On What There Is*

Difei Xu

School of Philosophy,Renmin University of China difeixu@163.com

Abstract.In this paper,I first clarify that Quine’s ontological commitments thesis cannot provide answers to logical theories.Then I explore Kit Fine’s criticism(2009)on the quantificational account of ontology.Although I agree that“ontological commitments”does not itself provide an explanation for distancing ordinary commitments from theoretical commitments,I disagree that the philosophical analysis underlying ontological commitments thesis is trivial or non-philosophical.I also discuss two kinds of realism formulated by Fine(2009)and Ye(2010),and argue that their conclusions are swiftly drawn.In comparison with Frege and Quine,I analyze the origin of Quine’s ontological commitments and Frege’s comments on the quantificational account of ontology.

1 Introduction

Ontology,as etymology suggests,is the study of what there is,especially for answering typical ontological questions like,“Are there abstract objects,such as numbers,sets,etc?”“Apart from particular entities,are there universals?”Philosophers provide different answers to these questions.For example, a kind of physicalism alleges that only concrete objects in time-space exist,while,in the philosophy of mathematics,Platonism holds that mathematical objects exist.If claims are just starting points in their respective philosophies,we will be disappointed because we expect philosophers to provide a neutral starting point in ontology and to explain how we accept that concrete physical objects exist or how we accept or deny abstract objects.Meta-ontology concerns itself with the nature and method of ontology and especially explores neutral ways to answer ontological questions.The question of ontological commitments asks,“When we accept a theory, what entities should we then accept?” The question of ontological commitments is prima facie,simpler than ontological questions,since it concerns entities in theory,but not the general question of what there is.Inquiry on ontological commitments belongs to meta-ontology,and philosophers attempt to find a neutral way to answer the question of ontological commitments,despite it being much more complicated than at first sight.

As one of the most influential philosophers,Quine proposed his famous dictum that“to be is to be the value of a variable.”This means that to see what entities a theory commits,we need to regiment the theory using the syntax of individuation,predication,and quantification;then the theory’s range of bound variables constitutes its ontological commitments.In the study of ontological commitments,Quine’s method is analysis of language,weighing heavily on first-order language.He attempts to show that quantifiers and first-order variables provide the criterion of a theory’s ontological commitments.According to Quine,higher-order quantifications are not seen as bellwether for a theory’s ontological commitments.In this paper,I first argue that,taking model theoretic interpretation,Quine’s quantificational account is deficient.Then I comment on Fine’s recent work([8]),which claims that ontology’s quantificational account is either trivial or non-philosophical.Third,I briefly comment on Ye’s naturalism([16])and show that ontology is related to truth.Fourth,I provide a brief introduction to Frege’s conceptual realism and then compare Quine’s ontological commitments with Frege’s ontology.

2 Three Counterexamples in Logic

2.1 Pure Logic and Its Model Theoretical Interpretation with Empty Domain

As we know,for technological convenience,we require that the domain of firstorder logic’s model theoretical interpretation is non-empty.At the same time,it is widely admitted that logical truths are universally true,meaning they are true in every model,in other words,a “neutral topic.”Apparently,when a model’s domain is allowed to be empty,some originally logical,valid formulas are not valid anymore.True,an empty domain has little or no importance in application of logic.However,if we reflect on philosophical problems,such as the problem of the neutral topic of logic,the interpretation with an empty domain is worth studying because,intuitively,a logical truth being universally true should also be true in a model with empty domain.Recently,finding a suitable treatment for such a more inclusive logic has aroused some curiosity.In this section,therefore,I apply a treatment from Mendelson’s work to show that quantifiers are not so direct an explanation of ontological commitments as Quine had thought.

The language of pure logic does not contain individual constants or function symbols.This restriction involves lack of clarity in how to interpret individual constants and function symbols when the interpretation’s domain is empty.Like typical first-order language,primitive symbols are individual variables,connectives,and a quantifier.

All axioms of axiomatic system ETH are sentences(without free variables)and are in the following schema:

Axiom scheme 1:A→(B→A)

Axiom scheme 2:(A→(B→C))→((A→B)→(A→C))

Axiom scheme 3:(?A→?B)→((?A→B)→A)

Axiom scheme 4:?P(?xA(x)→A(y)),whereyis free forxinA(x),andxis free inA(x),andPis a sequence of variables that includes all variables free inA(and possibly others).

