Logic and Ontology

First published Mon Oct 4, 2004; substantive revision Mon Mar 13, 2023

A number of important philosophical problems are at the intersectionof logic and ontology. Both logic and ontology are diverse fieldswithin philosophy and, partly because of this, there is not one singlephilosophical problem about the relation between them. In this surveyarticle we will first discuss what different philosophical projectsare carried out under the headings of “logic” and“ontology” and then we will look at several areas wherelogic and ontology overlap.

1. Introduction

Both logic and ontology are important areas of philosophy coveringlarge, diverse, and active research projects. These two areas overlapfrom time to time and problems or questions arise that concern both.This survey article is intended to discuss some of these areas ofoverlap. In particular, there is no single philosophical problem ofthe intersection of logic and ontology. This is partly so because thephilosophical disciplines of logic and of ontology are themselvesquite diverse and there is thus the possibility of many points ofintersection. In the following we will first distinguish differentphilosophical projects that are covered under the terms‘logic’ and ‘ontology’. We will then discuss aselection of problems that arise in the different areas ofcontact.

‘Logic’ and ‘ontology’ are big words inphilosophy, and different philosophers have used them in differentways. Depending on what these philosophers mean by these words, and,of course, depending on the philosopher’s views, sometimes thereare striking claims to be found in the philosophical literature abouttheir relationship. But when Hegel, for example, uses‘logic’, or better ‘Logik’, he means somethingquite different than what is meant by the word in much of thecontemporary philosophical scene. We will not attempt to survey thehistory of the different conceptions of logic and of ontology, nor thehistory of the debate about their relationship. Instead this articlewill look at this issue fairly top down, with an emphasis on areas ofoverlap that are presently actively debated. For more historicalinformation, see Kneale and Kneale 1985. Nonetheless, twohistorically important figures, namely Gottlob Frege and ImmanuelKant, will make repeat appearances below.

2. Logic

There are several quite different topics put under the heading of‘logic’ in contemporary philosophy, and it iscontroversial how they relate to each other.

2.1 Different conceptions of logic

On the one hand, logic is the study of certain mathematical propertiesof artificial, formal languages. It is concerned with such languagesas the first or second order predicate calculus, modal logics, thelambda calculus, categorial grammars, and so forth. The mathematicalproperties of these languages are studied in such subdisciplines oflogic as proof theory or model theory. Much of the work done in thisarea these days is mathematically difficult, and it might not beimmediately obvious why this is considered a part of philosophy.However, logic in this sense arose from within philosophy and thefoundations of mathematics, and it is often seen as being ofphilosophical relevance, in particular in the philosophy ofmathematics, and in its application to natural languages.

A second discipline, also called ‘logic’, deals withcertain valid inferences and good reasoning connected to them. Theidea here is that there are certain patterns of valid inferences whichare both an object of study in itself as well as connected to certainpatterns of good reasoning. How this connection between inference andreasoning is to be understood more precisely and to what extent itobtains is controversial, and beyond the scope of this survey.However, see Christensen 2005 for more. In any case, logic does notcapture good reasoning as a whole. That is the job of the theory ofrationality. Rather it deals with inferences whose validity can betraced back to the formal features of the representations that areinvolved in that inference, be they linguistic, mental, or otherrepresentations. Some patterns of inference can be seen as valid bymerely looking at the form of the representations that are involved inthis inference. Such a conception of logic thus distinguishes validityfrom formal validity. An inference isvalid just in case thetruth of the premises guarantees the truth of the conclusion, oralternatively if the premises are true then the conclusion has to betrue as well, or again alternatively, if it can’t be that thepremises are true but the conclusion is false.

