[reposted January 12, 2002] Preliminary Comments Generation and Validity in Sentential Logic

Preliminary Comments on Logic and Validity

T IS frequently possible for one person, without using rules, to make subtle distinctions which another person makes only by using an explicit set of rules. Typically, the intuitive capacity of the first person is acquired through training and apprenticeship in discriminating between actual instances displaying the subtle differences; often a person who acquires an expert, intuitive capacity is not able to articulate very helpfully the bases for his (reliable) discriminations. A wine taster, for example, may have learned to distinguish reliably between vintages from different geographical areas or periods of time without being able to articulate those characteristics which enable him to make his distinctions. And a person may learn to speak a language fluently, to recognize grammatical flaws and to say how to correct them, without being able to articulate a systematic set of rules for this purpose. But sometimes it is possible, by studying the discriminations made intuitively by an expert, to devise a systematic and comprehensive set of rules which can be used to make the same distinctions, and with equal reliability, by someone who has not acquired the intuitive capacity of the expert. In hopes of such a result, chemical analyses of wines are made in various experiments designed to extract the chemical bases for a systematic set of rules which can account for and duplicate the expert's discriminations. And, closer in point, systems of grammar are devised to account for and duplicate the reliable, intuitive capacity possessed by a sophisticated user of a natural language.

People did not wait, of course, for the likes of Aristotle to distinguish between good and poor arguments, and to invent counter examples to highlight the flaws of poor ones. Sophisticated speakers of a natural language are aware of salient analogies of form between individual arguments; they make discriminations which display that awareness, and even explicitly draw attention to these analogies of form when they argue "by parity of reasoning." The sophisticated speaker displays what I have called an "intuitive capacity" for distinguishing valid from invalid arguments. What logicians have tried to do is to construct a set of rules whose systematic applications can replace the role of an expert's intuitive capacity. So even though the study of logic comes upon the scene later than the acquisition of intuitive capacities of experts, it is, like the study of grammar, of practical as well as theoretical importance. It is of theoretical importance because it allows us to identify and explain what it is that the expert gains intuitive mastery of—what makes one argument good and another poor, and what enables the expert to distinguish them. And it is of practical importance in helping to sharpen our own intuitive capacity, and as a tool to rely upon explicitly in those sometime cases where our intuitive capacity loses its way.

Logicians have identified a simple, precise procedure for determining when two arguments possess a salient analogy of form: a second argument has the form of the first when the two are related such that the second could be constructed from the first by means of consistent substitutions for the informational components of the first. This simple and precise notion, which we will call "generation," explicates the common sense notion of analogy of structure with astonishing success. But although the central notion is simple, the rules required to specify the nature of the substitutions in complex arguments can be extremely complicated to articulate, as we shall see when we turn later to Predicate Logic. (We begin the term with Sentential Logic.)

In symbolic logic, we construct a system in which we represent arguments. Since we are interested in whether an argument is valid or invalid, and since that is determined by the argument's structure rather than by its content, our schematic representations will ignore, or abstract from, an argument's content. And since validity depends upon an argument's structure, our schematic representations will represent the patterns of the informational components within arguments.

When one argument is valid, every other argument which shares the formal structure of the first is valid too: two arguments which share the same formal structure differ only in content, and that is irrelevant to validity. So we can depict any large class of arguments whose members share the same formal structure by means of a single schematic device; the different "interpretations" of the argument schema will specify the different members of this class of structurally identical arguments. As we shall see, a given argument schema in effect generates all of the members of the class of arguments which display its formal structure.

The function of an argument is to convince someone who already concedes the truth of its premises, that its conclusion is true. The definition of a valid argument is tailored to that function: by definition, its conclusion cannot be false so long as its premises are true. A valid argument thus provides a proof of a conclusion conditionally upon the acceptance of its premises: the conclusion of a valid argument must be true if its premises are. An argument is valid when, but only when, the joint assertion of its premises is inconsistent with the denial of its conclusion (i.e., when but only when it would be contradictory to suppose both that the premises are true and that the conclusion is false).[1] Symbolic logic allows us to understand how the features of valid argument patterns are responsible for making the arguments which share them valid.

