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## Mathematics, the empirical facts, and logical necessity

### Erkenntnis (1983-05-01) 19: 167-192 , May 01, 1983

### Conclusion

We have argued that mathematical statements are *a posteriori synthetic* statements because they are ultimately based on *empirical* facts or on *empirical* hypotheses. In some cases, mathematical statements can be verified by direct inspection of an empirical model of the relevant mathematical structure. In other cases, they must be inferred from the axioms of the relevant axiomatic theories, on the assumption that these axioms are *consistent*. Yet, the consistency of these axioms is always an *empirical* assumption: it may admit of empirical *verification* (by exhibiting an empirical model for these axioms), or at least it will admit of potential empirical *falsification* (by deriving contradictory implications from them).

On the other hand, mathematical statements are a posteriori synthetic statements of a very special sort, because they do not depend on the *contingent* features of the empirical world, but rather are logically *necessary* properties of the mathematical structures realized or potentially realized in the empirical world (as shown by the invariance of these properties under isomorphism). Moreover, even though they are synthetic statements, they resemble analytic statements in being logical consequences of the *axiomatic definition* of the mathematical structure they are dealing with. For this reason, we have proposed the term *structure-analytic* statements to describe them.

Their logical status as structure-analytic statements gives them a logical position *intermediate* between truly analytic statements and ordinary empirical statements. This explains the nontrivial and nontautological character of many important mathematical theorems, which often gives them the quality of *a priori* quite unexpected “brute facts.” This is an aspect of mathematics very hard to explain on the logical positivist assumption that mathematical statements are *truly* analytic.

We have also discussed some of the philosophic and mathematical problems posed by various limitational theorems. We have argued that for Peano arithmetic the danger of inconsistency can be minimized (though it cannot be fully eliminated), and the problem of noncategoricity can be fully overcome, by stating it in the form of a quantifier-free recursive theory.

On the other hand, in the case of set theory, we have argued that the Skolem paradox shows that we are logically free to *reject* the existence of absolutely nondenumerable sets, yet that, both on intuitive and on pragmatic grounds, it is preferable to *admit* their existence, as most set theorists do.

Finally, we have found that we are logically free to opt *either* for dualism (or for pluralism) *or* for monism in set theory. If we opted for the former position, this would mean only that the theory of finite sets can be extended to infinite sets in two (or more) different but equally admissible ways-just as other mathematical theories can often be generalized in more ways than one. (Moreover, if infinite sets have the nature of ideal elements, as Hilbert has suggested, it cannot really surprise us if we find ourselves to be logically free to invent two or more different but equally consistent stories about them.) Indeed, we have argued, again both on intuitive and on pragmatic grounds, that it seems preferable to admit a need for *both* Cantorian and non-Cantorian set theories.

## Carnap’s Relevance Measure as a Probabilistic Measure of Coherence

### Erkenntnis (2017-04-01) 82: 339-350 , April 01, 2017

Tomoji Shogenji is generally assumed to be the first author to have presented a probabilistic measure of coherence. Interestingly, Rudolf Carnap in his *Logical Foundations of Probability* discussed a function that is based on the very same idea, namely his well-known relevance measure. This function is largely neglected in the coherence literature because it has been proposed as a measure of evidential support and still is widely conceived as such. The aim of this paper is therefore to investigate Carnap’s measure regarding its plausibility as a candidate for a probabilistic measure of coherence by comparing it to Shogenji’s. It turns out that both measures (i) satisfy and violate the same adequacy constraints, (ii) despite not being ordinally equivalent exhibit a strong correlation with each other in a Monte Carlo simulation and (iii) perform similarly in a series of test cases for probabilistic coherence measures.

## Logical foundations of psychoanalytic theory

### Erkenntnis (1983-05-01) 19: 109-152 , May 01, 1983

### Conclusion

Freud told us that the theory of repression is the cornerstone of the entire psychoanalytic theory of unconscious motivations. And he claimed that his clinical evidence furnishes compelling support for this cornerstone. Thus, I was able to scrutinize the logical foundations of the psychoanalytic edifice by examining Freud's clinical arguments for the repression-aetiology of the psychoneuroses, and for the cardinal causal role of repressed ideation in committing parapraxes (”slips”) and in dreaming. The upshot of this scrutiny was that the reasoning by which he thought to justify the very foundation of his theory was grievously flawed.

Plainly, this conclusion leaves quite open whether some other, genuinely probative evidence will turn out to lend significant support at least to the repression-aetiology of psychoneuroses, which is *the* major pillar of the Freudian structure.

Though I have given a critique of the basic pillars of psychoanalysis, it might be asked: Why its anachronistic focus on Freud's reasoning to the exclusion of the modifications and elaborations by those post-Freudians whose doctrines are recognizably psychoanalytic in content rather than only in name? Latter day psychoanalytic theoreticians that come to mind are the very influential Heinz Kohut, who pioneered the socalled “selfpsychology”, and the so-called “object relations” theorists, who include not only the leading Otto Kernberg but also Harry Guntrip, W. R. D. Fairbairn, Donald Winnicott, and others. Thus, Heinz Kohut, for example, downgrades Freud's Oedipal, *Instinctual* factors in favor of pre-Oedipal, *environmental* ones as the sources of the purported *unconscious* determinants of personality structure. More generally, insofar as these post-Freudian theories are indeed recognizably psychoanalytic, they do of course embrace some version of the repression-aetiology. And, furthermore, they rely epistemically on free association in the clinical investigation of purported pathogens and other unconscious determinants of behavior, while lifting repressions to effect therapy.

