What is a Logical System? A Commentary

Lecture Notes in Computer Science, 1997

The notion of an Institution 5] is here taken as the precise formulation for the notion of a logical system. By using elementary tools from the core of category theory, we are able to reveal the underlying mathematical structures lying \behind" the logical formulation of the satisfaction condition, and hence to acquire a both suitable and deeper understanding of the institution concept. This allows us to systematically approach the problem of describing and analyzing relations between logical systems. Theorem 2.10 redesigns the notion of an institution to a purely categorical level, so that the satisfaction condition becomes a functorial (and natural) transformation from speci cations to (subcategories of) models and vice versa. This systematic procedure is also applied to discuss and give a natural description for the notions of institution morphism and institution map. The last technical discussion is a careful and detailed analysis of two examples, which tries to outline how the new categorical insights could help in guiding the development of a unifying theory for relations between logical systems.

Proceedings of the 7th International Conference on Category Theory and Computer Science, 1997

The notion of an Institution 5] is here taken as the precise formulation for the notion of a logical system. By using elementary tools from the core of category theory, we are able to reveal the underlying mathematical structures lying \behind" the logical formulation of the satisfaction condition, and hence to acquire a both suitable and deeper understanding of the institution concept. This allows us to systematically approach the problem of describing and analyzing relations between logical systems. Theorem 2.10 redesigns the notion of an institution to a purely categorical level, so that the satisfaction condition becomes a functorial (and natural) transformation from speci cations to (subcategories of) models and vice versa. This systematic procedure is also applied to discuss and give a natural description for the notions of institution morphism and institution map. The last technical discussion is a careful and detailed analysis of two examples, which tries to outline how the new categorical insights could help in guiding the development of a unifying theory for relations between logical systems.

The most general questions are what modern logic regards the logical role of compositionality, how it works in two-component logical semantics. After showing different versions of the compositionality of natural language we analyze the possible appearances of the principle of compositionality in two-component logical semantics. Finally, some of the most fundamental notions of intensional logical semantics are given in the way of maintaining the priority of compositionality concerning sense.

2006

This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum algebra construction, its generalization to a Lindenbaum category construction that includes proofs, and model cardinality spectra; these are used in some examples to show logics inequivalent. Generalizations of familiar results from first order to arbitrary logics are also discussed, including Craig interpolation and Beth definability. Mathematics Subject Classification (2000). Primary 03C95; Secondary 18A15, 03G30, 18C10, 03B22.

2015

I n this paper, I would like to point out the problems of the presently reigning mathematico-functional concept of the logical form of sentences, which presents itself as the final answer to the question of true logical form of sentences and, with this, the final basic scheme of logic. I am of the opinion that the present conception of the logical form of sentences is also a historical result, which in many ways surpasses and encompasses all former concepts in the history of logic, but which is not the only and absolute logical form of sentences, but also of logic itself. It therefore contains some immanent limitations which are, in my opinion, linked mainly to the »functional« concept of the sentence, the elementary predicative sentence, which is the foundation for all other sentence structures. Logic, with its use of letters for marking variables, which represent arbitrary actual terms, received the possibility of simultaneously treating a whole class of logically identical deductions. Mathematics also analogously received the means for executing general solutions for a whole class of related tasks. For instance, in geometry equations, make it possible to treat the properties of the most general classes of geometrically similar figures. The link between logic and mathematics is even more tight in the notion of deduction, as both branches are strict deductive sciences, and the ideal of deduction is most surely the axiomatic system. As Lukasiewicz showed, Aristotle's syllogistic can be partly presented as an axiomatic system, constructed from the modi of the so-called »first figures« of the syllogistic as axioms and additional rules of substitution and transformation of individual statements (J. Lukasiewicz, A r is to tle ' s Syllogistic, 1951).

As already mentioned in several places, in the search for a proof there are almost always many (depending on the calculus, potentially infinitely many) possibilities for the application of inference rules at every step. The result is the aforementioned explosive growth of the search space ( .1 on page 58). In the worst case, all of these possibilities must be tried in order to find the proof, which is usually not possible in a reasonable amount of time.

Applied Logic Series, 2008

Journal of Applied Logic, 2007

1974

The author presents an algorithm for enumerating the verifying truth-value assignments to the variables of a wff of two-valued propositional logic that purports to be significantly more economical than testing seriatim each possible value assignment. When 1 and 0 are truth and falsehood respectively and a fixed order of value assignments to variables is adopted, one can represent the full disjunctive normal form of a wff in n variables as a sequence of 2" zeros and ones which is called a selector. Ingenious matrix-like arrays abbreviate selectors by telescoping repetitive patterns together and reducing patternless stretches of digits to single digits. In outline, the algorithm is: Beginning with a wff B in variables p x. • • • ,p n , one associates with each p t a multi-tiered array for a wff in variables Pi .• • •. p n that is equivalent to p,. Rules are given for determining the array for B by operating successively on the tiers in the arrays for variables beginning w ith the bottom tiers. The result is an array, or a disjunction or conjunction of arrays, from which one can read off the selector for B. For wffs that fall exclusively under thesis (8), the algorithm is quite economical when large numbers of variables are involved. For other wffs, however, it is not evident that the algorithm is as economical as the method of reading off verifying value assignments from the branches of a semantic tableau.. The informality and brevity of the elegant exposition often frustrates comprehension. Algorithm rule S4, for example, can be read in several ways and none of the unintended readings is ruled out by the author's examples. (S4 should be read as stipulating that the ones that terminate calculation rows that are exactly alike with respect to occurrences of K, L, and M shall be grouped together to consitute, when gaps have been filled with zeros, a distinct outcome component.) The author's terminology is sometimes unexplained and misleading. For example, selectors are called mutually exclusive if they correspond to mutually exclusive wffs. But without further explanation the author speaks of mutually exclusive arrays which are not arrays that abbreviate mutually exclusive selectors but rather are arrays having mutually exclusive selectors on those tiers that cause the construction of an array to split into several arrays. Justification of the eleven theses on which the algorithm depends is reduced to two fundamental theses (6) and (7) which are formulated too restrictedly. They must be understood so that, if a and b are selectors of degree m and n respectively, then a(a) is a function of the first m variables while b(fi) is a function of the next /; variables. The unexplained function J on page 197 is NE. The variable u in the penultimate formula on page 203 should be v. None of the other misprints impedes comprehension. GERALD J. MASSEY and CARL J. POSY