Since an argument is valid or invalid solely by virtue of its formal structure, the members of a class of arguments which share the same formal structure are either all valid together or all invalid together. A particular invalid argument may happen to have true premises together with a true conclusion: nevertheless, it is invalid because its premises' being true do not guarantee the truth of its conclusion; a second argument could be constructed which shares the formal structure of the first and which has true premises and a false conclusion. And so it is possible for that sort of argument (i.e., for an argument belonging to the class of arguments sharing the formal structure of that—invalid—argument) to have true premises and a false conclusion. On the other hand, while a valid argument can happen to have a false conclusion, nevertheless, if its premises are true, then it is not possible for the conclusion of this sort of argument (i.e., an argument belonging to the class of arguments sharing the formal structure of this—valid— argument) to be false.

Thus, since an argument is valid or invalid by virtue of its formal structure, and since any argument shares its formal structure with a huge class of other arguments, when we speak of a particular argument as being valid or invalid, we are actually saying something about all the members of a huge class of arguments. And when we represent one member of that class, we thereby schematically depict all the members of the class.

Symbolic Logic is subject to an important limitation. In order to give a systematic and simple account of the subtle characteristics defining the forms of arguments, a system of logic must ignore many of the nuances "encoded" in the structure of a natural language—for examples, differences between "and," "but," and "yet;"[2] features such as tense and mood (indicative as contrasted with subjunctive; simple predication as contrasted with what is possible or necessary; or descriptive as contrasted with prescriptive).

Although special systems of logic have been devised to accommodate many of these features, no one system could handle all of them at once—for the resulting complexity would make the system unmanageable. As a practical matter, this means that a student of logic must be aware of the limitations of the system and recognize when an argument depends upon the sorts of feature which the system does not represent: in that case, his tool will not work.

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Generation and Validity in Sentential Logic

1. Symbols such as " ~ ", "v", " . ", "", and "", are logical connectives; they are "truth functional,"[3] as you will have learned from Copi.[4] Letters of the alphabet are used as sentential variables—to stand in for sentences. A single sentence letter, or a "well formed" sequence of sentence letters, connectives, and devices of punctuation, is a sentence schema[5]—a schematic representation of a sentence. A sentence schema will contain more than one sentence letter when we are representing a compound sentence as comprised of its simpler sub-sentences. When a sentence schema contains more than one sentential letter, it will also contain connectives, and will be punctuated by parentheses, brackets, and braces so that the connectives have a definite rank of dominance: one connective will be the major operator of the sentence schema; another, or perhaps two others, will be the second-rank operators, and so on.[6] An argument schema is a sequence of sentence schemata—a schematic representation of an argument.

2. Let us introduce an important technical word which we will use throughout the term. We will frequently speak of a given schema as generating another schema. It is important to realize that generation is determined precisely by the "mechanical procedure" of substitution. A given sentence schema, call it X, generates another schema, Y, when, beginning with X, we can construct Y by means of the following procedure: Take the first sentence letter occurring in X, and strike out all of its occurrences: then choose some one sentence schema—this may be any schema at all, compound or simple, even the sentence letter you just struck out—and insert it at every place the original letter occurred. Next take the second sentence letter in X and strike out all of its occurrences: in its places insert some one sentence schema—which may be any schema, even the one you just used for the first letter. Continue this procedure until you have substituted for all the sentence letters which originally occurred in X, changing the punctuation as needed in order to preserve the rank of dominance of the original connectives.

Notice that we may only substitute for the occurrences of sentential letters in X, never touching the connectives which occur in X. Notice also that when we substitute something for one occurrence of a sentence letter, we must consistently substitute the very same thing for all the occurrences of that letter. Every formula constructed by this procedure is generated by X. You must not think about what schema X means or represents when you see whether it generates Y: you are simply to make the substitutions "mechanically," the way a computer's find and replace function would do it. To say that X generates Y is to say something about the form of the two schemata; it is not to say anything about their content—about what they represent or mean. (In general it is not true that when X generates Y, X will imply Y.)

3. A given sentence schema will generate many other sentence schemata. (When X generates Y, Y is a "substitution instance" of X; Copi would say that Y has "the statement form" of X.)[7] Two schemata are congruent when each generates the other—when each can be constructed from the other by consistent substitutions for its sentence letters. (Copi would say in such cases that each has "the specific form" of the other.)

A given argument schema will generate many other argument schemata. An argument schema, A, generates another argument schema, B, when by a set of consistent substitutions for the sentence letters in A, each sentence schema in A generates the corresponding sentence schema in B. (When A generates B, B is a "substitution instance" of A; Copi would say that B has "the argument form" of A.)