Axiom scheme 5:?x(A→B)→(A→?xB),ifAcontains no free occurences ofx.

Axiom scheme 6:?x1...?xn(A→B)→(?x1...?xnA→?x1...?xnB)Together with all formulas obtained by prefixing any sequences of universal quantifiers to instances of axiom 1 to axiom 6.

Modus Ponens is the only rule of inference.

To provide the formulation of inclusive validity,we make a small revision in the“truth”of universal formulas in the form?xφ(x).Normal model interpretation for?xφ(x)is given as the following:

Because we make a restriction that the domain of an interpretation could be nonempty,ifφ(c)is a contradiction,then?xφ(x)is not satisfied in any model,or this kind of formula is not logically valid.But when we allow an interpretation’s domain to be empty,we should revise the satisfaction relation for?xφ(x)as the following:

Because the condition of satisfaction for?xφ(x)involves a conditional,when the domain is empty,the premise of the condition is false,and the conditional is true.Therefore,all formulas in the form of?xφ(x)are satisfied under the interpretation with empty domain.By “inclusive validity,”we mean a formula is satisfied under all interpretations,including the interpretation with empty domain.

With this revision,inclusive validity can be characterized by an axiomatic system,called ETH.This system is sound and complete with respect to inclusive validity,which is shown in[14,pp.141–147].

According to Quine’s ontological commitments thesis,a theory’s ontological commitmentsis the value of bounded individual variables.How about ETH? Does this theory commit any entities?As we saw in the formulation of inclusive validity,a model’s domain may be empty.In this sense,ETH does not commit anything.But consequences of ETH are satisfied in models with non-empty domain,and in this sense,ETH commits everything or every object.Therefore,Quine’s ontological commitments thesis gives no information in this case.

2.2 First-order Arithmetic as a Counterexample

Non-logical axioms of first-order arithmetic are these seven:

·For any variable,x1,...,xn,y,and any formulaφ,such that

the sentence?x1...?xn(φ(y/0)∧?(φ→φ(y/y+1)→?φ)).

Adding the seven non-logical axioms to the usual first-order logic,we get the first-order arithmetic system.This system has nonstandard models.

The language of first-order arithmetic is countable.On the one hand,according to the L?wenheim-Skolem theorem,first-order arithmetic has a domain with a countable domain,and at same time,it has one with any infinite cardinality.Now the question is,“What ontological commitments does the theory of first-order arithmetic make?”According to Quine,ontological commitments concern the values of bound variables.But since a theory may have different models with different cardinality,how can we decide a theory’s ontological commitments?

2.3 Th(n)As a Third Counterexample

The first-order arithmetic system has different structure,with different cardinality.If we consider natural numbers’standard model,and focus on the first theory of Th(n)(the set of true sentences in natural numbers’standard model),we find the theory Th(n)also has a nonstandard model with a countable domain.

By the standard arithmetic model,we mean the structure of natural numbers,with the beginning member as 0 and every number n followed by its successorn+1.The structure looks like this:

By a nonstandard model of arithmetic,we mean a structure that is elementarily equivalent,but not isomorphic to the standard model.The definition of“elementarily equivalent”is easy:two structures A and B of a language L are called elementarily equivalent if for every sentenceφof the languageL,A|=φiff B|=φ.Then by the compactness theorem and the L?wenheim-Skolem theorem,it is not difficult to prove the existence of a countable structure elementarily equivalent to,but not isomorphic to the standard structure of natural numbers.Th(n)∪{?x≡0,?x≡1,?x≡2,...}is satisfied in a nonstandard structure A.Therefore,the domain of A has a nonstandard member“a.”A similar question is raised here:Does the theory Th(n)commit“a”?Quine’s ontological thesis cannot provide an answer.

From these three examples,we can easily see,at least in some cases,that Quine’s ontological commitments do not guide us to find the answer to a theory’s ontological commitments question.If we abandon the model theoretical interpretation,Quine’s ontological commitments is also controversial.I share with Quine a fundamental point,which is that the mark of our commitments to entities is our acceptance of statements as true.The difference between Quine and me is that Quine’s ontological commitments considers only bound individual variables and rejects higher-order quantification.I do not see any good reason to reject higher quantification,and I think different categories of entities are explained by different kinds of expressions embedded in statements..HereIdo not discuss different categories of entities,but concentrate on first-order quantification.Above,I argued that if we take model interpretations,Quine’s ontological commitments thesis confronts difficulties.Kit Fine([8])thinks if we accept the quantification account of ontology,the answer will be trivial or non-philosophical:(1)If we accept the neutral way of answering the question of ontological commitments,it seems that the ontological question is trivial.We accept that electrons exist,numbers exist,and so on,because we accept the modern theory of physics as true and accept arithmetic as true,etc.(2)The ontological questions are philosophical,but we need a non-philosophical answer.In the next section,I discuss Kit Fine’s criticism of the quantificational account of ontology.