Validity so understood is simply a modal notion, a notion about whathas to be the case. Others might think of validity as involving a morefine grained hyperintensional notion, but in any case, validity sounderstood is not what logic is concerned with. Logic is concernedwithformal validity, which can be understood as follows. Ina system of representations, for example a language, it can be thatsome inferences are always valid as long as the representational orsemantic features of certain parts of the representations are keptfixed, even if we abstract from or ignore the representationalfeatures of the other parts of the representations. So, for example,as long as we stick to English, and we keep the meanings of certainwords like “some” and “all” fixed, certainpatterns of inference, like some of Aristotle’s syllogisms, arevalid no matter what the meaning of the other words in the syllogism.^[1] To call an inference formally valid is to assume that certain wordshave their meaning fixed, that we are within a fixed set ofrepresentations, and that we can ignore the meaning of the otherwords. The words that are kept fixed are the logical vocabulary, orlogical constants, the others are the non-logical vocabulary. And whenan inference is formally valid then the conclusion logically followsfrom the premises. This could be generalized for representations thatare not linguistic, like graphic representations, though it wouldrequire a bit more work to do so. Logic is the study of suchinferences, and certain related concepts and topics, like formalinvalidity, proof, consistency, and so on. The central notion of logicin this sense is the notion of logical consequence. How this notionshould be understood more precisely is presently widely debated, and asurvey of these debates can be found in the entry onlogical consequence.

A third conception of logic takes logic to be the study of specialtruths, or facts: the logical truths, or facts. In this sense logiccould be understood as a science that aims to describe certain truthsor facts, just as other sciences aim to describe other truths. Thelogical truths could be understood as the most general truths, onesthat are contained in any other body of truths that any other scienceaims to describe. In this sense logic is different from biology, sinceit is more general, but it is also similar to biology in that it is ascience that aims to capture a certain body of truths. This way oflooking at logic is often associated with Frege.

This conception of logic can, however, be closely associated with theone that takes logic to be fundamentally about certain kinds ofinferences and about logical consequence. A logical truth, on such anunderstanding, is simply one that is expressed by a representationwhich logically follows from no assumptions, i.e. which logicallyfollows from an empty set of premises. Alternatively, a logical truthis one whose truth is guaranteed as long as the meaning of the logicalconstants is fixed, no matter what the meanings of the other parts ina representation are.

And there are other notions of ‘logic’ as well. One ofthem is historically prominent, but not very widely represented in thecontemporary debate. We will briefly discuss it here nonetheless.According to this conception of logic, it is the study of the mostgeneral features of thoughts or judgments, or the form of thoughts orjudgments. Logic thus understood will for example be concerned withthe occurrence of subject and predicate structure that many judgmentsexhibit, and with other such general features of judgments. It willmostly be concerned with thoughts, and not directly with linguisticrepresentations, though, of course, a proponent of this conception canclaim that there is a very close connection between them. To talkabout the form of a judgment will involve a subtly different notion of‘form’ than to talk about the form of a linguisticrepresentation. The form of a linguistic representation, basically,was what was left once we abstract from or ignore the representationalfeatures of everything except what we keep fixed, the logicalconstants. The form of a thought, on the other hand, is oftenunderstood as what is left over once we abstract from its content,that is, what it is about. We will briefly pursue the question belowhow these notions of form are related to each other. This conceptionof logic is associated with Kant. Kant distinguished different notionsof logic (for example transcendental logic, general logic, etc.), butwe won’t be able to discuss these here. See the entry onImmanuel Kant for more.

One important philosophical aspect of logic, at least in the sensesthat deal with logical consequence and the forms of judgements, is itsnormativity. Logic seems to give us a guide how we ought to reason,and how we ought to draw inferences from one representation toanother. But it is not at all clear what guide it gives us, and how weshould understand more precisely what norms logic puts on ourreasoning. For example, logic does not put us under the norm “Ifyou believe \(A\) and you believe if \(A\) then \(B\), then you oughtto believe \(B\).” After all, it might be that I should notbelieve \(A\) and if \(A\) then \(B\) in the first place. So, inparticular I shouldn’t believe \(B\). A reductio ad absurdum isa form of argument that illustrates this. If I believe A and if A then\(0=1\), then this should lead me to abandon my belief in A, not leadto a belief that \(0=1\). The consequences of my beliefs can lead meto abandon them. Still, if I have some reasons for my beliefs then Ihave at least some prima facie, but not necessarily conclusive, reasonto hold the consequences of those beliefs. Logic might thus tell us atleast this much, though: whenever I have some reason to believe \(A\)and if \(A\) then \(B\), then I have a prima facie reason to believe\(B\). See Harman 1986 for the view that logic has no distinctivenormative role, and Field 2009 for a nice critical discussion ofHarman’s view and an argument why logic should be tied to normsof rationality. A survey of this an related issues can be found in theentry onthe normativity of logic.