4. An interpretation of a sentential argument schema is the consistent assignment of some sentence or other to each sentential letter in the argument schema, so that each letter corresponds to one particular sentence about the inhabitants of some chosen universe of discourse; the assigned sentence is the referent or "value" of its corresponding sentence letter in that argument schema. When we interpret an argument schema, we replace the logical connectives with correlative words and phrases from a natural language (e.g., we replace " . " with "and"); notice that the rank in dominance of the logical connectives in each sentence schema will be preserved in the corresponding rank of dominance of their counterparts in the compound sentence which results from an interpretation. This is due to an interpretation's assigning sentences solely to individual sentence letters, never to complex schemata; in interpretation, the logical connectives are left alone until they are replaced by correlative words or phrases in the natural language. A given argument schema has an unlimited number of potential interpretations. Under any particular interpretation, an argument schema represents a specific argument.[8]

Because each sentence assigned to a sentence letter will be either true or false, we can classify the possible interpretations of an argument schema by means of the combinations of truth values of the sentences which could be assigned to its sentential letters. The different lines of a truth table for an argument schema thus correspond to the different subclasses of interpretations for that schema. One such line thus provides a general sketch of, or "recipe" for, a class of interpretations of an argument schema; and together, the several lines exhaust all possible classes of interpretations of the argument schema—each possible interpretation corresponding to one of the lines in the truth table. [9]

5. An argument schema is either valid or invalid. To say that it is valid is to say that there is no interpretation under which it will represent an argument which has true premises together with a false conclusion. And so, just in case there is at least one interpretation of an argument schema under which it represents an argument having true premises and a false conclusion, the argument schema is invalid.[10] When we construct a truth table for an argument schema, we are surveying all of the possible interpretations for that schema: by that means we can determine whether there is a possible interpretation for the schema under which the resulting argument has true premises together with a false conclusion. If no line of the truth table permits true premises together with a false conclusion, then there is no such interpretation and the schema is valid. On the other hand, if even one line permits the premises to be true while the conclusion is false, then there is and the argument schema is invalid. Notice that because it represents an argument, a sequence of sentences, an argument schema may not be said to be true or false.

6. Now let's look at schemata generally—both sentence schemata and argument schemata. Whenever one schema generates another, the second will display the same patterns which determine the array of truth table assignments for the first schema—for those patterns of repetition of the symbols which carry information, around the same connectives having the same rank of dominance, are preserved by generation. This is an important point, one which will later help us understand why natural deduction works: Recall how the mechanical procedure of generation works: we leave the connectives of the original schema, together with their rank of dominance, untouched; we make our substitutions only for the sentential letters of the original. And when we make substitutions, we are required to do so consistently. As a result, the overall patterns of the original schema are automatically preserved. If we use identifiable shapes instead of the sentence letters of the original, we will see that in schemata which are generated, the schematic clumps are still arranged in the same basic patterns as were the original sentence letters; and it was these very patterns which determined the pattern of Ts and Fs on the truth table. And so, if the one schema's patterns make impossible some interpretation of it, then the same patterns, which are preserved by generation, will make such an interpretation impossible of the second [11] Because of this, an important principle of generation holds: If an argument schema is valid (i.e., if its patterns make impossible any interpretation under which its premises are true while its conclusion is false), then every argument schema which it generates must also be valid. And for the same reasons, if a sentence schema is a tautology, every sentence schema which it generates must also be a tautology. We can confirm these fundamental conclusions by exploring different considerations. It will be important to recognize the thoroughgoing parallel between a schema's generating another schema, on the one hand, and its representing a sentence or argument under some interpretation. Both generation and representing under some interpretation depend upon substitution: generation is the construction of a schema by consistently substituting schemata for the sentential letters of the original schema; representation under some interpretation is the construction of a sentence or argument by the consistent substitution of sentences for the sentential letters—together with the replacement of the connectives of the original schema with correlative words and phrases from the natural language.

Because of its dependence on substitution, generation is "transitive": if schema A generates schema B, and B generates schema C, then schema A must also generate schema C. We can see why this is so if we think about generation abstractly for a moment. Remembering that generation leaves untouched the original connectives and their rank of dominance, to imagine some schema we need only think about its sentential letters. Imagine that schema A uses sentential letters P and Q; now imagine also that A generates schema B under the substitution of (R-S) for P and (T-U) for Q; and imagine, further, that B generates schema C under the substitution of W-X for R, and Y-Z for S. Then we can see that schema A will generate schema C under the substitution of (W-X-Y-Z) for P and (T-U) for Q.