3 Quantificational Account of Ontology

3.1 Kit Fine’s Criticism on the Quantification Account of Ontology

In the last section,I argued that if we take model interpretations of a theory,Quine’s ontological commitments is not adequate to answer what objects a theory commits.Kit Fine([8])argues that the standard quantificational account,deriving from Quine,of ontological questions is mistaken.I agree with Fine’s conclusion,but I do not agree with his arguments.When philosophers ask,“Do numbers exist?,”“Do electrons exist?,”they are asking ontological questions.According to Quine,questions turn to”Is it true that?x(xis a number)? ”and”Is it true that?x(xis an electron)? ”Fine thinks that the quantificational account trivializes ontological questions.For example,that?x(xis a number)is a trivial consequence of the arithmetic truth that there is a prime number between 9 and 12.Since?x(xis a number)is true,numbers exist.But the philosopher’s question is not trivial.([8,p.158])At the same time,the account is mathematical,not philosophical.Philosophical questions need a philosophical account.

Fine’s argument is misleading.As I said before,the main question of ontology is what there is.As for objects, the ontological question is, “What objects exist?” The study of ontological commitments belongs to meta-ontology and attempts to clarify ontological questions and explore a neutral way to answer them.When we ask the ontological question,“What objects are there?”are we asking the question of ontological commitments?Are we asking what objects, we think (accept), exist? You may insist that these two questions are different.The ontological question is about things in the world,but not in our thinking.Although they are different questions,when we reflect on ontological questions,we find another question should be answered first,“What do we mean that some object like this table,or two,exist?”This question involves the nature of ontology and is a meta-ontological question.If we do not know the ontological questions,how can we answer one?1Frege provides an answer different from Quine’s;readers can see “Frege’s Realism”in this paper.Some philosophers admit physical objects exist only because they think that scientific methods are the most reliable and that scientific theories are up-to-now paradigms of human beings’knowledge.We should accept objects in scientific theories.Apparently,the ontological question,“If we accept a theory,what entities should we accept?”arises.Although the account of ontological commitments is not itself an account of what there is,the account is not trivial,and neither is it non-philosophical.

Frege does not explicitly say his account is meta-ontological,but it is.He distinguishes objects and properties through analysis of language.By his theory of meaning,Frege claims that if we accept an atomic sentence,embedding a singular term,as true,we cannot deny that the singular term’s referent is real.For example,if we accept as true that Zeus lives on Olympus,we could not deny that the referent of“Zeus”is real.So,Frege gives an account of in what sense we accept that an object exists.Frege’s theory of meaning,of course,is not trivial.Quine follows Frege and explicitly provides the way to answer the question of ontological commitments.Quine’s dictum “to be is to be the value of a variable”has its philosophical arguments,and although I disagree with him,the arguments are not trivial.In his philosophy,roughly speaking,three basic ideas support his dictum.2See the section “Set theory in sheep’s clothing”in Quin’s Philosophy of Logic.First,the expression next to a quantifier should be a name,but“F”in the higher-order quantification?Fis a predicate,not a name.Second,Quine has a famous slogan “no entity without identity,”and according to Quine,there is no sufficient and necessary condition for properties’identities.Properties cannot be entities.Third,the extension of a property is a set,and the extension of a property is an entity because we have the criterion of the identity of sets.Sets are objects.In this sense,we allow the higher quantification because “F”here is a name for a set.Quine’s three basic ideas on rejecting properties as entities and only admitting first-order quantification do not constitute lack of criticism.Boolos([1])and Hale([13])are good examples of criticism of Quine’s three basic ideas.

It is better not to go further with Quine’s ontological commitments,and seeing that study of ontological commitments is not trivial is not difficult.But the application is trivial.When philosophers ask,“Do numbers exist?,”they are asking an ontological question.Once they accept Quine’s claim and accept that arithmetic truths are strictly and literally true,they will accept that numbers exist.The conclusion using the premises is trivial,but that does not mean that the ontological analysis is trivial or non-philosophical.