And, of course, logic does not tell us how we ought to reason or inferin all particular cases. Logic does not deal with the particularcases, but only with the most generally valid forms of reasoning orinference, ones that are valid no matter what one reasons about. Inthis sense logic is often seen to be topic neutral. It applies nomatter what one is thinking or reasoning about. And this neutrality,or complete generality of logic, together with its normativity, isoften put as “logic is about how we ought to think if we are tothink at all” or “logic is the science of the laws that weought to follow in our thinking no matter what we think about”.There are well known philosophical puzzles about normativity, andthese apply to logic as well if it is normative. One is why it is thatthinkers are under such norms. After all, why shouldn’t I thinkthe way I prefer to think, without there being some norm that governsmy thinking, whether I like it or not? Why is there an“ought” that comes with thinking as such, even if Idon’t want to think that way? One idea to answer this is toemploy the notion of a ‘constitutive aim of belief’, theidea that belief as such aims at something: the truth. If so thenmaybe one could argue that by having beliefs I am under the norm thatI ought to have true ones. And if one holds that one of the crucialfeatures of logically valid inferences is that they preserve truththen one could argue that the logical laws are norms that apply tothose who have beliefs. See Velleman 2000 for more on the aim ofbelief. The normativity of logic will not be central for ourdiscussion to follow, but the topic neutrality and generality will be.^[2]

Overall, we can thus distinguish four notions of logic:

(L1): the study of artificial formal languages
(L2): the study offormally valid inferences and logical consequence
(L3): the study of logical truths
(L4): the study of the general features, or form, of judgements

There is, of course, a question how these different conceptions oflogic relate to each other. The details of their relationship invitemany hard questions, but we should briefly look at thisnonetheless.

2.2 How the different conceptions of logic are related to each other

How (L1) and (L2) relate to each other is subject of controversy. Onestraightforward, though controversial view, is the following. For anygiven system of representations, like sentences in a natural language,there is one and only one set of logical constants. Thus there will beone formal language that best models what logically valid inferencesthere are among these natural representations. This formal languagewill have a logical vocabulary that captures the inferentialproperties of the logical constants, and that models all otherrelevant features of the natural system of representation withnon-logical vocabulary. One especially important system ofrepresentations are our natural languages. Thus (L1) is the study offormal languages of which one is distinguished, and this onedistinguished language nicely represents the fixed and non-fixedfeatures of our natural languages, through its logical and non-logicalvocabulary, at least assuming that our natural languages are similarto each other in this regard. And validity in that formal language, atechnical notion defined in the appropriate way for that formallanguage, nicely models logical validity or logical consequence in ournatural language system of representations. Or so this view of therelationship between (L1) and (L2) holds.

This view of the relationship between (L1) and (L2), however, assumesthat there is one and only one set of logical constants for eachsystem of representations. A contrary view holds that whichexpressions are treated as logical constants is a matter of choice,with different choices serving different purposes. If we fix, say,‘believes’ and ‘knows’ then we can see that‘\(x\) believes that \(p\)’ is implied by ‘\(x\)knows that \(p\)’ (given widely held views about knowledge andbelief). This does not mean that ‘believes’ is a logicalconstant in an absolute sense. Given other interests, otherexpressions can be treated as logical. According to this conception,different formal languages will be useful in modeling the inferencesthat are formally valid given different set of ‘logicalconstants’ or expressions whose meaning is kept fixed.