Since representation under some interpretation is similarly dependent upon substitution, the ensuing "transitivity" will have this result: if schema A generates schema B (under some consistent substitution for its sentential letters), and schema B represents a sentence or argument (under some consistent interpretation of its sentential letters), then schema A will also represent that same sentence or argument under some interpretation of its sentential letters. In fact, the compound sentences specified in the interpretation of schema A will have, as their component, simpler sentences, those sentences specified in the interpretation of schema B.[12]

Thus, when one schema generates a second schema, anything which the second schema represents under an interpretation, will also be represented by the first schema under some interpretation of its sentential letters. Therefore, once again, if one argument schema is valid (i.e., there is no interpretation under which it represents an argument having true premises together with a false conclusion), then any second schema which it generates must also be valid; for since the first can represent every argument which the second can represent, if the second could represent a particular invalid argument, then so could the first. Conversely, if a given argument schema does generate a second one which is invalid, then the first schema must be invalid also: a valid argument schema cannot generate an invalid schema.

7. When we say that an argument schema is valid or invalid, it is obvious, I hope, that we are talking about a class of arguments—for after all we are talking about all the possible arguments represented under any interpretation of the schema. But what is not so obvious is this: even when we say of a specific argument that it is valid or invalid, although we appear to speak very particularly about this piece of prose, we are implicitly talking about the whole class of arguments which are relevantly similar to it. For in speaking about whether it is valid, we look to the possible arguments which also must be valid if it is valid.

We say of a valid argument that "if its premises are true, its conclusion must be true." When we puzzle out what this means, we realize that it is not satisfied by an invalid argument which merely has true premises together with a true conclusion; for it is "possible" for this (invalid) argument to have true premises and a false conclusion—that is, under some other interpretation of its sentences, it could then have true premises and a false conclusion. And thus, in talking about an argument's validity, we are not merely referring to the "standard" interpretation (i.e., the standard meaning of its sentences referring in the normal way), but any consistent interpretation of (the sentences of) the argument. (Notice how we implicitly treat the argument as we treat a schema.) More intuitively, we can see this implicit generality in common sense terms by realizing that any argument is liable to attack by demonstrating that it is exactly analogous to another argument which admittedly has true premises and a false conclusion. Indeed, we will later exploit this bit of common sense by using those truth table assignments which show an argument to be invalid as a means of devising a counter example to the argument—a counter example which would be convincing even to a layman (possibly even to a roommate!).

Thus, symbolic logic is based upon the identification, and the relatively simple and precise representation, of those features of an argument which are relevant to defining the class of arguments to which it belongs—the sorts of formal characteristics that must be shared by two particular arguments in order for them to be instances of the same pattern of argument. This class of relevantly similar arguments may also be referred to as the class of arguments specified under all possible interpretations of the corresponding argument schema. And so, the original argument is valid just in case no interpretation of its argument schema specifies an argument having true premises and a false conclusion. Remember that a valid argument can have a false conclusion and that an invalid argument can have true premises (even together with a true conclusion); it is the combination of true premises with a false conclusion which a valid argument cannot have and which a invalid argument can have.

8. Unless there is some problem in pairing an argument with its argument schema, then if either one is valid, then so is the other; and if either one is invalid, so is the other. (When we "symbolize" an argument, we normally represent it by means of an argument schema with which it is in effect "congruent;" our proof that the argument schema is valid or invalid is the basis for our saying that the original argument is likewise valid or invalid.)

But we need to note that the accuracy and usefulness of a Sentential argument schema as a representation of an argument in a natural language depends upon the extent to which our truth functional connectives accurately reflect the functions of the corresponding logical words in that argument, and the extent to which the argument's structure is chiefly a matter of the patterns of the simpler sentences within its compound sentences. (For example, subjunctive conditionals are not accurately represented in Sentential Logic: the sentence, "If Boardman were a frog, he would have wings," is not true-by-virtue-of-its-false-antecedent, but instead is false because frogs don't have wings. Again, Sentential Logic is inadequate as a system in which to assess the validity of syllogistic arguments—for their validity depends upon features of sentences which Sentential Logic cannot represent.

Thus, Sentential Logic would not be an adequate system for assessing the validity of arguments that depend on the subjunctive mood or on syllogistic elements.)

So we need to look upon symbolic logic as a tool for understanding what makes a particular sort of familiar argument in a natural language valid or invalid, and as a tool for aiding us in determining whether some argument in a natural language is valid. But because the machinery of a system of symbolic logic is never as subtle and rich as the machinery of a language such as English, it is not a foolproof tool. (This will become evident much later when we begin to symbolize moderately difficult, though commonplace, arguments.)