AN OUTLINE OF LOGIC AND LOGICAL SYSTEMS

INTRODUCTION TO CONCEPTS

Categorical propositions; immediate inference; categorical syllogisms; other argument forms; symbolic logic; inductive logic

SYSTEMS OF LOGIC

FORMAL LOGIC

Propositional and Predicate Calculi – lower and higher-order predicate calculi

Modal Logic

Set Theory

METALOGIC

Its Nature, Origins, and Influences

Syntax and semantics; the axiomatic method; logic and metalogic; semiotics – the study of signs and sign using behavior

Nature of a Formal System and of its Formal Language

Discoveries About Formal Mathematical Systems

Gödel’s two incompleteness theorems; decidability and undecidability; consistency proofs

Discoveries About Logical Calculi

Calculi of formal logic

Propositional calculus; the first-order predicate calculus; Löwenheim-Skolem theorem – any system which can be formalized in the first order predicate calculus, if it has a model, will have an enumerable model; the completeness theorem; the undecidability theorem and reduction classes

Model Theory

Is the branch of metalogic in which the interpretation of theories formalized in the framework of formal logic are studied

Background and typical problems

Satisfaction of a theory by a structure: finite and infinite models; elementary logic; non-elementary logic and future developments

Characterizations of the first-order logic; generalizations and extensions of the Löwenheim-Skolem theorem; ultrafilters, ultraproducts, and ultrapowers – constructs that are useful in studying models

APPLIED LOGIC

The Critique of Forms of Reasoning

Correct and defective argument forms; kinds of fallacies - material, verbal and formal fallacies

Epistemic Logic

The logic of belief

Theory of belief

The logic of knowing; the logic of questions

Practical Logic

The theory of reasoning with concepts of practice—of analyzing the logical relations among statements about actions and their accompaniments in choosing, planning, commanding, permitting, and so on—this is the domain of practical logic

The logic of preference – or the logic of choice, also known as proairetic logic

Symbolization and approach taken in proairetic logic; construction of a logic of preference

The logic of commands; deontic logic – the permitted, the obligatory, the forbidden, or the meritorious are the deontic modalities

The systematization and relation to alethic modal logic – what is commonly called modal logic, the modal logic or logics of truth [and falsehood]; alternative deontic systems

Logics of Physical Application

Temporal logic

Classic historical treatments; fundamental concepts and relations of temporal logic; systematization of temporal reasoning

Mereology – founded by Stanislaw Lesniewski, mereology clarifies class expressions and axiomatizes the relation between parts and wholes

Basic concepts and definitions; axiomatization of mereology

Computer Design and Programming

There is a clear connection between two valued logics and the 0 / 1 binary elementary computer states. However the details of computer design depend more on lattice theory than on logical theory. There is, however, a strong connection between computer programming and logical theory. Additionally, there is a connection between machine states and the possible worlds used in the semantics of modal logic. This connection, extended to include dynamic logic – the logic of dynamic or non-static descriptions of the world, temporal logic and process logic – the logic or argumentation that is applicable to all kinds of processes, has been used to study the properties and behavior of computer programs – e.g., does a program stop after a finite number of steps?

Hypothetical Reasoning and Counterfactual Conditionals

These have to do with what would obtain if something that is not known to be true or known to be not true were true. I have read that developments of the theory have various practical applications, including science and history

FORMAL LOGIC AND THE VARIETY OF LOGICS

One of the purposes of formalization is to abstract and make precise the kinds of informal argumentation. In so doing, we get systems that are specialized, even compartmentalized. The price of precision is that some of the variety and interconnections are lost. At the same time formalization leads, in addition to greater precision, to the study and extension of formal systems. Obviously, care is needed in the selection of one or another formal system to a particular situation. While some logical systems have been constructed out of or as curiosities, the origin of formal logic is in the desire to make explicit and analyze the structures of inference

The purposes of study of logic and of the variety, here, are as follows. First, out of interest in the analysis of, the need for and the application of the various forms. Second, to consider what structural unities there may be among the various logics; is there a single logic from which all may be derived? Third, in an attempt to formulate a theory of logic: what is logic, what is its relationship to knowledge and the knowledge process, is there a single logic or concept of logic from which all the foregoing follow, and, finally, is there a Logic and, if so, what is its relation to logic?

How will such a logic or Logic be “generated?” The bases will include:

The kinds of category: substance, space, time, cause, part-whole…

The kinds of knowledge: body, iconic, linguistic – these correspond roughly to immersion, acquaintance and description

The kinds of speech act: assertive, directive, commissive, expressive and declarative

Variety of practical application

Unifying and differentiating principles applied to the foregoing

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