This debate thus concerns whether there is one and only one set oflogical constants for a system of representations, and if so, whichones are the logical ones. We will not get into this debate here, butthere is quite a large literature on what logical constants are, andhow logic can be demarcated. For a general discussion and furtherreferences, see for example Engel 1991. Some of the classic papersin this debate include Hacking 1979, who defends a proof-theoreticway of distinguishing logical constants from other expressions. Theleading idea here is that logical constants are those whose meaningcan be given by proof-theoretic introduction and elimination rules. Onthe other hand, Mauthner 1946, van Benthem 1986, van Benthem1989, and Tarski 1986 defend semantic ways to mark that difference.The leading idea here is that logical notions are ‘permutationinvariant’. Since logic is supposed to be completely general andneutral with respect to what the representations are about, it shouldnot matter to logic if we switch around the objects that theserepresentations are about. So, logical notions are those that areinvariant under permutations of the domain. Van Benthem 1989 gives ageneral formulation to this idea. See the entry onlogical constants for more.

The relationship between (L2) and (L3) was briefly addressed above.They seem to be closely related because a logical truth can beunderstood as one that follows from an empty set of premises, and Abeing a logical consequence of B can be understood as it being alogical truth that if A then B. There are some questions to be ironedout about how this is supposed to go more precisely. How should weunderstand cases of logical consequence from infinitely many premises?Are logical truths all finitely statable? But for our purposes we cansay that they are rather closely related.

The relationship between (L2) and (L4) on the other hand raises somequestions. For one, of course, there is an issue about what it meansto say that judgments have a form, and whether they do in the relevantsense. But one way in which this question could be understood directlyties it to (L2). If thoughts, and thus judgments, are realized byminds having a certain relation to mental representations, and ifthese representations are themselves structured like a language, witha “syntax” and a “semantics” (properlyunderstood), then the form of a judgment could be understood just likethe form of a sentence. Such a view of thoughts is commonly called theLanguage of Thought hypothesis, see Fodor 1975, and if it iscorrect then in the language of thought there might be logical andnon-logical vocabulary. The form of a judgment could be understoodalong the lines we understood the form of a linguistic representationwhen we talked about formally valid inferences. Thus the relationshipbetween (L2) and (L4) is rather direct. On both conceptions of logicwe deal with logical constants, the difference is that one deals witha system of mental representations, the other with a system oflinguistic representations. Both, presumably, would deal withcorresponding sets of logical constants. Even though mental andlinguistic representations form different sets of representations,since they are closely connected with each other, for every logicalconstant in one of these sets of representations there will be anotherone of the corresponding syntactic type and with the same content, orat least a corresponding inferential role.

But this conception of their relationship assumes that the“general features of judgments” or “forms ofjudgment” which (L4) is concerned with deal with something likethe logical constants in the language of thought. Here the judgment asa mental act is assumed to operate on a mental representation thatitself has syntactic structure. And the form of the judgment wasunderstood as the form of the representation that represents thecontent of the judgment, whereby the form of the representation wasunderstood along the lines of (L2), involving logical constants. Butwhat if we can’t understand “form of judgment” or“form of thought” that way? One way this could fail is ifthe language of thought hypothesis itself fails, and if mental statesdo not involve representations that have something like a syntacticform. The question then becomes, first how should we understand‘form of judgement’ more precisely, and secondly, how doeslogic, as the discipline concerned with forms of judgments in thesense of (L4), relate to (L2)?

One way to answer the first question is to understand “form ofjudgment” as not being concerned with the representation thatmight be involved in a judgment, but rather with the content of thejudgment, i.e. with what the judgment is representing to be the case.Contents of judgments can be seen as propositions, and these can beunderstood as entities that are structured, for example Russellianpropositions. Such propositions are ordered sets whose members areobjects and properties. How such a conception of (L4) relates to (L2)will in part depend on how one thinks of the logical constants inRussellian propositions. If they are higher-order properties orfunctions that are members of these propositions alongside otherobjects and properties then presumably the logical constants havecontent. But this seems to be in conflict with an understanding of(L4) as being concerned with the form that is left once we abstractfrom all content. If would seem that on such an understanding of (L4)one can’t closely associate ‘form of judgment’,understood as what’s left once we abstract from all content ofthe judgment, with logical constants if the latter have content.

