FUNDAMENTAL CONCEPTS IN LOGIC


In virtually every aspect of philosophy we are concerned with the reasons given for holding any particular view. Insofar as we are concerned with asking if the reasons given really do support the view in question, we are concerned with a logical inquiry.

Inference is the name given to the reasoning process by which we are led to assert or deny the truth of a conclusion on the basis of other beliefs assumed to be true.
Strictly speaking "inference" refers to a private, subjective process that goes on in the mind of the individual thinker, and thus is properly a subject for psychology rather than logic. However, corresponding to every inference, insofar as the beliefs in question can be expressed in language, one can formulate a group of statements as an argument leading to the conclusion in question.
An argument is a group of statements (or propositions) in which the truth of one statement, the conclusion, is claimed to follow from the assumed truth of the other(s), the premises.
An argument is not merely any group of statements. It must exhibit a flow of thought from the reasons given in the premises to the truth of the conclusion. Notice that every argument claims that its conclusion "follows" from its premises, or in other words that the assumed truth of the premises provides good grounds or reasons for the truth of the conclusion. However, while all arguments make this claim, all such claims are not equally justified; that is, not all arguments are equally good arguments. Some arguments are better than others, and some, of course, are altogether bad.
The primary goal of logic is the appraisal of arguments; the proper study of logic should equip you with techniques for distinguishing good arguments from bad ones.
Arguments are often used to persuade people to believe (or not believe) the truth of various claims; however, the study of techniques of persuasion is "rhetoric," not "logic." Rhetoric must inevitably involve considerations of psychological and social matters, for different persons and different cultures will differ in what they consider persuasive. Logic has no concern with such questions that lead to the formation of belief, but is concerned only with the argument itself, not with those who use it or may or may not find it persuasive. Obviously logic may be used as a device for persuasion, but it is easy to imagine situations in which purely logical considerations carry little persuasive force.

Since logic concerns itself with the analysis of arguments, and arguments are constructed of statements or propositions, a study of logic must begin with defining what is meant by "statement" or "proposition," which for introductory purposes may be considered to be synonymous.

A statement (or proposition) must satisfy two defining characteristics:
a) it is what is unambiguously asserted by a sentence, and
b) it has "truth value."
The first defining characteristic above, a), roughly identifies a "statement" as what is meant by a sentence, its "meaning." Since two or more sentences in a language, as well as different sentences in different languages, can have the same meaning, the same statement can be expressed by different sentences. Therefore, a"statement" cannot be identified with a "sentence." A "sentence" is in some particular language; a "statement" may be thought of what different sentences which mean the same thing have in common; but a statement is not in a particular language. (For many purposes "statement" and "proposition" may be considered synonymous; however, the two may be distinguished in that a "statement" may be thought of as the meaning of a sentence in a particular context of usage, whereas a "proposition" is abstracted from such a context. Since sentences often contain personal pronouns ("he," "she," "it," etc.) or demonstrative pronouns ("here," "there," "now," "then," etc.) the meaning of which can only be established by context of use, establishing the meaning of a sentence -the "statement"- may require understanding how the terms in the sentence refer to states of affairs in the world.)

The second defining characteristic further restricts the class delimited by the first characteristic. Not all meaningful sentences express "statements." Some "sentences" express commands, questions, or exclamations. Only those sentences which have a meaning which can be said to be "true" or "false" are those which express "statements."

A "statement" (or "proposition") must, by definition, have truth value; i.e., it must be either true or false.
(Note that having truth value is identified with being either true or false, not with human knowledge of truth or falsity. There may be many statements known to be true or false; there are certainly many others whose truth value is unknown. For statements there are only two possibilities: T or F, but for human knowledge there may be three: T, F, or Unknown. The "truth value" of a statement is a "metaphysical" matter, or in other words it depends on the way the world is; it is a function of reality. Whether or not we (or anyone) knows the truth value of a statement is an "epistemological" matter; or in other words it depends on the evidence available to particular persons at particular times; it is a function of our beliefs about the world rather than the world itself.)

The above definition of "statement" assumes that a sentence conveys a single meaning. Actually in all natural languages many, if not most, sentences can be "interpreted" in a variety of ways with subtle or often great differences of meaning from one "interpretation" to another. Such sentences are said to be "ambiguous." Since a statement is identified with a meaning, ambiguous sentences may actually express more than one statement.

