Abstract
This paper does not raise objections but spells out problems that I consider at present unsolved within Priest’s view on logic. In light of the state of scientific and other theories (Sect. 8.2) and in light of the character of natural languages (Sect. 8.3), Priest’s central arguments do not seem convincing. Next, I offer some six independent obstacles for defining consistency, identifying models and describing the semantics and metatheory of \(\mathbf {LP}\) (Sect. 8.4).
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Notes
- 1.
Newton’s achievement is not diminished by the fact that, from his days to the advent of relativity theory, several severe problems remained unsolved. Next, however significant the problems solved by early relativity theory, they were outnumbered by the problems solved by Newtonian mechanics. A sane view on scientific problem solving was elaborated by Laudan (1977).
- 2.
Where a fragment of a natural language is involved, there is a striking structural circularity: the fragment comprises certain occurrences of sentences; the sentences need to have certain forms; in the occurrence, certain words need to have the meaning fixed by the theory. The importance of the theory will depend on properties of the fragment. The theory may be invoked as a stipulative definition.
- 3.
But caution is advisable. The logic \(\mathbf {UCL}\) extends \(\mathbf {CL}\) in that a \(\mathbf {CL}\)-model is a \(\mathbf {UCL}\)-model iff, in the model, every predicate of rank r has either the rth Carthesian product of the domain or the empty set as its extension (interpretation). Although no \(\mathbf {UCL}\)-model ‘fits’ the real world, \(\mathbf {UCL}\) is a sensible technical entity in some adaptive logics of inductive generalization (Batens and Haesaert 2001).
- 4.
Priest argues that logica ens may be modified (Priest 2014c), but it seems to me that the dynamics of natural languages reside at a ‘deeper’ level than the change Priest has in mind.
- 5.
The English Wikipedia page clarifies that the application of syllogistic to the vernacular requires a set of extremely artificial constraints. Comparing with Wikipedia pages in other languages is even more instructive. On the Plattdüütsch (Low Saxon) page, “mütt” (should) is used as a copula in one of the examples.
- 6.
So, yes, methods and values belong to our knowledge. A few positivist fossils aside, everyone agrees on that.
- 7.
A theory will presumably be required (i) to be a set of statements from a fully unspecified language and (ii) to be presented in a further unspecified way judged acceptable by the future competent community. Had Aristotle anything more specific to delineate the set from which twenty-first-century theories would be taken? And “theory” is an easy case as it need not involve meanings of ‘referring’ terms. Try redoing the exercise for the conceptual systems of social psychology or string theory.
- 8.
As in all language use that relates to the world, there is an interaction between conventional elements and properly empirical elements in the empirical dimension of language. This is not the place to spell that out.
- 9.
Informal logic is meant in the traditional sense here. Carnap (1947) tried to push informal logic into meaning postulates. This heavily restricts the possible meaning relations because it requires a theory rather than an underlying algebraic structure.
- 10.
The central equivalence sign in the abstraction axiom will still be Weber’s, but every implication of the language may occur within the open formula in that axiom. This will impose restrictions on the interaction between the implications. Thus, where \(\rightarrow \) is Weber’s implication and \(\Rightarrow \) any implication Priest allows in his language, the inference from \(A\rightarrow (A\Rightarrow B)\) to \(A\Rightarrow B\) should be invalid.
- 11.
- 12.
- 13.
Allowing for negation gluts in an indeterministic semantics is simple enough: remove the clause “if \(v_M(A)=1\), then \(v_M(\lnot A)=0\)” and keep “if \(v_M(A)=0\), then \(v_M(\lnot A)=1\)”. In a deterministic semantics, the valuation function \(v_M\) is determined by the model, which here is \(M=\langle D,v\rangle \) with D a set and v an assignment function. I refer to literature for details (Batens 2016).
- 14.
This may be phrased in a perfectly unambiguous way (Batens 2016). As the reader expected, \(\mathsf {A}\) is a metametalinguistic variable. If you do not like meta talk: it is a variable for variables for formulas.
- 15.
Intuitively, there is an ambiguity in a model M when, in the superscripted language, non-logical symbols that merely differ in their superscript, like \(\xi ^i\) and \(\xi ^j\), have a different meaning in M.
- 16.
The same holds for many adaptive logics outside the family. Infinitely many logics allow for certain gluts or gaps and there are several strategies to minimize abnormalities.
- 17.
Consider a premise set that has a minimally abnormal interpretation allowing for negation gluts and another one allowing for existential gaps. Any adaptive logic (from the family) that moreover allows for further abnormalities will obviously also define a minimally abnormal interpretation. However, those interpretations will be less interesting. Their consequence sets will contain many disjunctions of formulas where adaptive logics that allow for a smaller selection of abnormalities will have the disjuncts as consequences.
- 18.
One of the border cases is where \(\Gamma \) is itself the trivial set.
- 19.
To explain in which way new concepts originate is a different matter. As \({\mathbf {CL\emptyset }^{\mathsf {m}}}\) is described here it is not helpful for that purpose.
- 20.
It is a matter of occurrence indeed. Imagine “Mary loves Bill” written on the wall, and underneath it “This sentence is false”, especially when written by another hand.
- 21.
True and false here obviously stand for true-in-a-model and false-in-a-model.
- 22.
Priest’s semantics has positive and negative extensions for predicates and these require no complications for the transition to the bivalent semantics. And the same trick would work for sentential letters.
- 23.
