| [reposted August 6, 2008] | |
Analytic Sentences & Valid Arguments in 1st order Predicate Logic
from Supplement for Symbolic Logic (c) Boardman
orresponding to tautologies in Sentential Logic are analytic sentence schemata in First Order Predicate Logic. You will remember that a tautology is a sentence schema which is true under any consistent interpretation of its sentential letters; no line of its truth table is false, each line sketching a class of interpretations (classified according to the combinations of truth values of the sub-sentences represented by the sentential letters). For a tautology is true independently of the content which its sentential letters might represent: its structure and the truth-functional definitions of its logical connectives are what make it true.
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The sentence schemata in Predicate Logic are used to represent sentences which make claims about the individuals inhabiting some universe of discourse--the subset of the world which we choose to represent--and their properties and relations to one another. The typical sentence schema is contingently true (or false), its truth value depending upon whether the assertion the interpretation assigns to it happens to be true in the chosen universe of discourse. But some sentence schemata in Predicate Logic--like the tautologies of Sentential Logic--are true no matter what their interpretation--that is, they are logically true, or analytic . (Tautologies are a specific sub-class of analytically true sentence schemata--ones which are truth functional.) Like tautologies, the analytic sentence schemata of Predicate Logic are made true by their structure and the definitions of their logical connectives and quantifiers rather than their content.
For "(x)(Fx ⊃ Gx)" to be contingently true is for there to be an interpretation of "F" and "G" into some domain of discourse under which "Fx ⊃ Gx" remains true for each substitution of a constant--i.e., for each substitution of a name for an individual in that domain: e.g., "Fa ⊃ Ga" is true and "Fb ⊃ Gb" is true, and so on. And if "(∃x)(Kx · Lx)" is contingently true, then under some interpretation of "K" and "L" into some domain of discourse, "Kx · Lx" is true for at least one substitution of a constant for "x": e.g., "Ka · La" is true or "Kb · Lb" is true, and so on. But if a sentence schema in Predicate Logic is logically true, then it will be true under any interpretation into any (non-empty) domain of discourse: for when a schema is logically true, its truth doesn't depend upon whether the chosen domain of discourse has just one individual or many more, on whether the individuals are humans or pigs or shapes or numbers, or on whether the properties and relationships are tangible or abstract--so long as the interpretation is followed consistently throughout the schema.
An interpretation of a sentence schema in Predicate Logic is the assignment of a non-empty domain, or universe, of discourse for that sentence schema such that each predicate letter and relation letter stands for some property or relationship within that domain, each individual constant stands for some specific individual within that domain, and the individual variables "range over" the individuals of that domain. On a given interpretation of a sentence schema, then, the schema will represent a sentence which is true or false about the individuals of a particular universe of discourse.1
Consider the universe of discourse which will now be displayed:
; our description of it,
"Do = {a, b, c}," says that the domain of objects consists of three individuals. We will let "a" stand for the left-most (the blue) triangular spot, "b" stand for the other (red) triangular spot, and "c" stand for the (green) square spot, "S-" stand for "- is square," "T-" stand for "- is triangular," and "R- -" stand for "- is to the right of -." Given our assignment, "Tb · Rbc" represents the sentence, "b is triangular and b is to the right of c;" under the interpretation provided, that sentence is false (since the second conjunct is false). On the other hand, given this same interpretation, "Sc · Rcb" is true, and so is "(x)[Tx ⊃ (∃y)(Sy · Ryx)]," which, under that same interpretation, says that every triangular spot has a square spot to the right of it. The sentence schemata exhibited so far are not analytic: while some are true under the interpretation we provided, they would be false under some other interpretation--either because, for example, we let "R- -" stand for "- is below -," or because we chose some different universe of discourse in which the corresponding sentences are false.But now consider the sentence schema, "(x)Tx ⊃ (∃y)Ty." Under our present interpretation, this schema represents "if everything is triangular, then something is triangular;" and that is true. More interesting for logicians, however, is the fact that this sentence schema will remain true under any interpretation into any non-empty domain of discourse. No matter what (non-empty) domain of individuals is considered, and no matter what property "T-" is allowed to stand for, it will be true that if all individuals have that property then at least one of them does. So this sentence schema is analytic. Or consider another schema: "Rca ∨ ~Rca;" under our present interpretation, it represent the sentence, "either c is to the right of a or it isn't." Since this schema is analytic, it will be true no matter what relationship we let "R- -" represent, and no matter what individuals we let "a" and "b" stand for.
If we have an "open" sentence schema, one which has at least one free variable, then it is analytic just in case--that is, if and only if --it is true for all interpretations into any non-empty domain. Thus, "Tx ⊃ (∃y)Ty" is analytic just in case "(x)[Tx ⊃ (∃y)Ty]" is analytic. (Logicians would call the second formula the "universal closure" of the first; and so, an "open" sentence schema is logically true just in case its universal closure is logically true.)
