Abstract
This chapter will look at some other semantic theories and the logics they generate. Mainly, these logics come about by allowing truth value gaps and truth value gluts. If a semantic theory allows for statements that are neither true nor false, then it allows for gaps. If, on the other hand, it makes room for statements that are both true and false, then it allows for truth value gluts.
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Notes
- 1.
Sometimes I will talk about “gappy” and “glutty” theories. The first kind are also known as “partial” theories.
- 2.
One common misconception is that paraconsistent logic is one particular logical system. In fact, there are a lot of them, cf. Priest (2003).
- 3.
That is, relevant logics form a proper subset of the set of paraconsistent logics. This shows that the not uncommon perception that paraconsistency is a more radical doctrine than relevance is completely unfounded. It rests, again, on the common confusion between paraconsistency and dialetheism.
- 4.
A scan of the chapter is available for download on Belnap’s homepage, http://www.pitt.edu/belnap/papers.html.
- 5.
There is no table for the conditional, simply because FDE does not have a conditional. This is actually the feature that gives First Degree Entailment its name: An entailment of the first degree is one in which the turnstile (\(\vdash )\) is the only entailment or conditional-like item. We will later see an example for how a conditional can be added to FDE.
- 6.
It is not essential to know what a lattice is to understand the following. Those who know what this means might benefit from knowing that the lattice is one of the de Morgan lattice variety. It also is the paradigmatic example of a bilattice, which in essence means that you can find two distinct lattice structures on the elements (although there are some more requirements, cf. Fitting (2002)). Additionally to the ordering that goes from bottom to top, you find the ordering that goes from left to right, that is, \(\mathcal {N}\) is the lowest element and \(\mathcal {B}\) the highest. The first is called the truth ordering (ascending the ordering means either gaining in truth or waning in falsity), the second the information order. Moving from left to right means to increase the amount of information available. This will become clearer in the discussion of the intuitive interpretation of the values below.
- 7.
Inferences such as \(A\wedge \lnot A\vdash A\wedge B\) or \(C\vdash C\wedge (A\vee \lnot A)\) show that parameter sharing is necessary, but not sufficient for a relevant consequence relation. Note, however, that these inferences are invalid in FDE as well.
- 8.
That means that FDE will not suit those relevantists who would like to keep some tautologies, but only reject that these tautologies follow from an arbitrary premise.
- 9.
For this reason, I try to avoid calling them “truth values.”
- 10.
I will leave off the “told-” prefixes for the rest of this section to increase readability.
- 11.
The first author of that paper and the author of the present study are the same person, different last names notwithstanding.
- 12.
“Now for an account which is close to the informal considerations underlying our understanding of the four values as keeping track of markings with told True and told False: say that the inference from A to B is valid, or that A entails B, if the inference never leads us from told True to the absence of told True (preserves Truth), and also never leads us from the absence of told False to told False (preserves non-Falsity). Given our system of markings, this is hardly to ask too much.” Anderson et al. (1992), p. 519.
- 13.
Bad for the proposition in question or for you, if you are reluctant to give up your belief in it. Belnap writes: “We note that in the logical lattice, each of the values None and Both is intermediate between \(\mathcal {F}\) and \(\mathcal {T}\), and this is as it should be, for the worst thing to be told is that something you cling to is false, simpliciter. You are better off (it is one of your hopes) either being told nothing about it or being told both that it is true and also that it is false; while of course best of all is to be told that it is true with no muddying the waters.” Anderson et al. (1992), p. 516.
- 14.
The logic has independently been described in Marcos (2011).
- 15.
This shows that the rule of proof “If \(A\vDash C\) and \(D\vDash C\), then \(A\vee D\vDash C\)” fails, which seems to stand in the way of a natural sequent calculus for this logic; the paper gives a Hilbert-style proof system instead.
- 16.
Here, there is a relevant difference between the two ways of giving the semantics. For we are dealing with a logic that, on the first interpretation, is a three-valued logic and therefore is not bivalent. However, one might argue that bivalence holds on the second interpretation, as there are only two truth values and every statement is either true or false. The only difference to classical logic is that the “either true or false” is an inclusive disjunction. Of course, one could hold that part of the idea of bivalence is that there shall be no gluts, that is, that the disjunction is an exclusive one. It is hard to guess what Dummett would have said, at least it does not clearly emerge from his extended discussion of terminology in the preface of TOE (p. xix).
On the other hand, \({\text {K}}_{3}\) is not bivalent, no matter how we choose to give the semantics.
- 17.
Including \((A\wedge \lnot A)\rightarrow B\), if the arrow is interpreted as the material conditional. This makes it quite obvious that modus ponens is not a valid rule for the material conditional in LP, and usually, LP is thought of as having no conditional (not even a defined one) at all.
- 18.
Cf. for example Blamey (1986).
- 19.
Strawson (1950).
- 20.
TOE, p. 23.
- 21.
TOE, p. 12.
- 22.
TOE, p. xviii.
- 23.
TOE, p. 14.
- 24.
For example in LBM, pp. 47–49.
- 25.
While I do not know of an a priori argument why assertoric content and ingredient sense might not turn out to be completely distinct, I would guess that in most theories, we will find the assertoric content somehow subsumed under the ingredient sense. The ingredient sense will have to answer; for example, how conjunctions are decided to be assertible, and it seems hard to answer that if we do not know the conditions under which the conjuncts were assertible on their own.
- 26.
However, we can see a different example in the Kripke semantics in which the assertoric content and the ingredient sense come apart: Assume that a statement receives value \(0\) at a world. Then, it is already settled that it is, at that world, not assertible. However, more is needed by way of information to decide whether the negation of that statement is assertible, namely the future development of our investigation.
- 27.
See, for example, Berto (2007) for more.
- 28.
The most important exposition and defense is Priest (2006a).
- 29.
Unfortunately, received terminology works a bit against clarity in this case: The law of non-contradiction is a semantic principle, like bivalence, and not a logical principle like the Law of Excluded Middle. As said above, I write semantic principles in lowercase letters in the hope to avert confusion.
- 30.
Cf. Chap. 16 of Priest (2006a).
- 31.
This is impressionistic. Counting is, as so often, difficult, as there is an infinite number of different paraconsistent systems.
- 32.
He argues against truth value gaps and intuitionism alike in Chap. 4 of Priest (2006a), though in the auto-commentary to that chapter that is supplied in the second edition of the book, he takes a slightly more lenient approach.
- 33.
If even the prospect of being committed to trivialism, the view that everything is true, cannot scare you off, then even Explosion cannot compel you to keep your reasoning contradiction free. Priest has tried to argue against an imaginary trivialist in Priest (2006b), and this turns out not to be an easy task at all.
- 34.
Beall and Ripley (2004).
- 35.
Beall and Ripley (2004), p. 30.
- 36.
Beall and Ripley (2004), p. 34.
- 37.
There is, quite obviously, also a view on which there are gluts but no gaps and according to which only non-falsities should be asserted. It should not come as a surprise that this view (which I do not think has a dedicated name) would give rise to \({\text {K}}_{3}\)’s consequence relation.
- 38.
If the underlying lattice is either of the three-valued ones, the consequence relation defined in terms of truth-and-non-falsity preservation will coincide with \({\text {K}}_{3}\).
- 39.
Indeed, Wansing (Wansing 2012) offers a notion of “non-inferentialist, anti-realistic truth” based on told-truth values.
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Kapsner, A. (2014). Gaps, Gluts and Paraconsistency. In: Logics and Falsifications. Trends in Logic, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-319-05206-9_4
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