Logic and Reasoning
The meaning of reason is to question, to call to account, to hold argument, discussion, discourse, talk or converse with another to employ reasoning or argument with a person in order to influence conduct or opinions; to think in a connected, sensible or logical manner; to employ the faculty of reason in forming conclusions.; to explain, support, infer, deal with by reasoning; to think out, to arrange the thought of in a logical manner.
Three categories of reason have been distinguished as evidencing, motivating and causally necessitating. In the context of evidencing, a reason for believing X is an item of evidence showing or tending to show that X is true.
The relation between logic and actual inferences is similar to that between the theory of a game and the actual playing of it. The analogy is worth pursuing in that it reveals where empirical considerations are, or should be, the concern of the logician. Deductive arguments aim to induce belief in their conclusions by force of reason; arguments which succeed in this being called logically valid. Logic in its narrower sense is the study of the principles of deductive inference, or of methods of deductive inference, or of methods of proof or demonstration.
Logic is a study which is not conducted by collecting data about the ways in which people argue, for logic is a theoretical rather than empirical science. It is the study of winning strategies in argument and of legitimate inferences and it is concerned with the possible means of establishing propositions. In the motivating context, a reason for doing something is a possible motive for that action deployed as purely prudential grounds for self persuasion.In context to causal necessitation is reasoning which suggests all the causes together which necessitated the eruption.
Logic and reasoning is found in philosophy, mathematics and science. Reading, writing, and mathematics are the traditional methods that people learned to think logically (i.e. correctly); science is a fourth method. Scientists and critical thinkers always use logical reasoning. Logic allows us to reason correctly. People can learn logical thinking by trial and error, but this method wastes time, is inefficient, is sometimes unsuccessful and often painful.
Reason is a word used in many various and often vague senses with complex and sometimes obscure connections. In one usage, it is sometimes marked with a capital. Reason is contrasted with such personified internal or external rivals as imagination, experience, passion or faith. Reason has been used to imply the access to answers through discovering what can and cannot be established by different forms of argument and what actions may or may not be commended as reasonable.
If an argument is to induce belief by force of reason it must be such that it shows that it would be irrational not to accept its conclusion after having agreed to its premises. ‘Irrational’ is taken here in the sense of ‘inconsistent’, so that denying the conclusion of a valid argument after having accepted its premises must amount to holding contradictory beliefs. The expression ‘ampliative argument’, was suggested by C.S. Pierce as an expression of inductive reasoning.
There is no generally accepted subclassification of inductive or ampliative arguments, though induction by simple enumeration is often thought to be the fundamental form. In inductive reasoning, when the premises are considered true, then the conclusion follows with a degree of probability attached. Induction is a method of reasoning by which a general law or principle is inferred from observed particular instances. The word ‘induction’ is derived from the latin translation of Aristotle’s ‘epagoge’, which seems in turn to have been taken from earlier Greek writers on military tactics.
With the growth of natural science philosophers became increasingly aware that a deductive argument can only bring out what is already implicit in its premises, and hence inclined to insist that all new knowledge must come from some form of induction. The term is employed to cover all arguments in which the truth of the premise, or premises, while not entailing the truth of the conclusion, or conclusions, nevertheless purports to constitute good reason for accepting it, or them.
Deductive reasoning holds that the conclusion is inherent in the premises and that the conclusion follows as a matter of course. The conclusion is generally considered non-false; whereas the unqualified premises are held to be neither true nor false. A deduction is a valid argument in which it is impossible to assert the premises and to deny the conclusion without thereby contradicting oneself. The word is not confined within the restrictions of the traditional syllogistic form (syllogism).
However, Sir Karl Popper maintained that Hume had succeeded in demonstrating the invalidity of induction and went on to say that respectable science advances by hypothetico-deductive rather than inductive methods.
The question to which the study of logic should provide some answers is how does one recognize an argument and construct one in support of a given proposition. The study of logic cannot answer these without being drawn to also take a look at what are accepted standards of rationality – at what inferential moves are regarded as being rationally compelling. One’s view of the nature of the science of logic will depend on one’s views on the nature of truth, knowledge and our cognitive capacities.
If knowledge is knowledge of an independent reality, and if the truth of a proposition consists in its presenting a picture that is an accurate representation of this reality, then laws of logic, as regulative principles governing the pursuit of knowledge and the construction of scientific theories, will appear as laws founded in the nature of the reality we seek to know.
Viewed in this way, logic is the most general of all sciences; to paraphrase W.V. Quine (Word and Object, 1960), the study of logic is the illumination (limning) of the most general traits of reality. To give the principles according to which debates should be conducted is thus to give a list of general rules for the construction of arguments that can establish the relationship between terms and of rules for refuting claims of this form. Aristotle gave expression to the four traditional forms which categorical propositions can take in the theory of syllogism in his Prior Analytics.
