Abstract
Logic’s historically central mission has been to provide formally precise descriptions of logical consequence. This was done with two broad expectations in mind. One was that a pre-theoretically recognizable concept of consequence would be present in the ensuing formalization. The other was that the formalization would be mathematically mature. The first expectation calls for conceptual adequacy. The other calls for technical virtuosity. The record of the past century and a third discloses a tension between the two. Accordingly, logicians have sought a reasoned, if delicate, rapprochement, one in which each expectation would be given its due, but well-short of free sway. Recent developments have imperiled this perestroika. One is logic’s massive and often rivalrous pluralism, and the cheapening relativism to which it beckons. This is exacerbated by the long-acknowledged part that the formal representations of logic distort the logical particles of natural language. The present paper discusses what might be done about this.
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Notes
Coinage of “entails” for the converse of “follows from” is G.E. Moore’s. See Moore (1922).
There is an analogous problem about how scientific models represent their target systems by way of idealizations that fail on the ground. For more of this, see Woods and Rosales (2010).
While everyone agrees that entailment is a principal target and that entailment and logical truth are interdefinable, there is some disagreement about which is the primary notion. Quine (1970/1986) opts for logical truth and Dummett (1973) for entailment. For present purposes it is unnecessary to settle this question one way or the other.
This is not to say that the sentential operators aren’t interesting. Jennings (1994) is a very good book on disjunction and Horn (1989) is the same for negation, as are Bennett (2003); Nute and Cross (2002) and Lycan (2001) for conditionality. The point is that the job of a theory of deduction is to get deduction right. It is not the job of a theory of deduction to pronounce upon negation, disjunction and conditionality, save in ways that may facilitate its progress with deduction (which in its semantic sense is what entailment is).
Russell (1905). There is also the large class of cases in which the “not” in ⌜Not-Φ ⌝ serves as the denial or commitment-retraction of Φ, short of affirming the truth of its negation.
In CFL, Φ entails ψ just in case there is no valuation making Φ true on any interpretation that doesn’t also make ψ true on that interpretation. More loosely, there is no possible way of making Φ true without making ψ true. This, of course, is the strict condition.
Of course, CFL doesn’t need “⊃”. It can get by perfectly well with “~” and “∧”, concerning which all agree that the fit with their respective English counterparts is a good deal more intuitive. Yes, this is so. But it doesn’t matter. It doesn’t matter because there is no sense of “if … then” in English for which ⌜~(Φ ∧~ψ) ⌝ says ⌜If Φ then ψ ⌝. So there is no English sentence ⌜If Φ then ψ ⌝ not misrepresented by ⌜~(Φ ∧~ψ) ⌝. Yet it remains true that CFL gets entailment right.
One of the more instructive early disputes about the tenability of “⊃” pits Hugh MacColl against Russell. See, for example, MacColl (1908a, 1908b) and Russell (1905). There also exists a significant correspondence between MacColl and Russell, cited in Rahman and Redmond (2008). Here is some wise instruction from MacColl: “Thus, Mr. Russell, arguing correctly from the customary convention of logicians, arrives at the strange conclusion that (among Englishmen) we may conclude from a man’s red hair that he is doctor, or from his being a doctor that (whatever appearances may say to the contrary) his hair is red” (MacColl, 1908a, p. 152). For further dissatisfactions with how CFL represents the logical particles of English, see Adams (1988), Stove (1986), McGee (1985) and Jacquette (1999). While there certainly are difficulties arising from CFL’s treatment of the particles, these particular criticisms are I believe largely misplaced. See here Woods (2004), chapter 3.
The year in which Frege’s Begriffsschrift was published.
See here Griffin (1991), pp. 272–273.
Of course, there are extreme cases, such as those purported to arise from the paradoxes. If, like Frege and Russell, you really believe that the Russell Paradox extinguished the very idea of sets; and if, like Tarski, you really believe that the Liar Paradox detonated the very idea of truth, then for post-paradox set theory and post-paradox semantics the requirement of even moderate conceptual adequacy lapses. A cornerstone of dialethic logic (the logic of true contradictions) is the judgement that this pessimism is excessive. Finding it so, doesn’t however require one to be a dialethist. See here Woods (2003), chapter 5 et passim, and (2005).
Of course, there is in principle what we might call the option of mindless nihilism, according to which since there are no facts of the matter about what entailment actually is, all logics of entailment are false (for there is nothing that could make them true). There are lots of nihilist logicians, but none of the mindless breed.
I don’t want to leave the impression that Riemann was simply playing about until one day, hey presto! the general theory of relativity popped into view. Riemann’s geometry was a formidable piece of mathematics. It generalized Gauss’s work on surfaces to n dimensions. When n = 3, Riemann’s geometry is the result. Riemann didn’t think the 3D case described physical space or—since scaling up is a dubious assumption in physics—that it ever would prove to do so. Still, he did think it a physically possible hypothesis, hence a description of how the world might conceivably be.