As for what objects exist,Frege turns to an atomic statement embedding a singular term,and Quine turns to an existential statement.Is there a conflict in their applications?No.Let us consider the question “Do numbers exist?”If we accept Frege’s theory,and accept that “2 is a prime,” is true, then we accept that numbers exist.If we accept Quine’s theory and accept‘?x(xis a number)′is true,we accept that numbers exist.The reason these do not conflict is that in logic,we could infer?x(Fx)fromF(c).As for what objects exist,Frege weighs much on atomic statements and singular terms,while Quine weighs on first-order quantification.Their theories on ontological commitments differ.In the second section,I showed the difficulties in Quine’s theory if we take a model theoretic interpretation.But this does not mean that Frege’s theory also has the same difficulties.Frege resorts to singular terms,not bound variables,so the difficulties do not fit Frege’s theory.

Every theory has to start somewhere.Frege’s ontology of objects and even Neo-Fregean ontology of objects depends on atomic statements’truth.Their ontological theories take “truth”as a primitive notion,and in terms of truth,explain ontological commitments.But how do we accept a statement as true or false?Well,this involves another philosophical question on truth.Neo-Fregeanist Crispin Wright and Frege himself also provide their own theory of truth.I do not see any good reason to deny Frege or a Neo-Fregean’s ontological commitments,only because their theories take truth as a primitive notion.Anyway,we should admit that“truth”and “ontology”have a very close relationship.

Carnap([2])distinguishes internal and external questions of what there is.Fine also agrees with Carnap’s negative claim,that philosophical questions cannot be internal,but must be external.Ever since Carnap’s “Empiricism,Semantics and Ontology,”it has often been supposed that,for any given area of inquiry,one should adopt one of these points of view to the exclusion of the other,either engaging in the enquiry itself or evaluating it from the outside.([8,p.174])

But how should we evaluate inquiry from outside?There have been a numberof attempts to clarify the idea of realism.Fine([7,8]),with other philosophers such as Chalmers([3]),Dorr([5]),and Sider([15]),identify what is real with what is fundamental.In the next subsection,I explore this reductionist argument and claim it is not sound.I also clarify accepting that the outer world is not the criterion for distinguishing realism and idealism.

3.2 Potential Difficulties in the Quantificational Account

Above,I argue against Fine’s conclusion that the quantificational account of ontology is either trivial or non-philosophical.But I agree with him that Quine’s thesis of ontological commitments itself does not explain how to distance ordinary commitments and philosophical commitments.Quine later put forward dispensability arguments for distancing ordinary commitments and philosophical commitments.The idea here is that proper application of scientific method often shows that ordinary commitments are dispensable,whereas ontological commitments or philosophical commitments are indispensable.But this argument is controversial.Feferman argues that even if we accept that numbers are indispensable in scientific theory,we cannot conclude that is a realistic position on numbers.

One view of PA is that it concerns natural numbers as independently existing abstract objects;that is again a platonistic view,albeit an extremely moderate one.Another view is that PA concerns the mental conception of structure of natural numbers,which is of such clarity that statements concerning these numbers have a determinate truth value and that their properties can be established in an indisputable intersubjective way; this is more or less the predicativisitic view.Or one can make use of the fact that PA is reducible to HA to justify it on the basis of a more constructive ontology.([6,p.296])

Feferman argues that indispensable arguments cannot answer the question of ontology.Numbers are indispensable in scientific theory,but whether numbers exist independently of our minds still awaits an answer.I think the point here is whether ontological commitments are commitments for what is real.As I said before,ontological commitments belong to meta-ontology,which involves the nature and methods of ontology,and a complete philosophical theory should answer what entities are real.Some antirealists of mathematics claim that the statements are not literal truths,but the concepts of mathematics have cognitive functions,and in this sense,their philosophies explain why mathematics could be applied in scientific theories without Platonism in mathematics.This kind of instrumentalism in mathematics,on the other hand,admits that physical objects are real.As I said before,this kind of philosophy should explain why we accept physical objects and why statements about them are literal truths.However,as far as I know,assumptions in this kind of philosophy are biased or not neutral.Carnap also observed existential statements in theories.He distinguished external and internal questions.In his opinion,“There is a prime number between 4 and 6”is a truth in arithmetic,but the ontological question is an external one.Although his positive arguments are controversial,they also show that after accepting some statements’truth,ontological questions are still there.Or,in other words,ontological commitments do not answer ontological questions.

