Deductive Systems in Traditional and Modern Logic

Logical Investigations, 2020

Our goal is to develop a syntactical apparatus for propositional logics in which the accepted and rejected propositions have the same status and are being treated in the same way. The suggested approach is based on the ideas of Ƚukasiewicz used for the classical logic and in addition, it includes the use of multiple conclusion rules. A special attention is paid to the logics in which each proposition is either accepted or rejected.

The main concern of this paper is to give an account of the nature and fundamental features of formal deductive systems in order to understand, precisely, what Kurt Gödel incompleteness theorems speak about them.

1999

There has been an increasing demand of a variety of logical systems, prompted by applications of logic in Al, logic prograrnming and other related areas. Labelled Deductive Systems (LDS)[Gab96] were developed as a flexible methodology to formalize such a kind of complex logical systems. In the last decade, defeasible argumentation [SL92, PV99, Ver96, BDKT97] has proven to be a confluence point for many approaches to formalizing commonsense reasoning.

Mandel, D.R. Ed. Assessment and Communication of Uncertainty in Intelligence to Support Decision-Making , 2020

Multi-valued logics offer useful schemata for reasoning in uncertain conditions. Such logics encompass propositions which are neither true nor false; they can also encompass changes in state across time, meaning propositions which are true but were false, or propositions once true but now false. Łukasiewicz's three-valued logic and Prior's system Q are proposed as deductive logics for the assessment of some inferences in practical reasoning. They provide an alternative to psychological accounts of enumerative induction and Bayesian inference, which, while popular as accounts of human reasoning in naturalistic settings, have also attracted substantive criticism. The characteristics of 3-valued and 6-valued logics are reviewed. Statements about historical changes in North American political geography are used in an initial application of such systems to practical reasoning. There is a small virtue in allowing that not all truths are sempiternal. There are reasons a statement can fall short of being true (or false). A change of state is one reason: what was true may be false later. There exist systems of rules-calculi, if you will-that encapsulate such wayward properties in logical terms. They evince how less than well-determined states, or how changes in state over time may be admitted to practical reasoning. This is not to say that rules for practical reasoning must describe how people do reason, or else how they should reason. As Łukasiewicz (1951, p. 12) puts it: "It is not true, however, that logic is the science of the laws of thought. It is not the object of logic to investigate how we are thinking actually or how we ought to think". These systems of rules provide a few standards-sometimes unaccustomed standards-for practical reasoning. As nets of logic, they may or may not suit the size of fish we might like to catch in psychological inquiry. The present article sets out two systems of logic as standards for the investigation of practical reasoning. A number of issues arise in the treatment of information by intelligence analysts. Two related issues are: the weighting of decisions about possible states of affairs by separate analysts, and decisions about the truth of hypotheses about present events from past information. The arrangement of

Deductive Method: Deduction Means reasoning or inference from the general to the particular or from the universal to the individual. The deductive method derives new conclusions from fundamental assumptions or from truth established by other methods. It involves the process of reasoning from certain laws or principles, which are assumed to be true, to the analysis of facts. Then inferences are drawn which are verified against observed facts. Bacon described deduction as a "descending process" in which we proceed from a general principle to its consequences. Mill characterised it as a priori method, while others called it abstract and analytical. Deduction involves four steps: (1) Selecting the problem. (2) The formulation of assumptions on the basis of which the problem is to be explored. (3) The formulation of hypothesis through the process of logical reasoning whereby inferences are drawn. (4) Verifying the hypothesis. These steps are discussed as under.

Classical and Nonclassical Logics, 2020

Typically, a logic consists of a formal or informal language together with a deductive system and/or a modeltheoretic semantics. The language has components that correspond to a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record arguments that are valid for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions for at least part of the language. The following sections provide the basics of a typical logic, sometimes called "classical elementary logic" or "classical first-order logic". Section 2 develops a formal language, with a rigorous syntax and grammar. The formal language is a recursively defined collection of strings on a fixed alphabet. As such, it has no meaning, or perhaps better, the meaning of its formulas is given by the deductive system and the semantics. Some of the symbols have counterparts in ordinary language. We define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion. The other sentences (if any) in an argument are its premises. Section 3 sets up a deductive system for the language, in the spirit of natural deduction. An argument is derivable if there is a deduction from some or all of its premises to its conclusion. Section 4 provides a model-theoretic semantics. An argument is valid if there is no interpretation (in the semantics) in which its premises are all true and its conclusion false. This reflects the longstanding view that a valid argument is truth-preserving. In Section 5, we turn to relationships between the deductive system and the semantics, and in particular, the relationship between derivability and validity. We show that an argument is derivable only if it is valid. This pleasant feature, called soundness, entails that no deduction takes one from true premises to a false conclusion. Thus, deductions preserve truth. Then we establish a converse, called completeness, that an argument is valid only if it is derivable. This establishes that the deductive system is rich enough to provide a deduction for every valid argument. So there are enough deductions: all and only valid arguments are derivable. We briefly indicate other features of the logic, some of which are corollaries to soundness and completeness.

Axioms, 2019

Using the defined notion of the inference with multiply-conclusion rules, we show that in the logics enjoying the disjunction property, any derivable rule can be inferred from the single-conclusion rules and a single multiple-conclusion rule, which represents the disjunction property. Also, the conversion algorithm of single- and multiple-conclusion deductive systems into each other is studied.

Educational Studies in Mathematics, 2008

This study examines ways of approaching deductive reasoning of people involved in mathematics education and/or logic. The data source includes 21 individual semi-structured interviews. The data analysis reveals two different approaches. One approach refers to deductive reasoning as a systematic step-by-step manner for solving problems, both in mathematics and in other domains. The other approach emphasizes formal logic as the essence of the deductive inference, distinguishing between mathematics and other domains in the usability of deductive reasoning. The findings are interpreted in light of theory and practice.

2007

The paper presents ATP - a system for automated theorem proving based on ordered linear resolution with marked literals, its putting into the base of prolog-like language and some applications. This resolution system is especially put into the base of prolog-like language, as the surrogate for the concept of negation as definite failure. This logical complete deductive base is used for building a descriptive logical programming language LOGPRO, which enables eliminating the defects of PROLOG-system (the expansion concerning Horn clauses, escaping negation treatment as definite failure), but keeping the main properties of PROLOG-language and possibilities of its expansions. Some features of the system when it is used as the base for time-table and scheduling, a technique for the implicational problem resolving for generalized data dependencies and intelligent tutoring system are described.

Educational Studies in Mathematics, 2000

A key issue for mathematics education is howchildren can be supported in shifting frombecause it looks right'orbecause it works in these cases' to convincing arguments which work ingeneral. In geometry, forms of software usually known as dynamicgeometry ...