Set-theoretic paradoxes

There are passages in Wittgenstein where he compares his method to psychotherapy and one or two where he seems to suggest that the ‘patient’ has the last word on his ‘illness’ and ‘cure’.  This paper tries to take these seriously, especially in connexion with the idea that victims of philosophical confusion talk nonsense.

Philosophical accusations of talking nonsense made in a polemical spirit are bound to fail, since the only way one can effectively criticise another’s utterance is by attaching some meaning to it.  But if the other acknowledges the possibility that his puzzlement arises from his having been taken in by nonsense, then perhaps a co-operative investigation of this suggestion might be fruitful.  Indeed self-diagnosis ought in principle to be possible.  Ought it not to have happened at least once in the history of philosophy if philosophical nonsense has ever been exposed? 

But there are difficulties.  For example:
a) We must ask whether or not philosophical nonsense, when exposed as such, is supposed to lose all appearance of meaningfulness.  Either answer is problematic.
b) The acknowledged similarity of the methodology under investigation to that of psychoanalysis exposes it to many of the familiar objections to the methodology of psychoanalysis, especially those that draw attention to the misleading influence of suggestion by an authority-figure.
c) The proposal that relentless cross-questioning of a suspected nonsense-talker will eventually lead her to ‘dry up’ (if she really is talking nonsense) seems to involve a version of the so-called Socratic Fallacy.

I originally entitled this paper ‘Why are there no uncontroversial examples of philosophical nonsense?’, but since this seemed apt to provoke rather superficial responses, I decided to re-title it.  In it I ponder the fact that, not only are there no uncontroversial examples of philosophical nonsense, there are no uncontroversial examples of what one might call ‘analogue philosophical nonsense’.  These would be non-philosophical cases of someone’s believing he means something when in reality he means nothing.  (Someone’s saying, ‘It’s 5 o’clock on the sun’ is sometimes offered as an example.)  I ask what conditions such cases would need to satisfy.  What would count as a compellingly clear case of an illusion of meaning?  Why has no one described one?

I will use paradox as a guide to metaphysical grounding, a kind of non-causal explanation that has recently shown itself to play a pivotal role in philosophical inquiry. Specifically, I will analyze the grounding structure of the Predestination paradox, the regresses of Carroll and Bradley, Russell's paradox and the Liar, Yablo's paradox, Zeno's paradoxes, and a novel omega plus one variant of Yablo's paradox, and thus find reason for the following: We should continue to characterize grounding as asymmetrical and irreflexive. We should change our understanding of the transitivity of grounding in a certain sense. We should require foundationality in a new, generalized sense, that has well-foundedness as its limit case. Meta-grounding is important. The polarity of grounding can be crucial. Thus we will learn a lot about structural properties of grounding from considering the various paradoxes. On the way, grounding will also turn out to be relevant to the diagnosis (if not the solution) of paradox. All the paradoxes under consideration will turn out to be breaches of some standard requirement on grounding, which makes uniform solutions of large groups of these paradoxes more desirable. In sum, bringing together paradox and grounding will be shown to be of considerable value to philosophy.

Edward Witherspoon distinguishes Wittgenstein’s conception of nonsense from Rudolf Carnap’s.  The latter does not fully take into account the fact that, if something really is nonsense, it has no more meaning than ‘Ab sur ah’ and therefore cannot have any logical properties.  He finds the Carnapian conception even in professed Wittgensteinians such as Peter Hacker and Gordon Baker who think of themselves as rejecting it.  My question is where this leaves the notion of PHILOSOPHICAL nonsense.  Could anything that was utterly devoid of content play the crucial deceptive role that nonsensicalists, including Wittgensteinians, assign it?  Could it ever be detected if it did?

Since the demise of logical positivism accusations of talking nonsense in philosophy have mainly been made by Wittgensteinians.  But some of these latter are as free with words like 'nonsense', 'senseless' and 'meaningless' as any logical positivist.  One such is Rupert Read, who is prepared to write off the whole of metaphysics as nonsense.  Condemnations of this sort have in recent years largely been ignored by other philosophers, which is surprising in view of their apparent importance.  It is equally surprising that the evident problems with accusations of talking nonsense have been largely ignored by those who are still prepared to make them.

Zahlen standen am Beginn des mathematischen Denkens und bilden bis heute einen syste-matischen Bezugspunkt im Aufbau und Verständnis der Mathematik. Für jede logische und philosophische Untersuchung der Mathematik spielt die Idee der Zahl eine zentrale Rolle. Im vorliegenden Buch wird anhand der natürlichen Zahlen, und später der reellen Zahlen, auf Fragen im Umfeld der Unendlichkeit der Zahlenreihe und der Eindeutigkeit des axiomatisch definierten Zahlbegriffs eingegangen, bis hin zur konkreten Einführung von Nichtstandardzah-len. An zentraler Stelle stehen Auseinandersetzungen um die Bedeutung und Tragweite der klassischen formalen Logik, der Mengenlehre und der modernen symbolischen (mathemati-schen) Logik als Darstellungsmittel mathematischer Zusammenhänge. Dies schliesst eine originelle philosophische Diskussion von Paradoxien und von verschiedenartigen Grenzen logischer und symbolischer Darstellbarkeit mit ein, die auch Paul Finslers Gedanken zur Grundlegung der Mathematik einbeziehen. Im Anhang findet sich ein Nachwort von Gregor Schneider, welches die vorliegenden Untersuchungen mit einigen modernen Tendenzen in der Philosophie der Mathematik konfrontiert, sowie ein bisher nicht auf Deutsch erschiener Aufsatz von Thoralf Skolem aus dem Jahre 1941 über die Grundlegung der Mengenlehre

I discuss Charles Pigden’s paper ‘Coercive theories of meaning, or why language shouldn’t matter (so much) to philosophy’ and ask whether theories of meaning whose primary purpose is to discredit rival philosophies as meaningless necessarily employ artificially narrow conceptions of meaningfulness.  In the course of my argument I ask whether the colloquial-rhetorical use of words like ‘nonsense’ and ‘meaningless’ has made it difficulty for philosophers to see just how novel and radical their own accusations of talking nonsense are.

I discuss Charles Pigden’s paper ‘Coercive theories of meaning, or why language shouldn’t matter (so much) to philosophy’ and ask whether theories of meaning whose primary purpose is to discredit rival philosophies as meaningless necessarily employ artificially narrow conceptions of meaningfulness.  In the course of my argument I ask whether the colloquial-rhetorical use of words like ‘nonsense’ and ‘meaningless’ has made it difficulty for philosophers to see just how novel and radical their own accusations of talking nonsense are.

Russell decided that the only way to avoid the paradox that bears his name was to reject certain combinations of words that were apparently about classes as being in fact nonsensical.  In doing this he was not accusing some rival school of philosophers of talking nonsense; only his own previous self.  This gave his position an air of non-polemical neutrality.  So long as no alternative was known to his treatment of the matter, it looked as though he had DISCOVERED that seemingly meaningful, well-formed sentences could be in reality nonsensical.  In spite of the fact that Russell made only a limited use of this idea in his own philosophy, I argue that he gave a decisive impetus to nonsensicalism by providing what seemed to be a museum exhibit of philosophical nonsense.

There is a principle in things, about which we cannot be deceived, but must always, on the contrary, recognize the truth – viz. that the same thing cannot at one and the same time be and not be": with these words of the Metaphysics, Aristotle introduced the Law of Non-Contradiction, which was to become the most authoritative principle in the history of Western thought. However, things have recently changed, and nowadays various philosophers, called dialetheists, claim that this Law does not hold unrestrictedly – that in peculiar circumstances the same thing may at the same time be and not be, and contradictions may obtain in the world. This book opens with an examination of the famous logical paradoxes that appear to speak on behalf of contradictions (e.g., the Liar paradox, the set-theoretic paradoxes such as Cantor’s and Russell’s), and of the reasons for the failure of the standard attempts to solve them. It provides, then, an introduction to paraconsistent logics – non-classical logics in which the admission of contradictions does not lead to logical chaos –, and their astonishing applications, going from inconsistent data base management to contradictory arithmetics capable of circumventing Gödel’s celebrated Incompleteness Theorem. The final part of the book discusses the philosophical motivations and difficulties of dialetheism, and shows how to extract from Aristotle’s ancient words a possible reply to the dialetheic challenge. How to Sell a Contradiction will appeal to anyone interested in non-classical logics, analytic metaphysics, and philosophy of mathematics, and especially to those who consider challenging our most entrenched beliefs the main duty of philosophical inquiry.

Die Frage, ob sich das menschliche Denken selbst begründen kann oder durch neurophysiologische Vorgänge bedingt ist, wird nach wie vor in der gegenwärtigen Philosophie kontrovers diskutiert. Hier werden einige grundlegende Einwände gegen das Unternehmen einer Selbstaufklärung des Denkens besprochen und gezeigt, dass diese nicht haltbar sind. Dies gibt Anlass zu einer detaillierten Untersuchung, was Selbstbezug in Mathematik, Logik und Erkenntniswissenschaft überhaupt bedeuten kann.