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Quantification by Sidney Felder the Truth-Functional Connectives

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Quantification by Sidney Felder the Truth-Functional Connectives Quantification by SidneyFelder The truth-functional connectivesconstitute the expressive and deductive “engine” of propositional logic. These combine and relate complete atomic sentences whose internal structures are givenno distinctively logical role whatsoever. From one formal point of view, quantificational logic,inall of its infinite varieties, is marked by its penetration to a sentence’ssub-atomic level, a levelfrom which sub-sentential elements that are common to multiplicities of sentences can be discerned and exploited. It has been found, specifically,that just by isolating those aspects of propositions that can be interpreted as making assertions about all or some individuals of one domain or another,a tremendously more powerful system of logic is created: In augmenting the propositional logic by the universal and existential quantifiers “for all x” (symbolized (x) or (∀x)) and “there exists an x” (symbolized (∃x)), and their associated apparatus, we both 1) enormously expand the sphere of abstract ideas and structures that can be givenprecise and distinctive expression in formal terms and 2) vastly increase the range of propositions that can be brought into non-trivial expressive and deduc- tive relationship with each other. We are nowgoing to define the vocabulary and concepts of what is variously called the FirstOrder Predicate Calculus, Predicate Logic,the Lower Predicate Calculus (LPC), and FirstOrder Logic (FOL). (Another (nowseldom used) older name is the FirstOrder Functional Calculus). First Order Logic is simultaneously the most elementary and the most standard of the quantificational log- ics. (There is, for example, Second Order Logic, Third Order Logic, Quantified Modal Logic, Epis- temic Logic, etc.). The First Order system whose axioms include only logical axioms (only axioms that are logically valid) singles out the class of logically valid formulae. However, there are manyother concepts and structures that can be expressed in a First Order language. Each augmentation of the logical axioms (which belong to all First Order systems) by a logically consistent set of non-valid axioms (proper axioms)defines a class of more definite structures. While manydifferent choices of axioms will single out the same class of structures, and while manyconceptually natural structures are undefin- able in First Order systems, the expressive power of First Order languages completely dwarfs that of sentential calculi. These notes are designed to provide a synoptic orientation to some of the simplest concepts, rela- tions, and structures that can be expressed in a First Order Language. In addition to the sentences and connectivesofpropositional logic, the basic elements of a First Order System include: 1) anon- empty domain or universe of individuals U; 2) aset of individual constants a, b, c,... each denoting some definite individual belonging to U; 3) aset of predicate constants A, B, C,... designating prop- erties and relations among the individuals of U (a monadic predicate corresponds to a subset of U and an n-adic predicate for n greater than 1 corresponds to a set of n-tuples of elements of U); 4) a set of variables x, y,z,... ranging overall the objects of U; 5) the quantifiers (∀x) and (∃x), repre- senting the expressions ‘for all x’ and ‘there exists an x’ respectively; and 6) the logical constant ‘≈’, invariably representing the relation of identity among individuals. While the designations of the predicate constants (predicates) and the individual constants (constants) vary from interpretation to interpretation, the quantifiers and the the identity relation are (likethe truth-functional connectives) Course Notes Page 1 Quantification treated as logical constants, and their interpretations are fixed. (Although an infinite number of dis- tinct formulae can be composed from these linguistic elements, each well-formed formula of the lan- guage is assumed to be finite in length (i.e., to be composed of a finite number of symbolic tokens)). Nowsuppose that U is the set of physical objects and the predicate G is interpreted as the property green.Ifthe constant a is interpreted as ‘the largest leaf on Earth’ and the constant b is interpreted as the sun, the sentence G(a)isinterpreted as the sentence “The largest leaf on Earth is green”, and the sentence G(b)isinterpreted as the statement “The sun is green”. It is a consequence of the interpretation chosen that G(a)istrue and G(b)isfalse. Or,suppose that U is the set of natural numbers {0,1,2,3,...