Another way to understand “form” as being concerned withwhat the judgment is about, rather than the judgment itself, is tothink of what it is about, the world, itself as having a form. In thissense we associate “form” neither with the representationthat is involved in the judgment, nor with the proposition which isits content, but rather with the world that is judged about. On such aconception the world itself has a form or basic structure. (L4) wouldbe concerned with this structure. How (L4) relates to (L2) is then asomewhat tricky question. One way, again, could be that the logicalconstants that (L2) is concerned with correspond to the structure ofwhat a representation in which they occur is about, but don’tcontribute to the content of that representation. This again seemsincompatible with the logical constants themselves having content. So,whether one associates form of judgment with the‘syntactic’ structure of a representation that is involvedin the judgment, or with the content of that representation, or withthe structure of what the representation is about, the relationshipbetween (L4) and (L2) will in part depend on whether one thinks thelogical constants themselves contribute to content. If they do, and ifform is contrasted with content, then a close association seemsimpossible. If the logical constants don’t have content, then itmight be possible.

Finally, the relationship between (L1) and (L4) either comes down tothe same as that between (L1) and (L2), if we understand ‘formof thought’ analogous to ‘form of representation’.If not, then it will again depend on how (L4) is understood moreprecisely.

Thus there are many ways in which (L1), (L2), (L3), and (L4) areconnected, and many in which they are quite different.

3. Ontology

3.1 Different conceptions of ontology

As a first approximation, ontology is the study of what there is. Somecontest this formulation of what ontology is, so it’s only afirst approximation. Many classical philosophical problems areproblems in ontology: the question whether or not there is a god, orthe problem of the existence of universals, etc.. These are allproblems in ontology in the sense that they deal with whether or not acertain thing, or more broadly entity, exists. But ontology is usuallyalso taken to encompass problems about the most general features andrelations of the entities which do exist. There are also a number ofclassic philosophical problems that are problems in ontologyunderstood in this way. For example, the problem of how a universalrelates to a particular that has it (assuming there are universals andparticulars), or the problem of how an event like John eating a cookierelates to the particulars John and the cookie, and the relation ofeating, assuming there are events, particulars and relations. Thesekinds of problems quickly turn into metaphysics more generally, whichis the philosophical discipline that encompasses ontology as one ofits parts. The borders here are a little fuzzy. But we have at leasttwo parts to the overall philosophical project of ontology, on ourpreliminary understanding of it: first, say what there is, whatexists, what the stuff of reality is made out of, secondly, say whatthe most general features and relations of these things are.

This way of looking at ontology comes with two sets of problems whichleads to the philosophical discipline of ontology being more complexthan just answering the above questions. The first set of problems isthat it isn’t clear how to approach answering these questions.This leads to the debate about ontological commitment. The second setof problems is that it isn’t so clear what these questionsreally are. This leads to the philosophical debate aboutmeta-ontology. Let's look at them in turn.

One of the troubles with ontology is that it not only isn’tclear what there is, it also isn’t so clear how to settlequestions about what there is, at least not for the kinds of thingsthat have traditionally been of special interest to philosophers:numbers, properties, God, etc. Ontology is thus a philosophicaldiscipline that encompasses besides the study of what there is and thestudy of the general features of what there is also the study of whatis involved in settling questions about what there is in general,especially for the philosophically tricky cases. How we can find outwhat there is isn’t an easy question to answer. It might seemsimple enough for regular objects that we can perceive with our eyes,like my house keys, but how should we decide it for such things as,say, numbers or properties? One first step to making progress on thisquestion is to see if what we believe already rationally settles thisquestion. That is to say, given that we have certain beliefs, do thesebeliefs already bring with them a rational commitment to an answer tosuch questions as ‘Are there numbers?’ If our beliefsbring with them a rational commitment to an answer to an ontologicalquestion about the existence of certain entities then we can say thatwe are committed to the existence of these entities. What precisely isrequired for such a commitment to occur is subject to debate, a debatewe will look at momentarily. To find out what one is committed to witha particular set of beliefs, or acceptance of a particular theory ofthe world, is part of the larger discipline of ontology.

