Abstract
After having argued that Prawitz’s notion of ground for \(A\) is not epistemically transparent when \(A\) is an empirical statement (Sect. 18.2), a non-factive and defeasible notion of \(C\)-justification (“\(C\)” for “computational”) for empirical statements is defined and proposed as the key notion of the theory of meaning. \(C\)-justifications for \(A\) are conceived as cognitive states (Sect. 18.3), and are defined by recursion on the logical complexity of \(A\). In the atomic case (Sect. 18.4) they are defined in terms of two other concepts: the one of \(C\)-authorization to use a name to refer to a given entity, and the one of \(C\)-authorization to use a predicate in order to apply an accessible concept to objects. In the logically complex cases (Sect. 18.5) the meaning of the logical constants as applied to empirical statements is discussed, and the necessity is shown of Nelson’s strong negation besides intuitionistic negation. In the conclusion (Sect. 18.6) it is argued that the notion defined is epistemically transparent and not exposed to traditional objections.
I am most grateful to Beatrice Bernazzi, Luca Tranchini and Giuseppe Varnier for helpful comments to earlier versions of this paper. This work was partially supported by the MIUR fund n. 20107738C5_002.
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Notes
- 1.
- 2.
- 3.
More precisely, by each member of a class of arguments having a common normal form.
- 4.
I use here “is in a position to know” in the sense defined by Williamson:
To be in a position to know p, it is neither necessary to know p nor sufficient to be physically and psychologically capable of knowing p. No obstacle must block one’s path to knowing p. If one is in a position to know p, and one has done what one is in a position to do to decide whether p is true, then one does know p (Williamson 2000, p. 95).
- 5.
I have replaced “ground” with “evidence”, since grounds—as explained below in the text—are precisely the formal counterpart of what is intuitively called “evidence”.
- 6.
- 7.
Cp. Prawitz (1977, [Sect. III.3]).
- 8.
See for example Prawitz (2011, [p. 18]).
- 9.
Notice that in Heyting (1974) the notion of general method of construction is explicitly mentioned as one of the primitive notions of intuitionistic mathematics.
- 10.
This is an antirealist thesis, as is clear from the following argument.
- 11.
Another price is put into evidence by Cozzo (2015, Sect. 5): as a matter of fact, we often infer conclusions from wrong premises, for which we have no grounds; since, according to Prawitz’s definition of inference, an inference act must involve (conclusive) grounds for the premises, our acts are not inferences.
- 12.
For a more detailed discussion of the following argument see Usberti (2004).
- 13.
The role of the notion of best explanation will be clarified below.
- 14.
According to Williamson,
even grasping a proof of a mathematical proposition is a defeasible way of having warrant to assert it. One can have warrant to assert a mathematical proposition by grasping a proof of it, and then cease to have warrant to assert it merely in virtue of gaining new evidence about expert mathematicians, utterances, without forgetting anything.
(Williamson 2000, p. 265)
This is plausible if “proof” is understood as meaning a written proof or something similar; if a proof is conceived as a whole cognitive state, and having that proof is equated to being in that state, Williamson’s view becomes much less plausible. If the subject \(s\) of Williamson’s example has been in a mental state \(\sigma _1\) that is a proof of \(A\) and later is in a mental state \(\sigma _2\) that is not a proof of \(A\), then either \(s\) has forgotten \(\sigma _1\) when he is in \(\sigma _2\), or he has chosen to trust the mathematicians instead of his own memory; the first case can be neglected: subjects are necessarily idealized, and idealized subjects have no limitation of memory, attention, etc.; in the second case s has made a mistake, hence he is not justified in believing that he has not warrant to assert \(A\).
- 15.
The problem is evidently a vast one, because the variety of atomic statements of a natural language is immense. I will be concerned only with a restricted number of cases, but I will select them in such a way that they are representative of a fairly large class of cases. In particular I will keep present, besides mathemathical ones, observational statements and several other empirical statements in the present tense and in the third singular person.
- 16.
This idea could be motivated with considerations analogous to Gareth Evans’ ones leading to his Generality Constraint:
For we cannot avoid thinking of a thought about an individual object x, to the effect that it is F, as the exercise of two separable capacities; one being the capacity to think of x, which could be equally exercised in thoughts about x to the effect that it is G or H; and the other being a conception of what it is to be F, which could be equally exercised in thoughts about other individuals, to the effect that they are F.
(Evans 1982, p. 75)
Analogously, we cannot avoid thinking of a justification for \(P(n)\) as the result of two separable components; one being a \(C\)-authorization to use \(n\) to refer to a given entity, which could equally be a component of a justification for \(G(n)\) or \(H(n)\); and the other being a \(C\)-authorization to use \(P\) to apply an accessible concept to objects, which could equally be a component of a justification for \(P(m)\) or \(P(k)\).
- 17.