We can now observe some important parallels to things which we learned earlier in Sentential Logic. In Predicate Logic, it continues to hold that if a sentence schema is analytic, then every sentence schema which it generates must also be analytic. Thus, since "(x)[Tx ⊃ (∃y)Ty]" is analytic, so is
"(x)[(Tx ∨ Sx) ⊃ (∃y)(Ty ∨ Sy)]," and so is "(x)[Rxa ⊃ (∃y)Rya]," and so, also, is"(x)[(∃z)Rxz ⊃ (∃y)(∃z)Ryz]." Recall § 6 in "Generation and Validity in Sentential Logic" and also the information you will find at its footnotes, [9], and [11]: these same principles of generation's transmitting validity and logical truth and the principles due to the parallel of generation to interpretation (since both depend upon substitution ), continue to hold. Consider, for purposes of illustration, the following two quantified sentence schemata:
(x)[Mx ⊃ (∃y)My] generates (x){[Fx · (∃z)Rxz] ⊃ (∃y)[Fy · (∃z)Ryz]}.
Imagine that we specify an interpretation of the second schema: we assign to it some (non-empty) domain in which its variables range over the individuals of the domain, and in which "F-" stands for some property, call it F*, and "R- -" stands for some relation, call it R*; under that interpretation, the second schema will represent a specific sentence, S*, which says something about the individuals of a particular domain. Since the first schema generates the second, there will be some interpretation under which the first sentence schema will represent that same sentence, S*: in this interpretation of the first schema, we assign the same domain as before, and we now specify that an individual is "M" when and only when that individual is both F* and bears relation R* to some individual. In this way, any sentence which can be represented by the second schema can also be represented by the first. Thus, since the first is analytic (it cannot represent a false sentence), the second schema must also be analytic: for if the second were able to represent a false sentence, then since the first schema generates the second, it would be able to do so as well.
One of the familiar and important principles which our example illustrates is the close parallel between generation and interpretation: the increased complexity in the procedure of substitution in Predicate Logic (as compared with Sentential Logic) is mirrored in the increased complexity in the procedure of interpretation. When we provide an interpretation in English of a quantified schema, we are in effect substituting a predicate from the natural language for a predicate letter, a relation from the natural language for a relation letter, a proper name for a constant; and when we specify that "M-" stands for "- is a mother," we are in effect substituting "- is female and - is the parent of someone" for "M-." (Note how this parallels our earlier formula, "Fx · (∃y)Rxy.") Thus, generation depends upon substituting one propositional function for another, while interpretation in effect substitutes the English analog of a propositional function for a schematic one.2 Thus, the abstract device,
() [
![[Phi]](Phi_14x13.GIF)
)
serves to represent the shared pattern of the sentence schemata used in the illustration above, in virtue of which the first schema generates the second. And underlying these patterns are the familiar patterns of Sentential Logic, enabling us to continue using the Sentential inference rules, the Sentential Equivalence rules, and Conditional Proof: if we take any analytic sentence schema and strip away all of its quantifiers and all of its variables and constants, we find a tautology (for example, "Just as an analytic sentence schema generates only analytic sentence schemata, so it continues to hold in Predicate Logic that a valid argument schema generates only valid argument schemata. A "quantified argument schema"3 is valid if and only if, for every interpretation into any non-empty domain, no argument is specified which has true premises and a false conclusion. And if one "quantified argument schema" generates a second, any argument specified under an interpretation of the second schema will be specified under some interpretation of the first--in the way we have illustrated two paragraphs above. So, if the first generates the second, then if the first is valid, the second cannot represent an argument which has true premises and a false conclusion. Furthermore, if for some argument schema we can specify an interpretation into some domain under which the resulting argument has true premises and a false conclusion, we will have shown that argument schema to be invalid .
In Sentential Logic, we can prove an argument schema to be invalid by specifying a set of truth assignments to the sentential letters which results in true premises and a false conclusion; we thereby show that one line of the argument schema's truth table allows an interpretation having true premises and a false conclusion. In Predicate Logic, an argument schema typically consists of sentence schemata which are not truth functional: quantifiers, not truth functional connectives, are the major operators of the typical "quantified argument schemata." And quantifiers are not truth functional operators since they may represent an infinite number of individuals; the truth value of a quantified sentence schema is therefore not a function of the truth values of any finite number of simple sentence schemata. Nevertheless, we can test the validity of a quantified argument schema indirectly by constructing and testing its truth functional proxy for some (non-empty) domain of a specified (finite) number of individuals; each of the premises, and the conclusion, in the original schema will be equivalent in that domain to its truth functional counterpart in the proxy. Because it is comprised of truth functional sentence schemata, a proxy may be tested for validity by the short-cut method of truth value assignment, or by means of a truth table. And if a proxy proves to be invalid, it will provide a "recipe" for constructing an interpretation of the corresponding quantified argument schema into the same domain which will serve as a counter example, or refutation, to that argument schema. Thus, if the original quantified argument schema is valid, then all of its corresponding proxies must also be valid. If any one of the proxies corresponding to a quantified argument schema is invalid, then since it is therefore possible for the schema to have an interpretation into some domain under which its premises are true while its conclusion is false, the schema itself is invalid. Note that even though one particular corresponding proxy is valid, the original quantified argument schema might nevertheless be invalid: to be valid, every corresponding proxy (for every non-empty domain) must be valid.