When we say something is real,what do we mean exactly?When I was a student,I repeatedly met with an explanation of realism as“something exists independent of our mind and language.Physical objects are real because they exist independent of our mind and language,and we know them from our sensations.Abstract objects are not real because they are products of our thinking.”But this na?ve answer that physical objects are real is rather arbitrary.At that time,I found that the na?ve negative answer to abstract objects,was essentially based on the following reasoning:

All reals can be known from sensations

Abstracts cannot be known from sensations

Therefore,abstracts are not real.

But this reasoning is not convincing,because the first premise of what is real still needs explanation.Later,I found that the difficulty lies in proving something in the external world.

Reading more,I found that the criterion for real is not the true color of realism.

This difficulty is highlighted by the fact that realists and idealists can agree on virtually every verbal response they give to specific questions of reality:Russell once noted that Berkeley would surely agree that tables are real;Kant granted the independence of physical objects from mind and endorsed the correspondence theory of truth;James endorsed the correspondence of thought and fact as the obvious essence of truth; and during his most intimate flirtation with phenomenalism, Carnap granted the existence and reality of every theoretical entity postulated by science.If these philosophers may legitimately talk like realists about reality and existence,what is it that divides them from the others?([4,p.94])

Idealism’s general thesis claims that ideas are true objects of knowledge;the main point is that ideas are prior to things and that ideas provide grounds of being to things.Idealism’s characteristic is that ideas have priority both metaphysically and epistemologically,and external reality,as we know it,reflects mental operations.Idealists admit the external world in itself is certainly mind-independent,but they insist that the world as conceived by us must be constructed by mind.Therefore,idealism does not conflict with realism on the reality of the external world.

From the above, we see Feferman’s reduction arguments that arithmetic is indispensable for scientific theory because mathematics applied in science is proof theoretically reduced to arithmetic.But Feferman admits that indispensable arguments do not answer the question of what is real.

Reductionism arguments for what is real could date at least to ancient Greece.Democritus believed there was nothing more to the world than atoms in the void.Thales believed that the world was wholly composed of water.

As Fine said,this is “an intelligible position,whether correct or not,”and “We know in principle how to settle such claims about the constitution of reality……Democritus would have to argue that there being chairs consists in nothing more than atoms in the void or to explain in some other way how existence of chairs is compatible with his world-view.”([8,p.176])

If philosophers wish to argue that atoms and the surrounding void are real, but chairs are not,Fine’s explanation is not reasonable enough.From Fine’s explanation,we could say that atoms in the void are fundamental constituents of reality,but we could not infer that only fundamentals are real,whereas things consisting of them,say,chairs,are not real.Why are not things consisting of what is real,also real?

From biology,we know creatures are composed of cells;from physics,we know things consist of atoms or even subatomic particles.However, we do not reject our beliefs that creatures are real;chairs are real;atoms are real;and so on.I do not see any good reason to reject that things composed of fundamental constituents are real.

In this section,I clarify two ideas:(1)idealism is consistent with realism in the external world,so it is not a characteristic of realism to insist some things external(out of our minds)are real;(2)reductionism is not sound,in that only fundamentals are real.

Antirealists attempt to explain that mathematical statements are not literally true,but in the next section,I argue that their arguments are not sound.

4 Fictionism in Mathematics

That ordinary commitments do not aim to capture the strict and literal truth is widely held,but that philosophers’commitments do,is not.Some fictionists in mathematics deny that arithmetic truths are not strictly and literally true;in their philosophical theory,mathematics is like fiction.Fictions are sometimes very useful,but they are not true.Therefore,they do not accept that numbers exist.On the other hand,Platonists in mathematics insist that arithmetic truths are strictly and literally true,and therefore numbers do exist．As I said before,only Quinean ontological commitments cannot afford an answer.Equipped with a theory of truth—only after giving a sound theory to explain what a strict and literal truth is—we may find an answer.Ontological commitments is not toothless,because it clarifies that ontological questions have a close relationship to truth,and it asks,if we accept the(strict and literal)truth of a theory,what entities we should accept.

From a position of physicalism, Ye admits only “naturalized truths” be taken as strict and literal.In his theory,only statements describing properties of or relations among concrete objects in time-space are truth-apt.He thinks that thoughts about arithmetic and logic have cognitive functions,but no corresponding states of affairs outside the brain.So in his opinion,arithmetic and logic statements are not strict and literal truths.But I think his theory is not coherent.Let us consider an example provided by Ye.([16，p.94])

Let us consider the statement“3 atoms plus 2 atoms equal 5 atoms.”

Ye says this statement has a truth value,and it expresses a relation among physical objects.But does this statement really express a relation among concrete objects?By concrete objects,we mean particulars in space-time.Still,this statement does not mention any particulars,but instead,asserts a general truth,that any 3 atoms plus any 2 atoms equal any 5 atoms.In what sense could we say this is a truth about concrete objects?Ye also insists that this statement is a truth about atoms.But what properties of atoms contribute this statement’s truth?Ye may reply that number concepts have structures,so that they can combine natural concepts,say the concept of atoms,to form a thought,which has a corresponding state of affairs.If we follow Ye’s line,this corresponding state of affairs must involve numbers and operation of numbers,but why is this statement a truth about atoms,but not about numbers?A thinker cannot observe all the situations of 3 atoms plus 2 atoms equal 5 atoms to come to a conclusion;absolutely he cannot do so because,in his whole life,he cannot observe all atoms.More importantly,a child who has no idea about atoms may assert the statement through arithmetic.How does a thinker obtain a truth about atoms when he knows nothing of atoms?Concepts of numbers have structures,as do concepts of atoms;otherwise,how can we imagine they could somehow be combined?Therefore,having structure is not a characteristic of a number concept.If we endow concepts of arithmetic with cognitive function,I do not see any good reason to deprive of cognitive functions the concepts of natural things.I do not say Ye’s theory is mistaken,but at the least,it is not neutral.His conclusion that the statement is about atoms is too swiftly drawn.

As far as I know,no philosophical theory in the spirit of antirealism in the philosophy of mathematics gives a satisfying explanation why the statement in arithmetic is not truth-apt.

5 Frege’s Realism

5.1 Thoughts are Objective

In contrast with some kinds of idealism,Frege’s realism emphasizes that thoughts are not subjective,but objective.Subjective idealism,associated with Berkeley,argued that qualities are mind-dependent.The content of knowledge is mind-dependent in the position of subjective idealism.Kant called his own philosophical position“critical idealism.”He claimed that knowledge is limited to the phenomenal world and cannot inform us about things in themselves.In his philosophy,our knowledge must conform to a priori intuitions of space-time and categories of understanding.Therefore Kant’s idealism does not reject the content of knowledge as subjective.Frege argues that the content of a thought is objective.In the first part of “Logical Investigation,”Fregeclaims,“Athought neither belongs to my inner world as an idea,nor yet to the external world,the world of things perceptible by the senses.”

But not everything is an idea.Frege gave four reasons to show how ideas are distinct from things of the outer world.Firstly,ideas cannot be seen,or touched,or smelled,or tasted,or heard.Secondly,ideas are something we have.We have sensations,feelings,moods,inclinations,wishes.An idea that someone has belongs to the content of consciousness.Thirdly,ideas need an owner.Things of the outer world are,on the contrary,independent.Fourthly,every idea has only one owner;no two persons have the same idea.

The first reason shows that the idea of a physical thing differs from the thing.The other three reasons could infer that thoughts differ from ideas.

I can acknowledge a science in which many can be engaged in research.We are not owners of thoughts as we are owners of our ideas.We do not have a thought as we have,say,a sense impression,but we also do not see a thought as we see,say,a star.…In thinking,we do not produce thoughts—we grasp them.([12,p.363])

Thoughts are not ideas because they do not have owners,and different people may grasp the same thought.But thought cannot be seen,so it differs from physical things.Thoughts belong neither to our inner worlds as idea,not yet to the external world,the world of things perceptible by the senses.Thoughts belong to the third realm.Science is to reveal the external world.Only with sense impression,everyone would remain shut up in his or her inner world.

What must still be added is not anything sensible.And yet this is just what opens the external world for us.…So perhaps,since the decisive factor lies in the non-sensible,something non-sensible,even without the cooperation of sense impressions,could also lead us out of the inner world and enable us to grasp thoughts.([12,p.365])

Science is for true thoughts,which are in the third realm.Even though natural sciences are for true thoughts of perceptible things,just sense perception is not sufficient for this aim.We should also add something non-sensible,thoughts,to reveal the outer world of perceptible things.

