evidence” for the correctness of 4 and 5 as logics of broadly logical necessity. In this paper, I describe weaker propositional identity logics than  that accommodate core intuitions about identity and I argue that Cresswell’s informal arguments do not firmly and without epistemic circularity justify accepting 4 or 5. I also describe how to formulate standard modal logics (, 2, and their extensions) with strict equivalence as the only modal primitive. 1 T    Cresswell [2, 3] constructs two formal logics of propositional identity,  and , and informally argues for the correctness of 4 and 5 as logics of broadly logical necessity on the grounds of their respective deductive equivalence to  and . I will describe weaker propositional identity logics than  that accommodate core intuitions about identity, and I will argue that Cresswell’s informal arguments do not firmly and without epistemic circularity justify accepting 4 or 5. I myself will not argue for or against the correctness of 4 or 5. ∗A version of this paper was presented at the 2003 Northwest Philosophy Conference at Reed College where I benefited from comments by Peter Hanks and further discussion with Anthony Anderson and Mark Hinchliff. I also received helpful comments on various versions and ancestors from Bernard Linsky, Peter Loptson, Ernest Sosa, Karen Wendling, and anonymous readers. “Propositional Identity and Logical Necessity”, Australasian Journal of Logic (2) 2004, 1–10 http://www.philosophy.unimelb.edu.au/ajl/2004 2  is  + {, , 1, 2, 3}.1  is the classical non-modal propositional calculus formulated with uniform substitution () and modus ponens () as basic inference rules. > is an arbitrary -tautology. ‘=’ is a binary modal operator and is the only modal primitive in . ‘α = β’ is read as ‘that α is the very same proposition as that β’.2 () ` (p = q) ⊃ (α ⊃ β), provided that α differs from β only in having p in some of the places where β has q. () If ` (α ≡ β) then ` (α = β). (1) 2α =df (α = >) (2) (α↔ β) =df 2(α ≡ β) (3) 3α =df ∼2∼α  has theorems (, ) formally expressing the contentious metaphysical view that () strict equivalence is propositional identity and logical necessity is identity with a tautology.3 () (p = q) = (p↔ q)