Besides it not being so clear what it is to commit yourself to ananswer to an ontological question, it also isn’t so clear whatan ontological question really is, and thus what it is that ontologyis supposed to accomplish. To figure this out is the task ofmeta-ontology, which strictly speaking is not part of ontologyconstrued narrowly, but the study of what ontology is. However, likemost philosophical disciplines, ontology more broadly construedcontains its own meta-study, and thus meta-ontology is part ofontology, more broadly construed. Nonetheless it is helpful toseparate it out as a special part of ontology. Many of thephilosophically most fundamental questions about ontology really aremeta-ontological questions. Meta-ontology has not been too popular inthe latter parts of the 20th century, partly because onemeta-ontological view, the one often associated with Quine, had beenwidely accepted as the correct one, but this acceptance has beenchallenged in recent years in a variety of ways. One motivation forthe study of meta-ontology is simply the question of what questionontology aims to answer. Take the case of numbers, for example. Whatis the question that we should aim to answer in ontology if we want tofind out if there are numbers, that is, if reality contains numbersbesides whatever else it is made up from? This way of putting itsuggest an easy answer: ‘Are there numbers?’ But thisquestion seems like an easy one to answer. An answer to it is implied,it seems, by trivial mathematics, say that the number 7 is less thanthe number 8. If the latter, then there is a number which is less than8, namely 7, and thus there is at least one number. Can ontology bethat easy? The study of meta-ontology will have to determine, amongstothers, if ‘Are there numbers?’ really is the questionthat the discipline of ontology is supposed to answer, and moregenerally, what ontology is supposed to do. We will pursue thesequestions further below. As we will see, several philosophers thinkthat ontology is supposed to answer a different question than whatthere is, but they often disagree on what that question is.

The larger discipline of ontology can thus be seen as having fourparts:

(O1): the study of ontological commitment, i.e. what we or others arecommitted to,
(O2): the study of what there is,
(O3): the study of the most general features of what there is, and howthe things there are relate to each other in the metaphysically mostgeneral ways,
(O4): the study of meta-ontology, i.e. saying what task it is that thediscipline of ontology should aim to accomplish, if any, how thequestions it aims to answer should be understood, and with whatmethodology they can be answered.

3.2 How the different conceptions of ontology are related to each other

The relationship between these four seems rather straightforward. (O4)will have to say how the other three are supposed to be understood. Inparticular, it will have to tell us if the question to be answered in(O2) indeed is the question what there is, which was taken above to beonly a first approximation for how to state what ontology is supposedto do. Maybe it is supposed to answer the question what is realinstead, or what is fundamental, some other question. Whatever onesays here will also affect how one should understand (O1). We will atfirst work with what is the most common way to understand (O2) and(O1), and discuss alternatives in turn. If (O1) has the result thatthe beliefs we share commit us to a certain kind of entity then thisrequires us either to accept an answer to a question about what thereis in the sense of (O2) or to revise our beliefs. If we accept thatthere is such an entity in (O2) then this invites questions in (O3)about its nature and the general relations it has to other things wealso accept. On the other hand, investigations in (O3) into the natureof entities that we are not committed to and that we have no reason tobelieve exist would seem like a rather speculative project, though, ofcourse, it could still be fun and interesting.

4. Areas of overlap

The debates about logic and about ontology overlap at various places.Given the division of ontology into (O1)–(O4), and the divisionof logic into (L1)–(L4) we can consider several issues wherelogic, understood a certain way, overlaps with ontology, understood acertain way. In the following we will discuss some paradigmaticdebates related to the relationship between logic and ontology,organized by areas of overlap.