}, that the symbol > represents, as usual, the relation ‘numerically greater than’, and that the constants 2 and 3 represent, as usual, the numbers 2 and 3. Under this interpretation, the sentence >(0,5) (more commonly written 0>5) ,which represents the statement “0 is greater than 5” is false, but the sentence 5>0, which represents the statement “5 is greater than 0”, is true. On the other hand, the simple predicate formula G(x) is neither true nor false categorically: G(x) is true for some assignments of objects to x and false for others. Adopting a slightly different point of view, wesay that the formula G(x) is satisfied by some assignments of values to x and is not satis- fied by others. Thus if we interpret the predicate G as the property green (or,equivalently,the set of all green objects), G(x) will be satisfied by all green objects and will fail to be satisfied by any non-green object. Analogously,x>y is true for some pairs of natural numbers and is false for oth- ers. Specifically x>y is satisfied by all ordered pairs of natural numbers whose first terms are numerically greater than their second terms, and fails to be satisfied by anyordered pair whose first term is not greater than its second term. We now describe howthe quantified formulae are to be understood. Giventhe non-empty domain of discourse U of all physical objects, and supposing for the present that the predicate letter G is interpreted as the color green, the formula (∀x)G(x) states that all objects of U possess the property green, and the formula (∃x)G(x) states that there exists at least one object in U that possesses the property green. In both of these formulae, all occurrences of x are bound,and we say that x appears in this formula only as a bound or apparent variable.The variables embedded in the quan- tifiers themselves, (∀x) and (∃x), are always classified as bound, and the other occurrences of the variable x in the above formulae are bound because (as indicated by the pattern of parentheses) these occurrences fall within the scope of the quantifiers. Because we normally desire the proposi- tion (∀x)G(x) to imply the proposition (∃x)G(x) (i.e., because we want (∃x)G(x) to be true whenever (∀x)G(x) is true), it is standard to arrange things so that the formula (∀x)G(x) has existential import,meaning that the proposition that all objects are green is interpreted in such a way as to imply that there exists at least one green object. This is accomplished not by making anyparticular assumption about the meaning of the universal quantifier (∀x), but rather by imposing the constraint that the universe of discourse U can neverbeempty,that is, by imposing the demand that all admis- sible domains contain at least one object. Under the assumption that U has at least one object, the truth of the statement that “All objects are green” presupposes that at least one object in U is green, meaning that (∀x)G(x) is true in no domain in which (∃x)G(x) is false. If we admitted the possibil- ity of a domain U with no objects, (∀x)G(x) would be vacuously true, and we would be admitting a situation in which (∀x)G(x) was true and (∃x)G(x) was false. (The reader should be aware that there are variations of the standard formalism, called free logics,inwhich empty domains are admit- ted). What about a sentence of the form G(c), which states that the specific object designated by the Course Notes Page 2 Quantification constant c is green? Such a statement is intermediate in logical strength between (∀x)G(x) and (∃x)G(x) (i.e., is logically implied by (∀x)G(x), logically implies (∃x)G(x), and is logically equiv- alent to neither). Forexample, the statement that all objects are green is logically implies the state- ment that Joe’scoat is green, and the statement that Joe’scoat is green implies the statement that there exists at least one green object. (In the peculiar but logically irreproachable universe whose only object is Joe’scoat, the following statements are all simultaneously true: All objects are green; Joe’scoat is green; Some object is green; Nothing other than Joe’scoat is green). Now, consider anydefinite domain U. If it is not the case that all objects in U are green, there must exist some object that is not green. Symbolically,the sentence ¬ (∀x)G(x) is equivalent to the statement (∃x)(¬ G(x) (¬ G is of course the predicate ‘is not green’). Analogously,itisclear that if there does not exist evenasingle green object in U, all objects in U (and there has to be at least one object in U) must be non-green. In symbols, ¬ (∃x)G(x) is equivalent to (∀x)(¬ G(x). This means that it is possible to define universal quantification in terms of existential quantification, and vice-versa.Ifwe takeexistential quantification as primitive,the universally quantified formula (∀x)G(x) corresponds to the formula ¬ (∃x)¬ G(x). In words (interpreting the predicate constant G as green), the assertion that all objects are green is logically equivalent to the assertion that there does not exist evenasin- gle non-green object.
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