Thus before any logical analysis of an argument can begin, the sentences in which the argument is expressed must be given a single unambiguous meaning. For purposes of logic we may simply stipulate, by an arbitrary decision, which meaning is to be accorded the ambiguous sentence. In this context we need not be worried about whether this is the "right" or "correct" meaning. However, for the purpose of interpreting a text, the concern of hermeneutics, then of course it is quite possible that one interpretation would be better than another. Generally speaking the task of interpretation requires great sensitivity to meaning and cannot be reduced to simple rules; the result is usually controversial. However, the logical analysis of the argument which results once the interpretive process gives each sentence an unambiguous meaning can often be reduced to a rule determined procedure, the result of which is uncontroversial.

From the fact that logic is concerned with evaluating arguments, it follows that we must consider the types of relationships which can hold between the premises of an argument and its conclusion, the relationship of "...provides reasons for...". Many different relationships are possible, but that one which has received the most attention is the one expressed by saying that the truth of premises necessitates the truth of the conclusion, or in other words, if the premises are all true, the conclusion must be true. Arguments which exhibit this type of relationship are called "deductively valid arguments." All arguments which present themselves as making the claim that the truth of the premises necessitates the truth of the conclusion may be considered "deductive arguments," but only those for which this claim is in fact true are valid deductive arguments.

Deductive arguments are those arguments which claim that the truth of all the premises of the argument implies that the conclusion must necessarily be true.
Deductive logic studies techniques for determining the validity of deductive arguments.
The concept of "validity" is the central concept of deductive logic. Notice first of all that it refers to a property of arguments, not statements. Thus a premise or a conclusion is not said to be "valid" or "invalid," but "true" or "false." Nor is validity to be identified with the property of an argument having a true conclusion. Valid arguments may have false conclusions. This is so because saying that the truth of the premises necessitates the truth of the conclusion amounts to saying that if the premises are true, then the conclusion cannot be otherwise than true. The emphasis here is on the "if"; no claim that the premises are in fact true is made in claiming that an argument is valid. All that is claimed is that the conclusion necessarily follows from the premises, or in other words that should the premises turn out to be all true, then the conclusion would have to be true. Another way to put this is that it is impossible for a valid argument to have all true premises and a false conclusion. If it can be shown that an argument does have all true premises and a false conclusion, then, ipso facto it is invalid.

As far as the analysis of the validity of any deductive argument is concerned, the premises of the argument are simply assumed as true. The logician is asking if we knew all the premises to be true, what would we know about the truth of the conclusion. But the job of determining the truth of the premises is not a logical matter, but a matter of concern to whatever field deals with the subject matter of the argument. To know that the conclusion of an argument is in fact true, we must know both that its premises are in fact true and that it is valid. The logician is concerned only with the latter.

A sound deductive argument satisfies the two criteria of being both a) valid and b) having all true premises.
Determining soundness requires both a logical task of determining validity and the task of determining the truth of premises carried out by whatever field is concerned with the subject matter of those premises. An argument may be "unsound" if it fails on either a) or b) above (or, of course, both).

There are several reasons why logicians restrict their attention to validity: First, determining the truth of the premises requires determining the meaning of the sentences in which the argument is expressed. This is always open to "interpretation" and thus cannot always be decided unarbitrarily. Logicians cut through all of this by asking, if the premises turn our to be true, what then? Second, we often want to reason "hypothetically" from assumptions the truth value of which is unknown, as a way of deciding whether the premises are true or false, or whether or not, if they concern matters under human control, to make them true or false. Third, we may want to reason "contrary to fact" from premises known to be false as a way of deciding what could have been true had other conditions prevailed.

Although a great deal of attention is focused on deductive arguments, much reasoning cannot be presented as making the strong claim of deductive necessity. When an argument claims to provide some grounds for the truth of the conclusion but not to necessitate that conclusion, it is considered an inductive argument.

Inductive arguments claim that the truth of the premises provide grounds for the likelihood or probability of the truth of the conclusion.
Inductive logic is concerned with techniques for the appraisal of inductive arguments.
Inductive arguments are not usually said to be "valid" or "invalid," but according to the degree of support which the premises do provide for the conclusion, they may be said to be "strong" or "weak" over a spectrum of varying degrees of likelihood. Although many have attempted to formulate a theory of "inductive probability" whereby one could determine how probable is the truth of a conclusion given certain premises, such attempts in general have met with little success. For the most part, inductive logic provides only rather loose and informal guidelines for assessing the relative strength or weakness of one inductive argument over others.