4.1 Formal languages and ontological commitment. (L1) meets (O1) and (O4)

Suppose we have a set of beliefs, and we wonder what the answer to theontological question ‘Are there numbers?’ is, assuming(O4) tells us this is the ontological question about numbers. Onestrategy to see whether our beliefs already commit us to an answer ofthis question is as follows: first, write out all those beliefs in apublic language, like English. This by itself might not seem to helpmuch, since if it wasn’t clear what my beliefs commit me to, whywould it help to look at what acceptance of what these sentences saycommits me to? But now, secondly, write these sentences in what isoften called ‘canonical notation’. Canonical notation canbe understood as a formal or semi-formal language that brings out thetrue underlying structure, or ‘logical form’ of a naturallanguage sentence. In particular, such a canonical notation will makeexplicit which quantifiers do occur in these sentences, what theirscope is, and the like. This is where formal languages come into thepicture. After that, and thirdly, look at the variables that are boundby these quantifiers.^[3] What values do they have to have in order for these sentences all tobe true? If the answer is that the variables have to have numbers astheir values, then you are committed to numbers. If not then youaren’t committed to numbers. The latter doesn’t mean thatthere are no numbers, of course, just as you being committed to themdoesn’t mean that there are numbers. But if your beliefs are alltrue then there have to be numbers, if you are committed to numbers.Or so this strategy goes.

All this might seem a lot of extra work for little. What do we reallygain from these ‘canonical notations’ in determiningontological commitment? One attempt to answer this, which partlymotivates the above way of doing things, is based on the followingconsideration: We might wonder why we should think that quantifiersare of great importance for making ontological commitments explicit.After all, if I accept the apparently trivial mathematical fact thatthere is a number between 6 and 8, does this already commit me to ananswer to the ontological question whether there are numbers outthere, as part of reality? The above strategy tries to make explicitthat and why it in fact does commit me to such an answer. This is sosince natural language quantifiers are fully captured by their formalanalogues in canonical notation, and the latter make ontologicalcommitments obvious because of their semantics. Such formalquantifiers are given what is called an ‘objectualsemantics’. This is to say that a particular quantifiedstatement \(\lsquo \exists x\,Fx\rsquo\) is true just in case thereis an object in the domain of quantification that, when assigned asthe value of the variable \(\lsquo x\rsquo\), satisfies the open formula\(\lsquo Fx\rsquo\). This makes obvious that the truth of aquantified statement is ontologically relevant, and in fact ideallysuited to make ontological commitment explicit, since we need entitiesto assign as the values of the variables. Thus (L1) is tied to (O1).The philosopher most closely associated with this way of determiningontological commitment, and with the meta-ontological view on which itis based, is Quine (in particular Quine 1948). See also van Inwagen1998 for a presentation sympathetic to Quine.

The above account of ontological commitment has been criticized from avariety of different angles. One criticism focuses on the semanticsthat is given for quantifiers in the formal language that is used asthe canonical notation of the natural language representations of thecontents of beliefs. The above, objectual semantics is not the onlyone that can be given to quantifiers. One widely discussed alternativeis the so-called ‘substitutional semantics’. According toit we do not assign entities as values of variables. Rather, aparticular quantified statement \(\lsquo \exists x\,Fx\rsquo\) istrue just in case there is a term in the language that whensubstituted for \(\lsquo x\rsquo\) in \(\lsquo Fx\rsquo\) has a truesentence as its result. Thus, \(\lsquo \exists x\,Fx\rsquo\) is truejust in case there is an instance \(\lsquo Ft\rsquo\) which is true,for \(\lsquo t\rsquo\) a term in the language in question,substituted for all (free) occurrences of \(\lsquo x\rsquo\) in\(\lsquo Fx\rsquo\). The substitutional semantics for the quantifiershas often been used to argue that there are ontologically innocentuses of quantifiers, and that what quantified statements we acceptdoes not directly reveal ontological commitment. Gottlieb (1980)provides more details on substitutional quantification, and an attemptto use it in the philosophy of mathematics. Earlier work was done byRuth Marcus, and is reprinted in Marcus 1993.

Another objection to the above account of determining ontologicalcommitment goes further and questions the use of a canonical notation,and of formal tools in general. It states that if the ontologicalquestion about numbers simply is the question ‘Are therenumbers?’ then all that matters for ontological commitment iswhether or not what we accept implies ‘There are numbers’.In particular, it is irrelevant what the semantics for quantifiers ina formal language is, in particular, whether it is objectual orsubstitutional. What ontological commitment comes down to can bedetermined at the level of ordinary English. Formal tools are of no,or at best limited, importance. Ontological commitment can thusaccording to this line of thought be formulated simply as follows: youare committed to numbers if what you believe implies that there arenumbers. