The Evolution of Syntax in Wittgenstein's Philosophy

I will examine the shift in Ludwig Wittgenstein's position regarding
the
nature and status of logical truth from the __Tractatus__ to the early
1930s,
when he was in conversation with Friedrich Waismann and Moritz Schlick
of
the Vienna Circle. In the __Tractatus__, logical truths are merely
tautologous propositions. By 1930, though, Wittgenstein had
significantly expanded his notion of logical truth to include the
relations among senses of words. This changed position regarding the
nature of logical truth seems to force a change in the status accorded
to
these truths. Thus the question is raised for Wittgenstein as to
whether
logical truthand the syntax comprised thereofis a matter of
convention.
In the __Tractatus__, syntax is far from conventional; logical truth shows
the formal structure of the world. Yet a much-expanded notion of
logical
truth brings with it the possibility of different syntaxes, which in
turn
raises the question of how syntax is determined. Given the nature of
Wittgenstein's views at the time, convention seems the natural
answer.
Characteristically, his remarks fail to settle the issue explicitly, but
they are, of course, extremely suggestive. He has certainly moved
toward
conventionalism since the __Tractatus__, but at least in some moods he
seems hesitant to embrace a full-blown conventionalism. I suggest
that his
position might be seen as an unwillingness to accept the wholly
arbitrary
determination of syntax. Instead, he may have in mind a combination
of
limitations due in part to the logical properties of the world and in
part to the pragmatic selection of syntax that is nonetheless, on some
levels, a matter of convention.

Preservationism meets modal logic; a property based approach to modal logic.

In this paper we apply techniques and methods of preservationist logic to the problem of axiomatizing logics of finite relational frames. The strategy is to encode a finite frame as a preservationist matrix and then to axiomatize this matrix using well-known and rather straightforward preservationist methods. To this end, we will introduce a series of new modal operators, the Delta_i-operators, an operator for each world in the frame. $\Delta$ is a naming operator, and the sentence $\Delta_i \phi$ means simply that $\phi$ holds at $i$, or as we prefer to put it, $\phi$ has the property $i$. We will explore the relation between this approach and the other main approach that allows for world naming, namely that of hybrid logic. As the main technical result, we prove completeness of the axiomatization and show that adding delta-operators conservatively extends the logic in the underlying language of the frame.

Formal Consequence in Scotus and Ockham

Ockham famously disagrees with Scotus' in matters of ontology. In this paper by examining their discussions of a controversial theological question I show that he also disagrees radically with his logic.

Quadratic Reciprocity and the Theory of Congruences

The law of quadratic reciprocity states that if p and q are distinct odd primes, then the two congruences x^2 \equiv q (mod p) and x^2 \equiv p (mod q) are either both solvable or both unsolvable, unless p and q are both of the form 4k + 3, in which case one congruence is solvable and the other is not. Gauss is credited with the first proof of this law, given in Disquisitiones Arithmeticae. This work is also where he introduces his new theory of congruences, which is used in the proof of the law. In the discussion following his proof, he remarks on the fact that no one had thus far presented it in so simple a form. Furthermore, he finds it remarkable that Euler knew other propositions which depend on it and should have led to its discovery, yet did not prove the law himself. Legendre also worked on the problem, yet had an incomplete proof. Now, everything that can be stated in terms of congruences can also be stated in terms of divisibility. Yet in a letter to Schumacher, Gauss wrote that new calculi can play a significant role in problem-solving, though in general, one cannot attain anything by them that could not also have been attained without them. On the face of it, the theory of congruences looks like a particularly innocuous example of a new calculus, but we can see Gauss as crediting it with his proof of the law of quadratic reciprocity. In this paper, I will look at the relationship between Gauss' new theory of congruences and the proof he discovered, as part of a more general question about the epistemological relevance of representation in mathematics.

The Epistemic Value of Purity

A proof of a proposition P is pure, roughly speaking, if it is comprised of propositions that are definitions of terms in P or are deductive consequences of those definitions. We consider two specific proposals for purist projects in mathematics. We then examine and evaluate each project's epistemic value.

Last modified: Tue Oct 07 09:26:10 PDT 2003