But, I submit, precisely to the extent that these outgrowths of Freud's ideas are thus recognizably psychoanalytic in content as well as in method of inquiry and therapy, my epistemic critique of Freud's original hypotheses applies with equal force to the aetiologic, developmental and ther-apeutic tenets of these successors. How, I ask, for example, can Kohut possibly claim better validation for his species of unconscious determinats than Freud can for the sexual ones? Moreover, it is just ludicrous to pretend with Flax (1981, p. 564) that my focus on Freud in appraising psychoanalytic theory epistemically is akin to the anachronistic procedure of “throwing out physics because there are unresolved problems in Newton's theory”. For this purported analogy suggests misleadingly that the epistemic difficulties which beset Freud's original formulations have been overcome by the much vaunted post-Freudian formulations of self-psychology and object relations theory. And it overlooks the logical incompatibility of the most influential of these versions: As Robbins (1980, p. 477) points out, Kohut's and Kernberg's views are “fundamentally antagonistic” to one another, being rooted in a schism between Melanie Klein and W. R. D. Fairbairn. (See Grünbaum, 1983b for details).

True, there are elements in some of the post-Freudian theories that give less emphasis to repression, both aetiologically and therapeutically, than the received doctrine. For example, self-psychology gives significant aetiologic weight to the absence of empathic mirroring in early childhood. Yet, as Morris Eagle has shown (private communication, and in press), these ingredients of the post-Freudian theories are at least as flawed epistemologically as the repression model that was found seriously wanting in this essay. Hence it is futile to adduce these modifications, as the disciples of self-psychology and object relations theory are wont to do, as improvements upon Freud's original hypotheses, whose articulations were more lucid and more amenable to scrutiny.

Indeed, there is not even agreement among the post-Freudians in regard to the probative value that may be assigned to *One and the same case study material*; While Kohut claimed clinical support for his theory from his re-analysis of Mr. Z.-a patient whose prior analysis had been a traditional one—the contemporary Chicago analyst Gedo (1980) harshly discounts the scientific quality of Kohut's case study material. And he concludes (p. 382) that the “theoretical inferences” drawn by Kohut from his clinical observations “fail to carry scientific conviction.” A similarly negative assessment is reached by the psychoanalytic psychologist F. J. Levine (1979), an ardent exponent of psychoanalytic methods of investigation and therapy. On the other hand, Ferguson (1981, pp. 135–36) believes that Kohut's case history of Mr. Z. is “a crystalline example of the *fact* that a progressive theory change [in L. Laudan's sense] has taken place in psychoanalysis.” But, Ferguson then seems to damn it with faint praise, saying “the case of Mr. Z. provides something of a ‘confirming instance’ of the new theory.”

No wonder that the psychodynamically-oriented psychologists Fisher and Greenberg (1977) reached the following verdict: “The diversity of the secondary elaborations of Freud's ideas is so Babel-like as to defy the derivation of sensible deductions that can be put to empirical test” (p. ix).

I am indebted to Benjamin B. Rubinstein and to Rosemarie Sand for their valuable comments on the first draft. And I thank the Fritz Thyssen Stiftung of Cologne for its support of my research.

## Pluralism and Proofs

### Erkenntnis (2014-03-01) 79: 279-291 , March 01, 2014

Beall and Restall’s *Logical Pluralism* (2006) characterises pluralism about logical consequence in terms of the different ways *cases* can be selected in the analysis of logical consequence as preservation of truth over a class of cases. This is not the only way to understand or to motivate pluralism about logical consequence. Here, I will examine pluralism about logical consequence in terms of different standards of *proof*. We will focus on sequent derivations for classical logic, imposing two different restrictions on classical derivations to produce derivations for intuitionistic logic and for dual intuitionistic logic. The result is another way to understand the manner in which we can have different consequence relations in the same language. Furthermore, the proof-theoretic perspective gives us a different explanation of how the one concept of negation can have three different truth conditions, those in classical, intuitionistic and dual-intuitionistic models.

## Pfychopathologie funktioneller Störungen

### Erkenntnis (1932-12-01) 3: 229-230 , December 01, 1932

## ‘A logical reconstruction of pure exchange economics’: An alternative view

### Erkenntnis (1983-07-01) 20: 115-129 , July 01, 1983

## Propensity representations of probability

### Erkenntnis (1987-05-01) 26: 335-358 , May 01, 1987

## The Real Combination Problem: Panpsychism, Micro-Subjects, and Emergence

### Erkenntnis (2014-02-01) 79: 19-44 , February 01, 2014

Taking their motivation from the perceived failure of the reductive physicalist project concerning consciousness, panpsychists ascribe subjectivity to fundamental material entities in order to account for macro-consciousness. But there exists an unresolved tension within the mainstream panpsychist position, the seriousness of which has yet to be appreciated. I capture this tension as a dilemma, and offer advice to panpsychists on how to resolve it. The dilemma is as follows: Panpsychists take the micro-material realm to feature phenomenal properties, plus micro-subjects to whom these properties belong. However, it is impossible to explain the generation of a macro-subject (like one of us) in terms of the assembly of micro-subjects, for, as I show, subjects cannot combine. Therefore the panpsychist explanatory project is derailed by the insistence that the world’s ultimate material constituents are subjects of experience. The panpsychist faces a choice of giving up her explanatory ambitions, or of giving up the claim that the ultimates are subjects. I argue that the latter option is preferable, leading to neutral monism, on which phenomenal qualities are irreducible but subjects are reducible. So panpsychists should be neutral monists.