A Pedagogical Approach to a Foundation for the Definition of Validity in First-order Logic

Matthew McKeon

Central Connecticut State University
102005.1722@CompuServe.com

Introduction

An argument is valid if and only if (iff) it is not possible for the conclusion to be false while all the premises are true. Depending on what meaning is attached to "possible", the following arguments may be valid or invalid.

(A) John kissed Beth on Tuesday, so they were not 100 miles apart all day Tuesday.
(B) John is a bachelor, so John has no wife.

By the criterion of possibility developed in formal logic, neither is valid; in each case, it will be possible for the conclusion to be false while the premise is true. However, textbooks typically introduce the subject in terms of a notion of validity which conflicts with formal validity, according to which the conclusion of a valid argument cannot be false if the premises are true, because of the logical structure of the premises and conclusion.

For example, in, Introduction to Logic and Critical Thinking by Merrilee H. Salmon, (B) is offered as an instance of a valid argument since "the truth of the conclusion depends solely on the truth of the premiss and conventional linguistic meanings." (pg.71) In, Elementary Logic by Benson Mates, we are told that "An argument is [valid] iff every conceivable circumstance that would make the premises true would also make the conclusion true." (pg.5) Mates takes this to ground the validity of arguments whose conclusions are mathematical truths. Furthermore, on the basis of this characterization, it is claimed that "nobody has yet been able to make the discovery needed for deciding whether the one-premised argument,

(C) The number of stars is even and greater than four, so the number of stars is the sum of two primes
is valid." (pg.4) However, both texts go on to develop the standard account of validity in first-order formal logic which makes (A), (B), and (C) invalid.

Surely, there are different senses of "validity." Nevertheless, an account of the modal characteristic of formal validity would be nice in order to motivate students' interest in the standard formal logic textbook account of validity. This is important for several reasons not the least of which is that generally students do find arguments like (A) and (B) valid in some sense.

There are, of course, textbooks that introduce the subject by illustrating that formal logic makes validity turn on the form rather than the content of arguments. The general strategy seems to be as follows. Students are presented with arguments that are clearly valid, the notion of form or structure is introduced, and then (either explicitly or implicitly) the claim is made that the validity of the arguments is due to logical structure. For example, students perceive without any difficulty that it is not possible for the premises of the following arguments to be true while their conclusions are false.

(D) All men are mortal, Socrates is a man, therefore Socrates is mortal.
(E) All monkeys like bananas, Tom is a monkey, therefore Tom likes bananas.

Most students will accept that (D) and (E) both have the same form: All A are B, S is an A, therefore S is a B. But what needs to be unpacked (and quite often is not) is reason for accepting that the impossibility of true premises and false conclusion in (D) and (E) is due to the form of the arguments. That is, why is logical form relevant to an explanation of why the premises of (D) and (E) can't be true while their conclusions are false? Convincing students, by example, that standard first-order semantics delivers a correct list of valid arguments for (say) a fragment of English does not eliminate the qualms many (good) students have about the meaning and purpose of formal logic. The first-order semantic account of validity may be extensionally correct, but why think that it is intensionally correct?

In this paper, I present a way of motivating the criterion of possibility inherent in the concept of validity which makes (A)-(C) invalid, and which generates interest in logical form by showing how it plugs into a theory of validity. The goal is to represent the standard semantic account of validity as a natural development of a pre-theoretic view of (logical) possibility. I have found this introduction to formal logic to be accessible to introductory students, and refer to it from time to time throughout the semester when I feel students have lost the forest for the trees.

II. The Possible Use Account of Validity in First-order Logic

To understand validity (it's not possible for the conclusion to be false while the premises are true), we must understand how it is possible for sentences to have truth-values other than the ones they actually have. Since truth depends both on the use of words and the way the world is, there are two ways of understanding possible truth and falsity.

We can say that "Bill married Hillary" could be false and "Bill married Maureen" true if the words "Bill", "Hillary", "Maureen", and "married" are used ordinarily but the world were different (imagine that Bill met Ronald Reagan's daughter at a party, fell in love, and married her.) Alternatively, "Bill married Hillary" could be false and "Bill married Maureen" true if the world is as it is but the words "Bill", "Hillary", "Maureen", and "married" were used differently (imagine that "Bill" and "Hillary" refer to 1, "Maureen" to 2, and "married" to the less than relation.)

The first way leads to what might be termed the possible world (PW) account of validity: an argument is valid if there is no possible world in which the premises are true and conclusion false, words used normally. The second way leads to the possible use (PU) account: an argument is valid if there is no possible use of its constituent words under which the premises are true and the conclusion is false, the world remaining as it is.

The PW account of validity makes logic hostage to the resolution of difficult issues in metaphysics. For example, in order to ascertain the PW-validity of the argument whose conclusion is "John is human", we need to search out possible worlds where this is false. But if John is in fact human, is there such a world? I don't know. The PU approach makes such issues irrelevant to logic by counting possible uses for proper nouns and predicates as elements of logically possible situations to be countenanced in fixing the extension of validity. The PU account is preferable because it is easier to say what uses are possible for terms than it is to say what worlds are possible.

Possible uses for terms are not that strange. The connection between a word and its ordinary use, is, after all, a matter of convention: "married" didn't have to mean married. And many words have more than one use that is possible relative to the conventions of English (e.g., indexicals like "you" and "here" change their referent with the occasion of use; there are many people named "Bill"; "married" can mean was married to or performed the marriage of.)

If there were no constraints on possible use, then every argument would be invalid. Hence, the need to develop some constraints. Consider how we discover the truth-values of sentences in which words are used normally. To know the truth-value of, say, "Bill married Hillary", we must know (1) that "Bill married Hillary" is true iff the individual named by "Bill" stands in the relation named by "married" to the individual named by "Hillary", and (2) what these individuals and relations are.

Knowledge of type (1) is structural knowledge, i.e., we process it as a relational-subject sentence where the proper nouns refer to individuals and (married( to a relation between individuals. To obtain knowledge of type (2), we appeal to the conventions of English and to context to supply the individuals and the relation. The conventions of English narrow the possible relations to just a few (look up "marry" in the dictionary); we also know that "Bill" is used to refer to males and "Hillary" to females. Context then filters out all but one pair of individuals (or so we hope.) So, the syntactic classification of a term fixes a broad range of possible uses for it due to the fact that syntactic elements have more or less well-defined semantic roles. A name like "Bill" is broadly used to refer to objects, and a predicate (married( is broadly used to refer to a relation between objects. This range is narrowed by the specific conventions of English and dictionary definitions.

Formal logic takes structural constraints as basic in characterizing possible uses. We imagine that sentences have a structure which determines how its unstructured parts (or elements) may be used in relation to one another and how the truth or falsity of the sentence depends upon such a coordinated use of elements. A possible use will be any coordinated use of the elements of the sentences constituting an argument that is consistent with their structure. So, the range of possible uses for a term will be fixed solely by its syntactic classification; as such, these uses will go far beyond what is possible according to the conventions of English; e.g., Bill could refer to 1 and (married( to the less-than relation.

In terms of such a picture, an argument is valid if there is no possible use of the elements of the sentences of which it is comprised for which, in accord with their structure, the premises are true and the conclusion is false. In order to ascertain whether a given argument is PU-valid, analyze the constituent sentences down to their elements and then ask whether there is any use of these elements on which the premises are true and the conclusion false. If so, then the argument is invalid, the use constituting a counterexample in which the premises are true and the conclusion false. If not, the argument is valid. Obviously, this makes validity relative to the structures we are prepared to see in sentences. Hence the need to develop a catalog of the types of structures we're prepared to see in English sentences (e.g., truth-functional, subject-predicate, identity, descriptive, and quantificational structures.) Sentences from a standard first-order language represent the logical structures (or logical forms) of corresponding ordinary language sentences. The analog of the notion of a possible use of elementary English expressions is an interpretation of a formal language.

III. Conclusion

The existence of a satisfying interpretation for (or a model of) the premises and the denial of the conclusion establishes the invalidity of a formal argument only if there is reason to believe that the interpretation represents a logically possible situation. Also, the fact that no interpretation satisfies the premises and denial of the conclusion establishes the validity of a formal argument only if the class of interpretations corresponds (many-to-one) with the class of logically possible situations. Based on my teaching experience, establishing these claims by treating the model-theoretic notion of logical possibility as basic fails to bridge the gap in the minds of many students between doing formal logic and real reasoning. Juxtaposing the PU theory of validity with the ordinary language definition of validity and the model-theoretic one is an attempt to bridge such a gap.

As Quine remarks, "Models afford consistency proofs; also they have heuristic value; but they do not constitute explication. Models however clear they may be in themselves, may leave us still at a loss for the primary intended interpretation." (W.V. Quine, "Book Review of Identity and Individuation" Journal of Philosophy 69 (1972): 488-497, 492) Comprehension of what is supposed to be going on in model-theoretic semantics turns on seeing why models succeed as representations of logically possible situations. In terms of the picture sketched here, the existence of a model for a sentence shows that it could logically be true because (in accordance with the PU account of validity) a logically possible situation is a use for the (non-logical) terms occurring in a sentence which makes it (actually) true. I conclude by indicating three sources of tension for the PU theory of validity, which I try to bring to the attention of students at some point during the semester.

(1) Attempts at application reveal that analyzing the logical forms of sentences is difficult. One of the primary challenges of a course in formal logic is removing the ad hoc appearance of the uncovering of logical form. The case for the PU account of validity over the PW account rest on logical form being more accessible than modal reality. This is not obvious.

(2) Clearly, PW-validity is not co-extensive with the classical account of validity: it makes the validity of an argument turn on things other than its logical form. But what about PU-validity? Recall that possible uses for words are restricted only by a broad construal of what sort of meanings are appropriate to what sort of words. Prima facie, failure of reference seems to be a possible use of a term, but this conflicts with the standard requirement that domains be non-empty. If predicates can be used so that they are true of nothing, why can't names fail to refer?

(3) The PU account of validity, like the Tarski-Bolzano account, makes the upper bound of the cardinality of the world's domain the determinant of the upper bound of the domains we may appeal to in fixing the extension of validity. So, the PU account secures standard first-order logic only if there is in fact a denumerably infinite totality of individuals (e.g., the argument, (xFx, so (xFx, is PM-invalid only if Parmenides is wrong, i.e., there is in fact more than one thing.) The argument against the PW account is not that it makes logic metaphysical, but that it makes logic too metaphysical. Nevertheless a potential problem for the PU account is that it makes the extension of validity contingent on the make up of the actual world. The later conflicts with the idea that an argument is valid or invalid regardless of the way the world happens to, in fact, be.