Thoughts could be expressed by sentences.But not all sentences express thoughts.Only assertions can express thoughts.More importantly,what some assertive sentences express are mock thoughts,which do not have true value.Truth values are semantic values of sentences that express thoughts.The sentence “Scylla has six heads”is not true,and its negation “Scylla does not have six heads”is not true either.For either to be true,the proper name “Scylla”would have to refer to something.If a predicate in a sentence has no definite range,the sentence has no true value either.“The rose is beautiful.”The predicate “xis beautiful”has no definite range,and the speaker may think the rose is beautiful,but other people may think it is not.Nothing is beautiful in itself,but is beautiful only for persons.What these sentences express are not thoughts.

Scientific theories, mathematics, and logic concern thoughts, which have truth value.Statements in these theories are assertions about things themselves, and these assertions’contents have nothing to do with the subjective.A thought is the sense of assertive sentences.Laws of nature, mathematical laws, and historical facts are all thoughts.“True”or “false” applies only to thoughts.The sense of an expression belongs to the third realm.Whether a thought is true or false is independent of our recognition.

In Frege’s theory,thoughts,which belong to the third realm,are objective and external of our inner world.Components of a thought,which are senses of subexpressions of a sentence,are also in the third realm.Before Frege,the logic dominating philosophical analysis was Aristotle’s.Philosophers believe the basic assertion is in the form subjectpredicate.Idealists try to argue that the role of predicates represents our minds’traits,whereas subjects are external to our minds.In idealists’explanation,contents of assertionshave the color of our minds.Frege is known as the father of modern logic.differs from Aristotle’s.The discovery of quantifiers convinces us that subject-predicate form is only a superficial trait of our natural language;if we wish to study predicates further,we find the logical structure of a language,instead of grammatical structure.In light of Frege’s work,logic is not only axiomatic systems.Frege devoted considerable effort to separating his conceptions of“l(fā)ogic”from that of others,like Boole,Jevons,and Schr?der.Frege thinks that these people are engaged in the Leibnizian project of developing a calculus ratiocinator,but his aim,much more ambitious,is to design a lingua characteristic.He complains that Boole’s work entirely ignored content.Indeed,Boole’s work is to produce an algorithm for solving a logical problem,but in Frege’s opinion,Boole’s formula language presents only a part of our thought;the whole of it cannot be taken care of by a machine or replaced by a purely mechanical activity.([10,9])

With his discovery of logic, Frege thinks that predicates are much more complicated than what predicates presented in old logic, and properties (in Frege’s notion of concepts),as components of thoughts,are objective and do not belong to our inner worlds.In this sense,he is a proponent of conceptual realism,contrasted with idealism.

5.2 Quine vs.Frege

Quine accepts modern logic and tries to use first-order logic to study ontological problems.Although he is not a conceptual realist like Frege,he does not deny abstract objects,for he accepts the truth of mathematics.His philosophy starts from science and scientific methods,and he thinks that so far,scientific theories are our best knowledge of the outer world.He calls his philosophy “holism,”and claims logic and mathematics are the core of our web of beliefs.His empiricism accepts that mathematical theories are true,but their truth should be justified by scientific application.Apparently,Frege’s theory differs completely from Quine’s.

In Frege’s opinion,natural sciences are for natural laws,and logic is for the laws of laws.We do not need to and cannot justify the truth of logic or mathematics by its application in scientific theories.Truths of logic and mathematics are independent of our recognition,and even if we do not recognize them,they are there.Mathematical truths,or some of them,are logical although they are derived from logic.Because logic truths are the laws of laws,how can we imagine a natural law in conflict with the laws of laws?

As mentioned,one of Quine’s famous slogans is “no entity without identity.”The underlying idea in his ontological thesis is that objects have the criterion of identity,but properties or relations do not.I think the criterion in Quine’s mind is Leibniz’s law, which says that if an object“a”is identical with an object“b,”if and only if for all propertiesF,F(a)iffF(b).As for properties,we cannot tell the criterion for the identity of two propertiesF,G.If this is the case,Leibniz’s law,as the criterion for objects’identity,involves quantification over properties;this is what Quine rejects.True,in Frege’s work,he does not state the criterion for properties,but he does not take Leibniz’s law as a criterion for the identity of two objects.In hisFoundation of Arithmetic,he does not take this law for the identity of numbers.We could also imagine that two physical objects could have the same properties,but that they are two different objects.In his recent workNecessary Beings,Hale proposes a criterion for objects and also for properties.If he is right,we could accept properties as entities in terms of Quine’s slogan.