Actually, the implications in both definitions are different. For readers not familiar with the matter, just neglect that for the present paper.
- 24.
Actually, they are equivalent—for all \(\Gamma \) and A, \(\Gamma \vDash A\) iff \(\Gamma \vDash ^c A\)—provided Priest’s semantics is consistent with respect to \(\vDash \): there is no \(\Gamma \) and A such that \(\Gamma \vDash A\) and \(\Gamma \nvDash A\).
- 25.
That is: L is false and not true. Some will prefer “\(\ulcorner L\urcorner \) is false” in which \(\ulcorner L\urcorner \) is a name for L. I shall continue to use formulas as names for themselves as this causes no confusion in the present paper.
- 26.
The contradiction is an obvious \(\mathbf {LP}\)-consequence of \(\mathrm {Rel}(L,1)\vee \mathrm {Rel}(L,0)\), (SL), and the \(\mathbf {LP}\)-theorem \(\mathrm {Rel}(L,1)\vee \lnot \mathrm {Rel}(L,1)\).
- 27.
The numbering is of course conventional. In an A-type-x.y model hold all statements true on the classical description of the A-type-x model (but now read paraconsistently).
- 28.
Yet paraconsistent set theories have some inconsistent sets, viz. sets of which certain entities are members as well as non-members.
- 29.
So I shall never write expressions like \(\mathrm {Rel}(\mathrm {Rel}(A,1),1)\). Many results proven for \(\mathcal {W}\) are provable in general, but this is a worry for later.
- 30.
The proof stems from the first edition and obviously has to be adapted to the relational semantics. However, precisely this is impossible in general. As, for every A, \(\lnot \mathrm {Rel}(A,1)\) holds in some model M, there cannot be a method to turn M into a \(\mathbf {CL}\)-model in which A is false. Next, even meta-theoretic proofs new to the second edition are in terms of the old set-theoretic semantics.
- 31.
Formulas in which \(\mathrm {Rel}\) is iterated are required. Remember indeed that we are devising a theory about the semantics of English. The \(\mathbf {LP}\)-semantics is the part that handles the traditional logical terms.
- 32.
One of them still holds if \(A\leftrightarrow B\) is the ‘material equivalence’ \((\lnot A\vee B)\wedge (A\vee \lnot B)\).
- 33.
I am considering classical models. Yet, going paraconsistent does not remove the problem.
- 34.
Each formula to the left is entailed by the one to the right in Priest’s understanding.
- 35.
The matter may be phrased in terms of truth, falsity and untruth.
- 36.
- 37.
The Liar formula itself does not entail triviality because Contraction (or Absorbtion) is not valid for the arrow. Triviality is engendered by (i) the decision that arrow-bottom formulas hold in the A-consistent models and (ii) the fact that the arrow is detachable.
- 38.
Where \(\mathord {\sim }\) is intuitionistic negation, the expressions hold for finitistic A.
- 39.
In the presence of \(\bot \), with its intended meaning that \(\bot \vDash A\) holds, \(A\supset \bot \) defines classical negation in case the implication is classical and defines intuitionistic negation in case the implication is intuitionistic. In the absence of \(\bot \), it is possible to have a detachable classical or intuitionistic implication without having the disadvantages of an explosive negation. This is illustrated by the work of da Costa (da Costa 1963, 1974; da Costa and Alves 1977), Jaśkowski (1969), and others including myself. Priest and other dialetheists (and most relevantists) consider this a minor detail.
- 40.
Obviously \(\mathord {\sim }\lnot A \vDash A\) and \(\mathord {\sim }A\vDash \lnot A\). Moreover, the matter may be phrased in terms of classical implication and bottom. I leave it to the reader to figure out this as well as the intuitionistic case.
- 41.
It seems to me that the separation between object-language and metalanguage can be avoided, at least to some extent, provided one is willing to accept that theories are formulated in restricted fragments of a language.
- 42.
The difference, viz. that \(A\vee \mathord {\sim }A\) is valid whereas \(A\vee (A\rightarrow \bot )\) is not, is immaterial for this purpose.
- 43.
And fortunately so. If it were true, then, given what it is talking about, it should be valid. So it should be true in every model, which it is not.
- 44.
The expression \(M\Vdash A\)—Priest actually writes \(\mathcal {I}\Vdash \alpha \)—is read as “A holds in M” by Priest. The same expression occurs as classical terminology in the very first table of the present section.
- 45.
The easiest way to understand the rest of the paragraph is to read it as a statement in classical terminology—so if a model both verifies and falsifies the same formula, then it is trivial.
- 46.
If the \(\mathbf {LP}\)-semantics is indeed part of the semantics of English, then those models are simply the models from the \(\mathbf {LP}\)-semantics. Yet, I shall keep italicizing “models” for perspicuity.
- 47.
A model of \(\Gamma \) is minimally inconsistent iff the set of inconsistencies that hold in it is not a proper superset of the set of inconsistencies that hold in another model of \(\Gamma \).
- 48.
In more interesting cases the premises require certain but not all contradictions to be true. The \(\mathbf {CL}\)-consequence set is then trivial, but the inconsistency-adaptive consequence set is, a few odd premise sets aside, non-trivial and moreover considerably richer than any consequence set defined by a paraconsistent Tarski logic. See the literature for details (Batens 2015).
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I am grateful to the two referees for their helpful comments.
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Batens, D. (2019). Looting Liars Masking Models. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_8
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