SUBSTITUTION QUANTIFICATION

Reified formulas are supposed to be a technique for transforming into and for explaining the meaning of natural language sentences through quantification logic syntax and semantics. Whether such a transformation is possible is itself a dispute in philosophy. This dispute is sometimes referenced as a distinction between Platonic and nominalist interpretations of syntax or semantics. In practice, reification is usually done by ordered pair reified formulas by there is an option for a less known substitution technique exemplified by the following taken from Kit Fine and Steve Kuhn’s Logic Elements and Foundations.:

Suppose we regard the reified formula A[δ], not as the ordered pair <A, δ>, but as the result A(d1,…, dn/x1,…, xn) of substituting the objects d1 =δ(x1), … , dn = δ(xn) for the variables x1, … , xn that occur free in A. Let d1 be Mont Blanc, d2 be the expression v1, d3 the expression v1v2, d4 the expression v2v3, and d5 the expression v3.

 

How might one make sense of the idea of substituting an object for an expression? What are P1v1(d1/v1), P1v1(d2/v1), P2v1v2(d2,d4/v1,v2), and P2v1v2(d3,d5/v1,v2)? How do these examples give rise to difficulties for the proposed account of reified formulas? After giving a summary of the conceptual problems at issue, I will argue that such substitution makes sense because it provides a means to deal with the ontological nature of singular terms such as Mount Blanc. The difficulties presented are that though in general there is no difference in truth conditions for ordered pair reified formulas from the truth conditions of the substitution technique for reified formulas, there will be a difference in truth conditions where some objects in the infinite domain of an ordered pair do not have a singular term or other linguistic expression in the countable object language used for the substitution. In such a situation, a sentence of the form ∃xA may be true in the former but false in the latter by the lack of a true substitution instance of A(dn/xn).

 

I. Historical Background

The best-known founders of the modern logical calculi are George Boole and Gottlob Frege who did not see their logic as providing for reification or as a technique for transformation of natural language into logic nor for explaining its relationship to what exists. They saw their logic calculi as a normative creation of a new language that would explain what exists and that would allow for reasoning from existing self-evident true premises to necessary conclusions in the same way as they saw mathematics doing. Boole sought to transform logical thought into mathematics; “‘A successful attempt to express logical propositions by symbols, the laws of whose combinations should be founded upon the law of the mental processes which they represent, would , so far, be a step toward a philosophical language’”. Frege sought to transform mathematics into logic; “the independence of logic from the mind … logic, he claimed, codifies the laws of truth, not of thought, the laws of how we ought to think if we wish to aim at the truth, not of how we do actually think”. Frege’s work improved upon Boole’s work by using the function analysis of mathematical terms as a richer and more exact expression of syntax in logic than the subject predicate analysis found in the traditional logic used by Boole. Frege thought of a function expression such as the addition of two numbers as containing gaps that allow places for arguments to be written; i.e., “2 + 3″ is not merely the relation “+” but the relation “( ) + [ ]” allowing for expression of a wider range of propositions and relationships between propositions. Frege in the semantics of his logic, as is true of all logicians trying to expand logic behind mathematics regardless of whether it be Aristotle or the Scholastics, ran into a problem of ontological commitment and in explaining the relationship between logic and reality.

In his logic, Frege made an absolute semantic distinction between objects and concepts. Objects have the following attributes: they name and pick out referents; can be described, defined, or predicated about; they are “saturated”, that is complete unto themselves, i.e. proper names such as Hesperus and Phosphorus. Concepts have the following attributes: they name and pick out properties of referents and therefore predicate upon objects but they do not name and pick out referents; concepts and their natures can be hinted at and used but not discussed directly; they are “unsaturated” or incomplete, i.e. “__ is the morning star”; “__ is the evening star”. (In natural language grammatical terms, objects are subjects and concepts are predicates.) An object and a concept together make a proposition or sentence that is true or false through the truth values contributed to the proposition or sentence by its component expressions. (Most logic now uses the term sentence instead of proposition. Sentences can change in truth value over time whereas propositions cannot. When such change occurs, sentences are expressing different propositions.) For Frege, there was no such thing as reification formulas given that ordinary natural language could not function as a means for truth conditions in his logic because it was “rife with vagueness, ambiguity, lack of logical perspicuity, and, indeed, logical incoherence”.
However, even without reification, he still had to posit a relationship between the subject/predicate semantics of his logic and of reality:

A definition of a concept (of a possible predicate) must be complete; it must unambiguously determine, as regards any object, whether or not it falls under the concept (whether or not the predicate is truly ascribable to it). Thus there must not be any object as regards which the definition leaves in doubt whether it falls under the concept; though for us men, with our defective knowledge, the question may not always be decidable. We may express this metaphorically as follows: the concept must have a sharp boundary. If we represent concepts in extension by areas on a plane, this is admittedly a picture that may be used only with caution, but here it can do us good service. To a concept without a sharp boundary there would correspond an area that had not a sharp boundary-line all around, but in places just vaguely faded away into the background. This would not really be an area at all; and likewise a concept that is not sharply defined is wrongly termed a concept. Such quasi-conceptual constructions cannot be recognized as concepts by logic; it is impossible to lay down precise laws for them. The law of excluded middle is really just another form of the requirement that the concept should have a sharp boundary. Any object A that you choose to take either falls under the concept or does not fall under it; tertium non datur. E.g., would the sentence ‘any square root of 9 is odd’ have a comprehensible sense at all if square root of 9 were not a concept with a sharp boundary?

 

“Language is not constructed from a logical blueprint, Frege insisted, and one can no more learn logic from studying grammar than on can learn how to think from a child.” Thus arises the famous problem of “Hesperus is Phosphorus” or “Hesperus the evening star is Phosphorus the morning star”. Both these expressed objects have the same singular referent and the expressed properties are the same and came be predicated to the exact same objects, therefore these propositions and sentences seem to create tautologies in the form of the identity function F = F but since the referents are the same they seem to create a necessarily true ontology as a matter of their logic. This is problematic. Neither sentence is in fact necessarily true; it is possible and it was the state of affairs for millennia that the truth conditions of both formulas were “false”.

Further, the object and concept distinction seems to create an ontology of non-existing referents or allows for a logic that violates the law of the excluded middle. “Pegasus does not exist” has a truth functional meaning of “true” but what is the referent of the object “Pegasus”? What does the referent of “Pegasus” contribute to the truth function? It appears the object Pegasus must have a referent and thus exist in order to prove the falsity of its non-existence which makes no sense. Or, there exist meaningful expressions predicating properties to objects that have no referents, but then how does the logic determine its truth condition? For example, by the law of the excluded middle one of the following propositions must be true: “the present king of France is bald” or “it is not the case that the present king of France is bald”. Propositions that predicate of no objects or have predication of degree zero are simply atomic formulas; Fv and ¬Fv are true or false if and only if the domain of the function is a non-empty set, i.e., there is some object to which to assign or to predicate the extension of the predicate “is bald” so as to determine whether it is true of false of the object. Since the object “present king of France” is not in the extension of all or of any predicates, we have simpliciter expressions or sentences F and ¬F for which there is no set of objects for determining their truth function.

In solution of these problems, Frege posited the existence of the sense or meaning of expressions as something distinct from any reference they may or may not have. It was this property of sense that contributed to the truth value of sentences and proposition. The sense of Hesperus and of the evening star is not the same as that of Phosphorus and the morning star and thus they cannot be represented in logic by the same sentence letter or formula regardless of having the same referent. Likewise, it is the sense of Pegasus that contributes the object and contributes to the truth function of logical sentences and formula. For Frege, existence became just another concept that can be predicated to objects through their sense for truth value:

.. Mont Blanc with its snowfields is not itself a component part of the thought that Mont Blanc is more than 4,000 metres high … The sense of the word ‘Moon’ is a component part of the thought that the moon is smaller than the earth. The moon itself (i.e. the Bedeutung of the word ‘Moon’) is not part of the sense of the word ‘Moon’; for then it would also be a component part of a thought. We can nevertheless say: ‘The Moon is identical with the heavenly body closest to the earth’. What is identical, however, is not a component part but the Bedeutung of the expression ‘the Moon’ and ‘the heavenly body closest to the earth’. We can say that 3+4 is identical with 8-1; i.e. that the Bedeutung of ‘3+4’ coincides with the Bedeutung of ‘8-1’. But this Bedeutung, namely the number 7, is not a component part of the sense of ‘3+4’. The identity is not an identity of sense, nor of part of the sense, but of Bedeutung …

This solution was unacceptable to many philosophers because it created this ontological Platonic world of the “sense” or of the meaning of words in violation of Ockham’s Razor that was seen to create more problems than it solved.

As a different solution of the semantic problems presented by the developing logical calculi, Bertrand Russell posited his famous Theory of Descriptions in which the singular objects of referring expressions are treated logically not as objects but as concepts or descriptions stating relationships among objects, i.e. singular objects become conjunctions of descriptions. Thus the statements “the present king of France is bald” or “it is not the case that the present king of France is bald” become: ∃x((KFx ∧ ∀y(KFy→x=y)) ∧ Bx); ¬∃x((KFx∧∀y(KFy→y=x) ∧ Bx)). The first formula is false and the second is true, thus the law of the excluded middle is preserved. The same modification can be done to singular terms such as Pegasus by making them a description: “x has wings”; “x is a horse”; “if y has wings and y is a horse, then y = x” or ¬∃x(Hx ∧ Wx ∧ ∀y((Hy ∧ Wy) → y = x)). Some logicians would use a verb description: “x pegasizes” or ¬∃x(Px) where “pegasusizes” is a predicate like “bald”, “tall”, or “winged” and thus no longer require a referent to existing entities to prove sentences of their non-existence to be true or false.

Thus for Russell, existence also became a second order function but for Russell it belonged to neither objects nor concepts but was a relationship among propositions. For Russell, it was logically possible for existential objects to be truth functional components of logic:

… Concerning sense and Bedeutung, I see nothing but difficulties which I cannot overcome … I believe that in spite of all its snowfields Mont Blanc itself is a component part of what is actually asserted in ‘Mont Blanc is more than 4,000 metres high’. We do not assert the thought, for this is a private psychological matter: we assert the object of the thought, and this is, to my mind, a certain complex (an objective proposition, one might say) in which Mont Blanc is itself a component part. If we do not admit this, then we get the conclusion that we know nothing at all about Mont Blanc. This is why for me the Bedeutung of a proposition is not the true, but a certain complex which (in the given case) is true. In the case of a simple proper name like ‘Socrates’, I cannot distinguish between sense and Bedeutung; I see only the idea, which is psychological, and the object. Or better: I do not admit the sense at all, but only the idea and the denotation. I see the difference between sense and Bedeutung only in the case of complexes whose Bedeutung is an object, e.g. the values of ordinary mathematical functions like ξ + 1, ξ2.

In various forms, analytic philosophy has been dealing with these existential problems ever since. In various forms, they are an unavoidable part of any reified formulas in logic.

 

II. Historical Development

In the following century after the work of Frege and Russell, the early conceptions of the logical calculus as an ideal syntax to which semantics adds an interpretation led to great developments and progress. Historians of philosophy describe this work and related work as analytic philosophy. The further work of many other logicians such as Ludwig Wittgenstein in his Tractatus Logico Philosophicus and especially of the logical positivists seeking to reduce all propositions to objects that can be verified by experience eventually lead to the creation of meta-logic through the recursive formal definition of truth created by Alfred Tarski in which a meta-language is used to formalize the syntax and semantics of an object language. Though Frege “viewed his formal notation as a novel language, self-contained and disjoint from every language”, modern logic has long forgotten this restriction and sees itself as the means for understanding all languages including natural languages. Thus the technique of reified formulas is a necessary attribute of meta-logic and with it there come issues of ontological commitment.