Quine goes further to claim that second-order logic is disguised as sheep of set theory.Here,I do not explore this thesis more.Instead,I introduce only Frege’s comments on the quantifier.A quantifier as a component of a predicate also contributes the sense of a sentence in which it occurs.Because it is part of a predicate,the quantifier itself is not a predicate,so a sentence“Leo Sachse exists”says nothing about Leo Sachse or says nothing about Leo Sachse’s property.Or in other words,any sentence including “Leo Sachse”implies “Leo Sachse exists.”As a matter of fact,“Leo Sachse exists”in logic language is not a legal sentence;and in Frege’s theory,it does not express a thought.In logic language,we could say,“There exists an object,such …”but cannot say,“There existsanobject.”Frege complained that natural language is not an appropriate vehicle for studying thoughts.But once we realize a sentence’s logical structure,we understand that existence cannot be a characteristic mark of a concept.I quote from Frege’s “Dialogue with Pünjer on Existence.”

We can say that the meaning of the word “exist”in the sentence “Leo Sachse exists”and “Some men exist”displays no more difference than does the meaning of“is a German”in the sentences“Leo Sachse is a German”and “Some men are Germans.”

The existence expressed by “there is” cannot be a characteristic mark of a concept whose property it is, just because it is a property of it.In the sentence“there are men,”we seem to speak of individuals that fall under the concept“man,”whereas we are talking about only the concept“man.”The content of the word “exist”cannot well be taken as the characteristic mark of a concept,because it is used in the sentence “Men exist,”which has no content.([11,p.66])

In the last paragraph of“Dialogue with Pünjer on Existence,”Frege concludes:

We can see from all this how easily we can be led by language to see things in the wrong perspective,and what value it must therefore have for philosophy to free ourselves from the domination of language.If one makes that attempt to construe a system of signs on quite other foundations and with quite other means,as I have tried to do in creating my concept-script,we shall have,so to speak,our very noses rubbed into the false analogies.([11,p.67])

Frege shows that ontological statements in ordinary language have no content or do not express any thought.But this does not mean that Frege was never concerned with ontological questions.As I showed above,his ontological study is hidden in the study of thought.If a proper name has no referent,then the sentence in which it occurs expresses mock thought.In other words,if a sentence expresses a thought,or it has truth value,then the proper name occurring in it has a referent.Quine seems to accept Frege’s logical language and advocates regimenting a scientific theory in first-order language to study ontology.But he departs from Frege where concepts(properties or relations)are real.

6 Conclusion

Quine’s ontological commitments thesis concerns values of first-order bound variables.Model theoretic semantics itself ca nnot provide philosophical explanations of what there is.There are axiomatic systems whose models are not categorical.Logical truths are universal,and if we decide the domain of a logical system,we cannot say the domain must be non-empty.Some logical systems allowing an empty domain could be valid in a broader sense.

Fine argues that if we accept the quantificational account of the ontological question,the answer becomes trivial or non-philosophical.I argue that the back theory supporting ontological commitments is not trivial.Even though we accept a trivial conclusion that there is a prime from “2 is a prime,”whether natural numbers exist is still a philosophical question as yet unanswered.The ontological commitments thesis results from a philosophical theory that itself is not trivial.However,the application of this result may seem trivial.

In modern times,disputation over realism and antirealism about abstract objects is a main issue in philosophy.Philosophers wish to provide a neutral analysis of this question.If a philosophical theory begins from an ontological premise,say,only concrete physical objects exist,it loses its neutral color.On the other hand,through the discovery of thoughts’logical structure,Frege argues that concepts are real,in contrast with idealism.He concludes that statements in ontology misuse the word “exist,”which is part of a concept,but is not itself a concept.Quine’s ontological thesis accepts modern logic’s result,but he denies concepts as entities.His argument against the entity of a concept is controversial.I do not say that all of Frege’s theory of meaning is gold,but apparently it lies far from Quine’s.If Frege were asked to answer what objects are committed to in an accepted scientific theory,he might first say that we should be clear what“something exists”means.Secondly,he might say that what objects are committed to in a scientific theory,are referents of proper names in the theory.