The Elementary Epistemic Arithmetic of Criminal Justice

This paper propounds the following theses: 1). that the traditional focus on the Blackstone ratio of errors as a device for setting the criminal standard of proof is ill-conceived, 2). that the preoccupation with the rate of false convictions in criminal trials is myopic, and 3). that the key ratio of interest, in judging the political morality of a system of criminal justice, involves the relation between the risk that an innocent person runs of being falsely convicted of a serious crime and the risk of being criminally victimized by someone who was falsely acquitted.

Reflecting in epistemic arithmetic

An epistemic formalization of arithmetic is constructed in which certain non-trivial metatheoretical inferences about the system itself can be made. These inferences involve the notion of provability in principle, and cannot be made in any consistent extensions of Stewart Shapiro's system of epistemic arithmetic. The system constructed in the paper can be given a modal-structural interpretation

Flagg realizability in arithmetic

Epistemic arithmetic—that is, first-order arithmetic with S4 as the underlying logic—was introduced by Shapiro in [7] and independently by Reinhardt in [6]. Shapiro showed that intuitionistic first-order arithmetic HA can be embedded in epistemic arithmetic EA. Moreover he showed that some of the basic proof-theoretic facts about HA, such as the existence and disjunction properties, can be extended to EA. In [3] we showed that the interpretation of HA in EA is faithful. (G.E. Mine has independently also proved this theorem.) Finally, in [2], Flagg showed that a suitable form of Church's thesis is consistent with EA. (Carlson [1] has announced another proof of this result.) Flagg's argument involves an ingenious realizability notion for EA which, as it stands, is not very perspicuous. The purpose of the present paper is to give a more transparent treatment of Flagg realizability. We obtain a new version of Flagg's proof of the consistency of Church's thesis with EA. Our main new result is that, in a sense to be made precise below, Flagg realizability coincides on HA embedded in EA with Kleene's 1945 realizability (e.g. see [5, pp. 501–516]). Thus it turns out once more that methods and results proved for EA can be viewed as extensions or generalizations of well-known methods and results for HA.

Epistemic arithmetic is a conservative extension of intuitionistic arithmetic

Questions about the constructive or effective character of particular arguments arise in several areas of classical mathematics, such as in the theory of recursive functions and in numerical analysis. Some philosophers have advocated Lewis's S4 as the proper logic in which to formalize such epistemic notions. (The fundamental work on this is Hintikka [4].) Recently there have been studies of mathematical theories formalized with S4 as the underlying logic so that these epistemic notions can be expressed. (See Shapiro [7], Myhill [5], and Goodman [2]. The motivation for this work is discussed in Goodman [3].) The present paper is a contribution to the study of the simplest of these theories, namely first-order arithmetic as formalized in S4. Following Shapiro, we call this theory epistemic arithmetic (EA). More specifically, we show that EA is a conservative extension of Hey ting's arithmetic HA (ordinary first-order intuitionistic arithmetic). The question of whether EA is conservative over HA was raised but left open in Shapiro [7].The idea of our proof is as follows. We interpret EA in an infinitary propositional S4, pretty much as Tait, for example, interprets classical arithmetic in his infinitary classical propositional calculus in [8]. We then prove a cut-elimination theorem for this infinitary propositional S4. A suitable version of the cut-elimination theorem can be formalized in HA. For cut-free infinitary proofs, there is a reflection principle provable in HA. That is, we can prove in HA that if there is a cut-free proof of the interpretation of a sentence ϕ then ϕ is true. Combining these results shows that if the interpretation of ϕ is provable in EA, then ϕ is provable in HA.