The term “object” in quantification semantics is taken “in the broadest possible sense. Thus, ordinary physical things, actions, sets, classes, properties, and any things of the kind one might predicate something of in ordinary language count as objects.” The truth value of a logical formula is defined with respect to a model interpretation. An interpretation assigns truth values to sentences letters and thus to the predication, functions, and constants in the formula. The intension of a sentence or formula does not matter only its extension and truth values determine the truth of the model and its interpretation.

The advantage of treating a reified formula A[δ] as the ordered pair <A, δ> is that it does not matter to its truth conditions what objects are in the domain of the model nor what properties among them the predicates of the language denote — all that matters is their extension. For all isomorphic models the exact same closed formulas will be true. Through use of the Lowenhiem-Skolem, it can be argued that there is no need to consider any models other than those whose domains consist of mathematical objects because every model is isomorphic to one of these — which was the goal all along because the language of science is the language of numbers. Thus, for example, logic can be used to support the Quine-Putnam Indispensability Argument for ontological commitment to numbers.

Treating instead a reified formula A[δ] as A(d1,…, dn/x1,…, xn), that is as the substitution of the objects d1 =δ(x1), … , dn = δ(xn) for the variables x1, … , xn that occur free in A, this treatment has different semantics. This formula is true iff all or some of the formulas you get by replacing the variables with the individual constant d1 … dn are true. Thus, there is no need for assignments. This interpretation associates with the variables not a domain of quantification but rather a set or sets of substitution of expressions from the object language. The truth conditions for this substitution are created in terms of the truth conditions of appropriate substitution instances of A(d1,…, dn/x1,…, xn) in which each occurrence of the variable has been replaced with a linguistic expression. The formulas P1v1(d1/v1), P1v1(d2/v1), P2v1v2(d2,d4/v1,v2), and P2v1v2(d3,d5/v1,v2) of Exercise 1[p] begins the substitution with the individual term Mont Blanc into various expressions and thus the truth of the formulas is dependent on the truth of that substitution and the following substitutions into the expressions. However, it does not matter whether the substitution is of a singular term such as Mont Blanc (d1) whereas the remaining substitutions may be of general terms, expressions, or other objects:

The distinction between general and singular terms may seem overrated. After all, it may be objected, the singular term differs from general terms only in that the number of objects of which it is true is one rather than some other number. Why pick the number one for separate attention? But actually the difference between being true of many objects and being true of just one is not what matters to the distinction between general and singular. This point is evident once we get to derived terms such as ‘Pegasus’, which are learned by description (§ 23), or such as ‘natural satellite of the earth’, which are compounded of learned parts. For ‘Pegasus’ counts as a singular term though true of nothing, and ‘natural satellite of the earth’ counts as a general term though true of just one object. As one vaguely says, ‘Pegasus’ is singular in that it purports to refer to just one object, and ‘natural satellite of the earth’ is general in that its singularity of reference is not something purported in the term. Such talk of purport is only a picturesque way of alluding to distinctive grammatical roles that singular and general terms play in sentences. It is by grammatical role that general and singular terms are properly to be distinguished.The basic combination in which general and singular terms find their contrasting roles is that of predication: ‘Mama is a woman’, or schematically ‘a is an F’ where ‘a’ represents a singular term and ‘F’ a general term. Predication joins a general term and a singular term to form a sentence that is true or false according as the general term is true or false of the object, if any, to which the singular term refers.

III. Sense and Difficulties

In general, this difference in truth conditions does not result in the truth conditions for ordered pair reified formulas to be any different from the substitution technique for reified formulas. This is true for the domain of numbers and mathematics because we have a name for each member of the intended domain. However, a difference in truth conditions may occur where some objects in the infinite domain of an ordered pair do not have a singular term or other linguistic expression in the countable object language. In such a situation, an existential sentence of the form ∃xA may be true in the former reified formula but false in the latter by the lack of a true substitution instance of A(dn/xn).
For example, take the claim “something is 4000 meters high”. On a reification using an ordered pair model interpretation, this formula ∃x(Hx) is saying that for some object in the domain, that object is 4000 meters high. On a substitutional reading, the formula is saying that for some d, “x is 4000 meters high” is true. In the case of d being Mont Blanc, the interpretation is true. However, if the object language had no name for the existing Mont Blanc nor for any object in the domain that is 4000 meters high, the former would still be true but the latter false.