It is this concentration on the relations between terms which characterizes both Aristotelian and scholastic logic. A syllogism, as defined by Aristotle, is ‘a discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so’. The pursuit of knowledge through rational debate is an ideal presented in the dialogues of Plato, but it was Aristotle who first engaged in a systematic and theoretical study of the principles according to which such debates should be conducted.
Due to a logically valid argument being one whose acceptance is a dictate of reason, the laws of logic have also been regarded as laws of thought – laws governing the operation of the intellect or the faculty of reason – and thus as having their foundation in the nature of the human intellect, rather than in any external reality.
The aim is to make explicit the rules which are implicitly recognized as rules according to which arguments ought to be constructed, and at the same time pointing out any anomalies that may appear in the process.
Practical reason has since Aristotle been distinguished from theoretical or discursive reason. David Hume proclaimed “Reason is, and ought only to be, the slave of the passions”. He went on to define ‘passion’ to include every conceivable motive for action, while ‘reason’, in a complementary sense, covered only inert and neutral application of what in fact is the case and what follows from what. It was only in the late 19th century that the work of Frege, whose approach was adopted by Russell and Whitehead, caused the focus of attention to be shifted away from terms to propositions and their relations.
The logic treating of rules governing the use of quantifiers is sometimes called quantification theory, but more usually, predicate calculus. The truth or falsity of a quantified statement (that is, one containing a quantifier) cannot be assessed unless one knows what totality of objects is under discussion, or where the values of the variables may come from. Even prior to the Fregean innovations, the study of logic had become increasingly mathematical since it is possible to treat a formal logical system as just another system of algebra giving rise to an algebraic structure that can be studied using mathematical techniques.
This enabled him to justify the logic of propositions (propositional calculus) with the study of those logical relationships which had previously been treated in the theory of the syllogism. The advance made in Frege’s system was the introduction of quantifiers and of the treatment of concepts by analogy with mathematical functions. Quantifier indicates what in traditional logic was called the quantity of a statement, namely whether it is universal, as in ‘All bats are blind’, or particular as in ‘Some swans are black’. Thus Frege made it possible to exhibit the relation between singular and universal propositions within a single system; that of predicate calculus. Frege’s system allows for the representation of relations and so makes possible a treatment of the logic of relations; something which is highly problematic in a term orientated logic.
Logic in its broadest sense is the study of the structure and principles of reasoning or of sound argument. Hence it is also a study of those relations in virtue of which one thing may be said to follow from or be a consequence of another.
Within this very loosely defined area one can distinguish various kinds of logic according to the kind of reasoning considered and the kind of sentences in which it is conducted. In the study of reasoning, which aims to establish the truth of propositions, the major distinction has been made between deductive and inductive logic. Another important use of reason occurs in practical reasoning where one seeks to establish what ought to be done. Such reasoning may be expressed using both indicatives and imperatives although the conclusions of of practical reasoning are characteristically expressed in the imperative. The study of this form of reasoning is termed deontic logic, the logic of norms, or the logic of imperatives.
George Boole, one of the pioneers to the mathematical approach to logic said “The laws we have to examine are the laws of one of the most important of our mental faculties. The mathematics we have to construct is the mathematics of the human intellect”.
They are concerned with the forms of arguments and not with their content. Thus a prerequisite of any theoretical approach to the principles of argument is the development of means of representing the logical form or structure of an argument. Abduction is a syllogism of which the major premise is true but minor premise is only probable.
Abduction also refers to the name given by C.S. Pierce to the creative formulation of new statistical hypotheses that explain a given set of facts. Argument by analogy infers that, because this is like that in some respects, this and that must therefore be similar also in others. As deductions such conclusions do not follow yet they may be sometimes usefully suggestive, because in fact true. A fourth method of reasoning is analogy. Reasoning by analogy is a reference to similarly or likeness.
The conclusion of an analogy is only plausible. Analogical reasoning is sometimes accepted only as an auxiliary method. A refined approach is case-based reasoning. The positive analogy between the two terms of a comparison – their likenesses – may be constrasted with the negative – their unlikenesses. Abductive reasoning, or inference to the best explanation, is a method which can involve both inductive and deductive arguments. The main characteristic of abduction is that it is an attempt to favour one conclusion above others by either attempting to falsify alternative explanations, or showing the likelihood of the favoured conclusion given a set of more or less disputable assumptions.
Whether the laws of logic are regarded as logically necessary truths or as laws of thought or of language, in being rules according to which arguments are to be judged, they must be independent of particular subject matter, for they must be applicable in general, no matter what the argument is about.
Boole’s attitude toward logic translated into the idiom of contemporary linguistic philosophy, would be that the laws of logic are laws governing the functioning of our language and thus have their foundation in socially instituted conventions for the use of words, in that conventions determine the nature of a language in that they provide the framework within which particular expressions of the language acquire meaning.