Psy 416: Reasoning and Problem Solving
Primer on Logic and Logical Concepts
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Erwin M. Segal
Department of Psychology and
Center for Cognitive Science
State University of New York at Buffalo

Logic was an attempt to describe normative or correct reasoning. A widespread belief in AI, Cognitive Psychology, and Cognitive Science is that there are procedures representing the way people think which can be implemented on computers using logic type rules directly (algorithms) or by finding other ways to implement logical principles. The study of reasoning in this case is the study of "natural" logic. Much reasoning is evaluated by the extent to which it conforms to logical analyses, and much reasoning research in psychology is aimed at explaining the errors people make when they attempt to solve logic type problems.

Logical reasoning focuses on one aspect of how human cognitive performance can be judged. Do the statements that are presented together in a single discourse hold together? Are the statements consistent with one another; and if any conclusions are drawn, do they legitimately follow from the other statements made? Are they logical or are they illogical? We hope to clarify what these terms and other terms mean, and give at least some simple examples of how to answer such questions. Logical analysis is based on the structural or the formal relations among the components of the arguments rather than the meanings of the components. If two arguments have the same logical form one cannot be logical and the other not, regardless of their differences in meaning.

There are many different logics with many similarities and some differences. Logics are formal systems that include principles which identify well-formed formulas, a set of axioms, and rules of derivation, which when correctly applied purportedly lead to correct conclusions. Different logics may reason about categories, propositions, relations, time, tenses, beliefs, necessity, events, etc.

I.  General concepts:

Deduction--Applying a sequence of rules of inference from a set of premises and from which a conclusion necessarily follows.
Induction--coming to a conclusion that is likely from the premises. Usually coming to a conclusion about a universal or a particular from previous instances of particulars.
Proposition: A concept that usually represents a statement, a sentence, or an assertion. (E.g. The moon is made of green cheese. 2+2 = 4. Five is greater than seven. Dr. Segal teaches Psy 416. All men are mortal. It's better to have loved and lost than never to have loved at all. Sammy Sosa didn’t hit any homeruns yesterday. Oedipus loved his mother) Sentences which represent propositions in most logics, should have a truth value; Many logics are called two-value logics, in that they assume that all propositions are either true or false. In formal logic different kinds of propositions are represented by well-specified structures or forms. In any formal logical system a given representation of (form for) a proposition should be clear and unambiguous. Such forms are often called well-formed formulas or wffs.
Premise: A proposition that is assumed to be true for the sake of a logical argument.
Argument: A sequence of propositions, starting with premises, which purportedly justifies another proposition, called a conclusion.
Syllogism: A set of (usually, but not necessarily, two) premises followed by a conclusion; the classical form of an argument.
Validity: A syllogism, or any argument, is valid if the conclusion(s) drawn logically follow from the premises; otherwise it is invalid. In a valid argument if the premises are true, the conclusion is necessarily true.
Soundness: A sound argument is a valid argument in which the premises are true.
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