Consider this argument schema: "(∃x)(Px · Bx) / ∴ (x)(Px ⊃ Bx)." Its corresponding truth functional proxy, for
Do = {a}, is"Pa · Ba / ∴ Pa ⊃ Ba." This proxy is a valid truth functional argument schema; one possible interpretation of it, into the universe of discourse of philosophers at Lawrence in 1998 who were born in Illinois, isa = Boardman, P = teaches Philosophy of Law, andB = is bald. Under this interpretation, both the original argument schema and its truth functional proxy have true premises and a true conclusion.Of course, you can readily see that, in a domain of one individual, if one individual has a given property, then "every" individual in the domain must have the same property. But, after all, if the original quantified argument schema were valid, the truth of its premises would guarantee the truth of its conclusion for any non-empty domain; and this it fails to do. For consider a second truth functional proxy, for
Do = {a, b}: "(Pa · Ba) ∨ (Pb · Bb) / ∴ (Pa ⊃ Ba) · (Pb ⊃ Bb)." An interpretation of it, into the domain of discourse of philosophers teaching at Lawrence in 1998 who revere Bertrand Russell, is a = Boardman, b = Ryckman, P = teaches Symbolic Logic, and B = is bald. Since this truth functional proxy has a true premise and a false conclusion under at least one interpretation (as Ryckman is not bald), it is invalid . So the quantified argument schema with which we began is thereby shown to be invalid , since at least one interpretation of it into one non-empty domain will have true premises and a false conclusion.Because of the multiplicity in the domain that may have to be specified, the procedure of constructing a truth functional proxy of a quantified argument schema is not a feasible procedure for proving such a schema to be valid; for in order to be valid, the corresponding proxy for a domain having a million, or even an infinite, number of members, must also be valid. Thus, testing a proxy is only useful as a procedure for proving invalidity; our failure to find an invalid proxy does not prove that the quantified argument schema is valid.4 To prove a quantified argument schema to be valid, we must show that we can derive the conclusion from the premises in a series of steps, each of which is generated by one of our original (Sentential) rules of inference or equivalence, by Conditional Proof, or by the additional inference rules and logical equivalences for "quantified schemata." In other words, to prove a quantified argument schema to be valid, we must derive its conclusion from its premises by means of Natural Deduction.
Finally, despite the greater complexity of First Level Predicate Logic, we need to be aware that the basic principles of logic underlie it as well as Sentential Logic. It is worth recalling the things which you read earlier at L 1-2. To say that a particular argument is valid is actually to be making a very general claim--that no argument similar to it has true premises and a false conclusion. When, in talking about arguments represented in Predicate Logic, we speak of "any interpretation into any domain," we are simply requiring that an argument's validity be applicable to all arguments which are relevantly similar to it. Thus, whether a given argument might possibly have true premises and a true conclusion is irrelevant to whether it is a valid argument.5 To be valid, it must be impossible for the argument--i.e., for any argument sharing these formal characteristics--to have true premises and a false conclusion.
FOOTNOTES:
1 An interpretation of an argument schema is an assignment of a non-empty domain which simultaneously provides an interpretation for each of the premises and for the conclusion of the argument schema; under a given interpretation, an argument schema will represent a specific argument whose sentences are true or false of the individuals in a particular domain of discourse. Parenthetically, we must restrict ourselves to non-empty domains because in an empty domain, every universally general schema is true, and every existentially general schema is false.
2 Throughout this procedure, we in effect make substitutions solely for the individual constants and variables, and predicate letters and relation letters, leaving the logical operators to be replaced by their functional counterparts in English. It is evident that the sorts of restriction on substitution in Predicate Logic are analogous to those in Sentential Logic--never substituting for complexes but solely for simple, informational elements, and then, only consistently. In that way, it is guaranteed that the form of the original schema is preserved both by generation and by interpretation. And, as we have already noticed earlier in the course, it is the form of sentence schemata (and argument schemata) which is responsible for logical truth (and validity).
3 A "quantified argument schema" is an argument schema containing at least one "quantified sentence schema;" a "quantified sentence schema" is a sentence schema containing at least one prefix.
4 As Copi points out on page 81 (and on p. 136), if a "quantified argument schema" uses only one-place predicates, the general decision procedure of testing a proxy having as many individuals as 2n, where n is the number of distinct predicates, is available; but if it contains relations, a finite domain may no longer do--for no general ("effective" or "mechanical") decision procedure of validity then exists.
5 Recall the example lately discussed: the argument, "Some Lawrence philosopher born in Illinois who teaches Philosophy of Law is bald; therefore, all Lawrence philosophers born in Illinois who teach Philosophy of Law are bald," has true premises and a true conclusion, but is, nonetheless, invalid.