Treating a reified formula A[δ] as A(d1,…, dn/x1,…, xn) may help with the existential and ontological commitment problems of quantification logic that have been around since Frege and Russell’s Theory of Description. For example, start with the usual suspects:

(1) Pegasus does not exist.
(2) Therefore, there is something that does not exist.

In the standard way, (1) is reified as ∃x(x does not exist) or ∃x(Px). The truth of (1) requires that there be an object in the domain that makes ‘x does not exist’ true when assigned to x. But this can only happen if the domain includes a nonexistent Pegasus or object. One way of resolving this problem is to accept the existence of nonexistent objects (#2 as true) and jump into the cloud of the continental philosophies equating nothingness with being. Another approach would be to deny that (1) is true by saying that ‘Pegasus’ does not refer to anything so we cannot assign this sentence truth conditions and thus, as before, violate the law of the excluded middle in our logic. We could go the Theory of Descriptions route and deny that these arguments are well-formed. The form of (1) is really ¬∃x(Px) in which following the work of W.V. Quine we treat the predicate as the verb “Pegasizes” and interpret or reify the sentence as really saying “nothing Pegasizes”; that is, there is nothing that has the properties wings, horse, flying, etc. Then the existential argument (2) is of the following form and it is invalid: ¬∃x(Px) ∴ ∃x¬(Px).

Treating a reified formula A[δ] as A(d1,…, dn/x1,…, xn) gives a further approach. Our problem is that we need to assign a value from the domain of x that makes ∃x(x does not exist) or ∃x(Px) true. If we find it, however, finding it proves the formula false. However, it is simple to find a substitution instance that makes the expression ‘x does not exist’ true; for example, substitute Pegasus to make it ‘Pegasus does not exist’ — A(d1/x1) but using the name Pegasus instead of Mont Blanc. This is enough to make the expression ∃x(Px) true. (Of course eventually, we need to find a way of understanding the extension of the name Pegasus or Mont Blanc and we may be back to the problems faced by Frege and Russell.)

 

IV. Conclusion

The relationship between logic and reality and even with reasoning and the ontological commitment of any semantic interpretations used in logic is an issue for philosophy regardless of whether a reified formula A[δ] is treated as an ordered pair <A, δ> or as the result A(d1,…, dn/x1,…, xn) of substituting the objects d1 =δ(x1), … , dn = δ(xn) for the variables x1, … , xn that occur free in A.

 

Bibliography:

Diggle, Phillip James. Vagueness : preliminary to a Wittgensteinian solution to the problems. PhD thesis, University of Leeds (2000). http://etheses.whiterose.ac.uk/id/eprint/2675.

Fine, Kit; Kuhn, Steve. Logic Elements and Foundations. Draft of Septemebr 19, 2016.

Frege, G. Philosophical and Mathematical Correspondence, ed. Gabriel, Hermes, Kambartek and Veraart. University of Chicago Press: Chicago (1980). http://www.logicmuseum.com/reference/fregerussell1904.htm.

Hacker, P.M.S.; Baker, G.P. Language, Sense and Nonsense. Blackwell: Oxford (1984).

Hacker, P.M.S.; Baker, G.P. Wittgenstein — Rules, Grammar, and Necessity. 2d ed. Wiley-Blackwell: Oxford (2009).

Potter, Michael. “Frege, Russell and Wittgenstein”. Routledge Companion to the Philosophy of Language, edited by Gillian Russell and Delia Graff Fara, 852-859. New York/Oxford: Routledge (2012).

Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. C. K. Ogden (trans.). Routledge & Kegan Paul: London (1922).

Quine, Willard Van Orman. From A Logical Point of View. Harvard University Press: Cambridge, MA. (1980).

Quine, Willard Van Orman. Philosophy of Logic. Harvard University Press: Cambridge, MA. (1986).

Quine, Willard Van Orman. Word and Object. Martino Publishing: Mansfield Centre, Connecticut (2013).