The choice of \(k\) varies according to the nature of the statement for which the cognitive state is a justification. IRS will be defined in Sect. 4.2.1.
- 18.
As a matter of fact, in this paper I will be concerned only with the second sense. I will take into consideration the first in a sequel to the present paper, which will be devoted to the notion of empirical warrant or truth-ground.
- 19.
This technical sense of “representation” is common in cognitive psychology; for instance, Chomsky (2000, p. 173) introduces it by saying that “there is nothing ‘represented’ in the sense of representative theories of ideas, for example.”
- 20.
Following common use in cognitive psychology, I use here “description” as a synonymous of “term” and “representation”.
- 21.
An analogous distinction is introduced and motivated in Bierwisch (1992, pp. 30–32).
- 22.
The need of associated information has also other reasons, which will be explained below.
- 23.
The classes belonging to this partition should not be confused with the ‘pre-linguistic’ ones corresponding to the criteria of identification associated to activated terms: two terms of IRS may be in the same class independently of their matching the epistemic content associated to any name; conversely, two terms of IRS may both match the epistemic content associated to the name \(n\) independently of their being in the same ‘pre-linguistic’ class.
- 24.
See also Dummet (1973, p. 402): “the notion of identifying a concept \([\ldots ]\) seems quite inappropriate”.
- 25.
It should be admitted that, while we have some hints about what recognizing a man or a dog amounts to in computational terms, more difficult is to give a computational analysis of actions and of the assignment of roles. On the other hand, it is a methodological assumption of computational psychology that it is possible to do it, and I see no a priori argument to the contrary.
- 26.
As it would be plausible to assume.
- 27.
I mean the horse and the man-pursuing-dog examples.
- 28.
As it might be thought if only cases similar to our first example were taken into consideration: in that case it might be suggested that the representation of a round disk is relevant because it is in some sense similar to the activated term, which is relevant because it is activated at the time of the cognitive state.
- 29.
Van Fraassen remarks that the requirement that \(K\) does not imply the denial of any presupposition of \(Q\) is very different from the requirement that all the presuppositions of \(Q\) are true: “K may not tell us which of the possible answers is true, but this lacuna in K clearly does not eliminate the question.” (van Fraassen 1980, p. 146)
- 30.
Nor is it constitutive of the meaning of “rain” or of “puddle”: it is a fact concerning the structure of our C-IS that there is a relation of relevance between the concepts denoted by “puddle” and “rain”, without this relation being constitutive of the two concepts. Of course, the existence of this relation can be seen as the result of an adaptation of our C-IS to the external environment; but this hypothesis plays no explanatory role in the theory of the structure of our C-IS.
- 31.
Hereafter I will write “\(\sigma (A)=1\)” to mean that \(\sigma \) is a justification for \(A\). The property of being a pair of \(cs^{\prime }\) \(\sigma _1\) and \(\sigma _2\) such that \(\sigma _1(A)=1\) and \(\sigma _2(B)=1\) is, in this case, the feature to be checked.
- 32.
“Yields” is to be understood as equivalent to “is known to yield”.
- 33.
For instance, a proof of “\(PRIME(n)\vee \lnot PRIME(n)\)” is a primality test for \(n\), i.e. an algorithm for determining whether \(n\) is prime. Such a test should not be confused with a general method consisting in applying to every number \(x\) a test for \(x\) (this general method is a proof of “\(\forall x(PRIME(x)\vee \lnot PRIME(x))\)”).
- 34.
“Belongs” is to be understood as equivalent to “is known to belong”.
- 35.
\(D\) is the cognitive domain (non-empty set of cognitive objects) over which \(x\) ranges.
- 36.
See footnote 35.
- 37.
Not in all. In the case of many empirical sentences, their negations cannot be construed but as intuitionistic. Consider the following example (due to Paolo Casalegno):
(13) Not all prehistoric men were black-eyed;
probably we will never be able to say, of a specific prehistoric man, that he was black-eyed; at the same time, it is quite plausible to say that we have justifications to believe that (13) is true; and if we reflect on the nature of these justifications, we realize that each of them can be verbalized as a reductio ad absurdum of the assumption that all prehistoric men were black-eyed, as the intuitionistic explanation of negation requires.
- 38.
Cp. Nelson (1949).
- 39.
It may be a question of discussion which negation is involved in a given natural language statement.
- 40.
“cs” abbreviates “cognitive state”.
- 41.
Prawitz (2009, p. 187).
- 42.
“Internalist” is therefore meant here in the purely methodological sense in which it is used, for instance, by Chomsky in (2000).
- 43.
This difficulty concerns Prawitz’s notion of open ground as well, and seems to be another reason against its epistemic transparency.
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Usberti, G. (2015). A Notion of C-Justification for Empirical Statements. In: Wansing, H. (eds) Dag Prawitz on Proofs and Meaning. Outstanding Contributions to Logic, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-11041-7_18
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