Showing papers in "Notre Dame Journal of Formal Logic in 1964"

Remarks on the completeness of logical systems relative to the validity-concepts of P. Lorenzen and K. Lorenz.

Abstract: The so-called semantics of elementary logic (predicate logic of first order) has some peculiarities which may be considered as disadvantages, at least if looked at from a certain point of view. First of all, it makes use of a very strong set-theoretical apparatus which is highly non-constructive. Strict constructivism may not be obtainable for a semantic foundation of logic; but even if one does not subscribe to constructivism one should expect that a weaker apparatus would be sufficient to define logical validity or logical implication on the elementary level. Secondly, this set-theoretical approach is restricted to classical logic and can therefore not be used, e.g. to define a concept of validity for intuitionistic logic or other logical systems differing from the classical one. This fact will not be considered as a drawback by those for whom classical logic is the only ‘real’ logic. On the other hand ‘classicists’ as well as ‘intuitionists’ should welcome an account of logic with the help of which different logical systems can be compared on the basis of semantical concepts alone — an account in which the deviations of logical systems from each other would be mirrored by different concepts of validity. Thirdly there are two characteristics of this semantics based upon the Bolzano-Tarski approach which in various contexts have been the main points of attack in the arguments of intuitionists and constructivists against this approach (which are mistakenly thought to be arguments against classical logic): (a) the procedure of introducing logical connectives as truth-functions tacitly presupposes that every meaningful sentence is either true or false (true-false-alternative). In view of such unproved and unrefuted sentences as ‘there is at least one odd perfect number’ one can reasonably doubt whether this assumption is correct. Namely if one decides to identify in mathematical contexts ‘true with ‘provable’ and ‘false’ with ‘refutable’ it is certainly not justified. And if one does not accept this identification then if one doesn’t want to destroy the meaning of ‘true’ altogether it seems hardly possible to do without some kind of ‘ontological hypothesis’ about the pre-existence of numbers or other kinds of mathematical entities; but the best that can be said about

Modal system ${\rm S}4.4$.

Abstract: The theses Gl and D2 are the proper axioms of the well-known systems S4.2 and S4.3 respectively, cf. [2], [l], [β], and [ l l j In [2], p. 263, Dummett and Lemmon have proved that Ml, i.e. their formula (8), does not hold in S4.3. Prior, [6], p. 139, pointed out that Geach showed that in the field of S4.2 theses Ml and Nl are equivalent. As one can easily notice βl and βZ verify also the following two formulas

Concrete and abstract properties.

Abstract: This paper is intended as a contribution to the present discussion concerning the "ontological commitment" of logical theories. It presupposes acquaintance with the distinction between "nominalism" and "platonism" as stated by N. Goodman and W. V. Quine. In my opinion both the nominalists as well as the platonists fail to explain how a predicate expression can be truly or falsely predicated of a given individual. A detailed analysis of the ways in which symbols can be related to what they stand for will suggest an interpretation of what the nominalists might intend when they say that predicate expressions function "syncategorematically." In the course of this analysis concrete properties and relations are distinguished from abstract properties and relations (from classes). The assumption of concrete properties and relations clarifies not only nominalistic semantics, it is also valuable for a platonist because he can prove that these concrete entities provide an adequate foundation for the construction of abstract entities. However the understanding of concrete properties and relations presents special difficulties some of which will be discussed here. 1. When we make statements about Peter, saying, e.g., that Peter is intelligent, that Peter is laughing, etc., we make use of the proper name 'Peter' to denote Peter. The expression 'Peter' stands for a concrete "thing". This is generally admitted and non problematic. But what of the expressions 'is intelligent,' 'is laughing/ and others, used in speaking of Peter or Paul? What do they stand for? Here opinions are divided. There are logicians, the so-called platonists, who consider predicate expressions almost as proper names, with this difference only that for them the entities which predicate expressions stand for, are not things but entities of a different type, namely classes or properties. Other logicians, called nominalists, say that although they have been looking for tfliese platonistic entities, all they have found are concrete things and "heaps" of concrete things. Thus for them a symbol stands either for a concrete thing like Peter or Paul or the "heap" made up of Peter and Paul together, etc., or else it functions "syncategorematically." 2. Both of these views, the platonistic and the nominatistic one, are in some respect unsatisfactory.

A liberalized system of quantificational deduction.

Abstract: In this note, we prove a liberalized version of the system of quantificational deduction of Section VI of Patton [2] to be sound. The rules of that system were Ul and El, where the instantial variable of an El step wasn't allowed to be one that was free in a previous line of the deduction. The system was designed to show a set of formulas to be inconsistent by deriving a truth-functionally inconsistent set of quantifier less lines from prenex normal form versions of these formulas. Soundness was proved by showing this to be impossible if the original set of formulas is consistent. The liberal system is like this one except that the premises of a deduction—the formulas of the set to be shown inconsistent—may be expressed as disjunctions of formulas in prenex normal form and Ul and El are supplanted by a rule that lets us drop from one to all leftmost quantifiers of the disjuncts of such a disjunction and replace all the variables they bound by a single variable that doesn't become bound in the line inferred. The El restriction is imposed here if an existential quantifier is dropped. (Thus every deduction of the old system is a deduction of this system too.) Now consider the system like this liberal one except that its rules are Ul, El, and a nonformal rule that allows passage from a given line to any formula equivalent to it. Despite the nonformal character of this system, the soundness proof of Patton [2] extends to it in obvious fashion. Our liberal system will be proved sound by showing that its every deduction D has a counterpart D in this nonformal system such that every line in D also occurs in D\\ D will have the same premises as D. Suppose that the first k-1 lines of D also occur in D and that line (&) in D comes from a previous line (j) thus (where Em, Fn, Gr, and Hs are formulas in which m, n, r , and s respectively occur free and a is neither free in a line previous to (k) nor bound in (&)):