# On Preserving: Essays on Preservationism and Paraconsistent Logic

Peter Schotch
Bryson Brown
Raymond Jennings
https://www.jstor.org/stable/10.3138/9781442688742

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Acknowledgments
(pp. ix-2)
4. One Introduction to the Essays
(pp. 3-16)
Peter Schotch

Instead it started in the summer of 1975, at Dalhousie University, when Ray Jennings¹ and Peter Schotch decided to write a ‘primer’ of modal logic - a project that never quite made it into the dark of print. So it was modal and decidedly not paraconsistent logic which was uppermost in the minds of Jennings and Schotch. In this connection one should keep in mind that neither had so much as heard the word ‘paraconsistent’ at the time we are considering.²

What got the ball rolling was the discovery by Jennings, which he communicated to Schotch, that so-called normal modal...

5. Two Paraconsistency: Who Needs It?
(pp. 17-32)
Ray Jennings and Peter Schotch

The classical account of consistency is inadequate in many situations in which we certainly require a notionlikeit. This is tantamount to saying that there are situations in which classical logic is not adequate. While such an assertion might have enjoyed a certain amount of shock value in the early twentieth century, things have changed since those heady days. Now most people agree that some changes are often necessary in the classical account of inference, though there is nothing approaching widespread agreement on the precise nature of those changes.

One can distinguish two broad strategies for fixing a misbehaving...

6. Three Weakly Additive Algebras and a Completeness Problem
(pp. 33-48)
Alasdair Urquhart

In a series of papers,¹ Ray Jennings and Peter Schotch have developed a generalized relational frame theory that goes beyond the standard approach in modal logic Schotch and Jennings (1980a,b); Jennings and Schotch (1981, 1984). The key idea is to generalize the usual truth condition for the necessity operator by using multi-place relations rather than the usual binary relation. Thus, ifRis an$(n + \;1)$-place relation defined on a setW, then Jennings and Schotch state the truth condition for the generalized necessity operator as follows. If$x\; \in \;W$, then

$x \vDash \;\square \,\alpha \; \Leftrightarrow \;\forall \,{y_1},\; \ldots ,\;{y_n}\,[Rx{y_1},\, \ldots ,\;{y_n} \Rightarrow \;\exists i\,({y_i} \vDash \;\alpha )]$.

This definition validates the scheme${K_n}$ofn-ary aggregation,

$\square \,{\alpha _0} \wedge \cdots \wedge \;\square \,{\alpha _n} \supset \;\square [\mathop V\limits_{0 \leqslant i < j \leqslant n} \;{\alpha _i} \wedge \;{\alpha _j}]$....

7. Four A Dualization of Neighbourhood Structures
(pp. 49-60)
Dorian Nicholson

The search for a completeness proof for Jennings and Schotch’s weakly aggregative modal logic lasted for nearly twenty years (Johnston, 1978; Schotch and Jennings, 1980a; Jennings and Schotch, 1981; Schotch and Jennings, 1980b) before the goal was attained in 1995 by Apostoli and Brown (1995), and also independently, algebraically, by Urquhart (1995). Apostoli and Brown’s proof was subsequently simplified by Nicholson, Jennings, and Sarenac (2000). But both proofs exploit the compactness of colouring for hypergraphs whose edges are finitely long.Chromatic compactness, also calledcolouring compactness, is the claim that$\forall H,$, ifHis a hypergraph thenHisk-colourable...

(pp. 61-84)
Kam Sing Leung and R.E. Jennings

The originating pulse of virtually all of the work presented in this volume lay in the inexpressibility, within the received Kripkean modal systems, of one or two deontically fundamental distinctions. The first was the distinction between deontic conflict, the common-enough moral pickle in which, through incompetence, inattention, or circumstance,we find ourselves forced to neglect one moral duty to fulfil another. The topic has confronted moral systematizers for at least two hundred years. John Stuart Mill evidently supposed that if obligations are objectively grounded, any apparent such conflict can be no more than the starting point of a moral calculation that...

9. Six Preserving What?
(pp. 85-104)
Gillman Payette and Peter Schotch

The (classical) semantic paradigm for correct inference is often given the name ‘truth-preservation.’ This is typically spelled out to the awestruck students in some such way as this:

An inference from a set of premises, Γ, to a conclusion, α, is correct, sayvalid, if and only if whenever all the members of Γ are true, then so is α.

This understanding of the slogan may be tried, but is it actually true? There is a problem: the way that ‘truth’ is used in connection with the premises is distinct from the way that it is used with the conclusion....

10. Seven Preserving Logical Structure
(pp. 105-144)
Gillman Payette

Faced with the possibility of logical pluralism, if not the reality of it, one should follow a methodology of research which is sensitive to the plurality of logics. What has not been dealt with so sensitively is the question of how best to deal with inconsistent sets if we don’t want to trivialize inference in every one of these cases.

In classical and intuitionistic logic the sets$\{ P \wedge \neg P\}$and$\{ P,\;\neg P\}$have the same deductive closure: Everything! In Latin the rule of inference is phrased asex falso quodlibet, which may be translated as ‘from the false whatever.’ This is the...

11. Eight Representation of Forcing
(pp. 145-160)
Dorian Nicholson and Bryson Brown

Once it is realized that X-level forcing¹ inherits only some of the principles of the underlying logic X, and in particular that an abridgement of the structural rule [Mon] of monotonicity is required,² the question of alternative presentations of the relation naturally arises. The goal is to give something like an axiomatization of the relation, and once that has been accomplished, we can, those of us who were worried, heave a sigh of relief. The project is not so urgent as the corresponding problem in so-called n-ary modal logic, but that shouldn’t be regarded as an excuse for not undertaking...

12. Nine Forcing and Practical Inference
(pp. 161-174)
Peter Schotch

What we might call thepureor perhapsgeneraltheory of the forcing relation is all very well in a theoretical setting. That shouldn’t come as much of a surprise since our treatment of forcing tends to be abstract and general. On the other hand, sometimes we want to study inference in something more closely resembling ordinary life, more ordinary than mathematical life at least. In these cases the general theory is too spare. In fact we shall distinguish two ways in which one might wish to come down from the Olympian heights of generality to no greater altitude than...

13. Ten Ambiguity Games and Preserving Ambiguity Measures
(pp. 175-188)
Bryson Brown

Brown (1999) began a line of work that has applied preservationist ideas to generate consequence relations first exploited by relevance and di-aletheic logicians. The starting point of this approach was the realization that, by treating certain sets of atomic sentences as ambiguous, we can produce consistentimagesof inconsistent premise sets. In this chapter we present several different approaches to using ambiguity and the preservation of ambiguity measures to arrive at some familiar consequence relations. The first of these approaches uses ambiguity to project consistent images of inconsistent premise sets. The second shifts the focus from the syntactic to the...

14. Nomenclature
(pp. 189-194)
15. References
(pp. 195-198)
16. Index
(pp. 199-202)
17. Contributors
(pp. 203-203)