A sociological observation: The Journal of Symbolic Logic used to make room in its reviews section for philosophical contributions to the subject. In time, this mandate was transferred to its sister journal, the Bulletin of Symbolic Logic, which was soon dominated by mathematical work. A further adjustment, intended to accommodate philosophical interests, was the Review of Symbolic Logic, which too, since its recent inception has been dominated by mathematical pieces. Meanwhile, even the Journal of Philosophical Logic which was the Association’s predecessor organ of the Review was rife with a mathematical emphasis. In a way, this is as it should be. It reflects the fact that in the past generation or so, logic largely ceased being a humanities discipline and has passed its remit to mathematics and computer science.
Consider here Gödel’s platonistic approach to sets. There is, says Gödel, a way in which sets actually are, hence a way of conceiving them as they actually are. However, owing among other things to the limiting contingencies of human thinking, there are different ways of conceiving of sets, none of which captures sets as they actually are.
Routley and Routley (1972). “First order” denotes a system in which entailment statements don’t themselves embed entailment statements.
Thus calling to mind Quine’s words about a different though related thing. Speaking of the dialethic approach to entailment, Quine writes: “To turn to a popular extravaganza, what if someone were to reject the law of non-contradiction and so accept an occasional sentence and its negation both as true? … My view … is that neither party knows what he is talking about. They think that they are talking about negation, ‘~’, ‘not’; but surely the notation ceased to be recognizable as negation when they took to regarding some conjunctions of the form ‘p &~p’ as true, and stopped regarding such sentences as implying all others” (Quine, 1982, p. 81).
Somewhat more technically, star worlds are susceptible to negation-inconsistency, but not absolute inconsistency. The inconsistency of a star world doesn’t make everything true there.
It might be objected that, owing to their susceptibility to inconsistency, we shouldn’t think of star worlds as possible worlds. We should instead use a more neutral name such as “set-up”. I don’t mind. Nothing I have to say against star negation is predicated on the idea that, except as a façon de parler, some possible worlds are actually inconsistent. Let us note that in the mainstream semantics for intensional logics, there is not one word of instruction about what possible worlds are, beyond their purely formal characterization as elements of a non-empty set governed by a binary relation having variously invoked abstract properties − properties such as reflexivity, symmetry, transitivity, extendability and—for weird systems such as S2 – quasi-reflexivity. On any such approach to modal logic, the invocation of worlds might tell us the meaning of “possible”, but certainly not that of “possible world”.
Which I borrow from Dagfinn Follesdal’s 1962 PhD thesis, reissued as Follesdal (2008).
Hughes and Cresswell (1968/1996) has a chart listing this multiplicity.
“Negational” as opposed to “emphatic”. The person who says, “I will never, never, never, never set foot in Dry Gulch again” is not announcing an intention to return. The innermost occurrence of “never” is negational. The other three are for emphasis.
See Burgess (2009), p. 65.
Burgess identifies truths by logical form alone as alethic modalities, and provabilities by logical form alone as apodictic modalities, concerning which he displays a caution appropriate to our present reflections: “… nothing said so far constitutes even an informal ‘proof’ that no formula not a theorem of S4 is correct for apodictic modality, or that no formula not a theorem of S5 is correct for alethic modality. And indeed there is no generally accepted informal argument for the first claim, though a convincing one can be given for the second claim” (Burgess 2009, p. 65; emphasis added to contrast argument with proof).
For state of the art information about the concept-project, see Margolis and Laurence (1999). We might note in passing that the chief backer of the “no identity, no entity”thesis is Quine, who has nary a word to say about what he takes a thing’s identity conditions actually to be. That is, for the concept of identity condition Quine himself furnishes no identity conditions. Even so, it is clear from Quine’s writings that the main targets of his criticisms are intensional entities—meanings, propositions, propositional attitudes, properties, concepts and modalities. On a plausible reading, we might take a thing’s identity conditions, the conditions that individuate it, to be its theory. A theory of something provides a principled basis for sorting what’s true of it from what’s false of it, which surely is one good way of saying what it is to be a thing of that kind. Quine also requires theories to be formulable in suitably interpreted first order languages. A first order language is one that excludes expressions for intensional entities. So an intensional entity lacks identity conditions just because it cannot be described in a language that excludes intensional entities!
Langford (1942).
According to Gresham’s Law, bad money drives out good.
Related is the all but universal proclivity of the mature natural sciences to subject their principal targets to conceptual change. Thus one gets to know something of what it is to be a C by, among other things, changing the meaning of “C”. See here Thagard (1992).
An earlier and considerably shorter version of this paper was delivered to the Ninth Conference of the Ontario Society for the Study of Argumentation in May, 2011. The generous suggestions and telling criticisms of my commentator, Nicholas Griffin, have been of material help in readying the paper for publication. I thank him most warmly for his assistance. Thanks are also owed and happily given to Harvey Siegel, and Geoffrey Goddu. For helpful instruction before and after the OSSA meetings, I am also grateful to Ivor Gratten-Guiness, Steven Savitt, Jean-Yves Béziau, Alirio Rosales, and the Editor of Topoi, Fabio Paglieri.
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Woods, J. Semantic Penumbra: Concept Similarity in Logic. Topoi 31, 121–134 (2012). https://doi.org/10.1007/s11245-012-9118-y
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DOI: https://doi.org/10.1007/s11245-012-9118-y