Online research in philosophy

This paper examines the quantification theory of *9 of Principia Mathematica. The focus of the discussion is not the philosophical role that section *9 plays in Principia's full ramified type-theory. Rather, the paper assesses the system of *9 as a quantificational theory for the ordinary predicate calculus. The quantifier-free part of the system of *9 is examined and some misunderstandings of it are corrected. A flaw in the system of *9 is discovered, but it is shown that with a minor repair the system is semantically complete. Finally, the system is contrasted with the system of *8 of Principia's second edition.

It is argued that, in the traditional subject-predicate sentence, two interpretations of the subject term coexist, one intensional and the other extensional, which explains the superficial difference between the traditional S-P relation and the predication of predicate logic. Data from psychological studies of syllogistic reasoning support the view that the contrast between predicate and argument is carried over to the traditional S-P sentence.

Intuitionistically. a set has to be given by a finite construction or by a construction-project generating the elements of the set in the course of time. Quantification is only meaningful if the range of each quantifier is a well-circumscribed set. Thinking upon the meaning of quantification, one is led to insights?in particular, the so-called continuity principles?which are surprising from a classical point of view. We believe that such considerations lie at the basis of Brouwer?s reconstruction of mathematics. The predicate ?α is lawless? is not acceptable, the lawless sequences do not form a well-circumscribed intuitionistic set, and quantification over lawless sequences does not make sense.

This paper consists roughly of three parts. In the first part, an attempt has been made to find some tenable interpretation of Hamilton's logic. This results in accepting that Hamilton's logic can be "saved" if it is understood as being an everday language version of Euler's relations, i.e., extensional relations between terms (classes). In the second part, the propositions of Euler and the propositions of Aristotle are compared and found to be interdefinable: every proposition of Aristotle can be defined by a disjunction of Euler's propositions, and every proposition of Euler can be defined by a conjunction of Aristotle's propositions. In the third part, extensional interpretation is applied to the traditional logic. As a result the 19 traditional syllogisms are reduced to 11.

Propositional logic -- Propositions and arguments -- Connectives and argument forms -- Truth tables -- Trees -- Vagueness and bivalence -- Conditionality -- Natural deduction -- Predicate logic -- Predicates, names, and quantifiers -- Models for predicate logic -- Trees for predicate logic -- Identity and functions -- Definite descriptions -- Some things do not exist -- What is a predicate? -- What is logic?

For various reasons several authors have enriched classical first order syntax by adding a predicate abstraction operator. “Conservatives” have done so without disturbing the syntax of the formal quantifiers but “revisionists” have argued that predicate abstraction motivates the universal quantifier’s re-classification from an expression that combines with a variable to yield a sentence from a sentence, to an expression that combines with a one-place predicate to yield a sentence. My main aim is to advance the cause of predicate abstraction while cautioning against revisionism. In so doing, however, I shall pursue a secondary aim by conveying mixed blessings to those who hold the view that in the logical sense of “existence” some existing object is such as to exist contingently. Advocates of this view must concede Williamson’s recent contention that the domain of unrestricted objectual quantification could not have been narrower than it is actually, but predicate abstraction affords them some hope of accommodating this concession.

Some natural language expressions –namely, determiners like every, some, most, etc.— introduce quantification over individuals (or, in other words, they express relations between sets of individuals). For example, the truth conditions of a sentence like (1a) are represented in Predicate Logic (PrL) by binding the..

In the Begriffschrift Frege drew no distinction—or anyway signalled no importance to the distinction—between quantifying into positions occupied by what he called eigennamen—singular terms—in a sentence and quantification into predicate position or, more generally, quantification into open sentences—into what remains of a sentence when one or more occurrences of singular terms are removed. He seems to have conceived of both alike as perfectly legitimate forms of generalisation, each properly belonging to logic. More accurately: he seems to have conceived of quantification as such as an operation of pure logic, and in effect to have drawn no distinction between first-order, second-order and higherorder quantification in general.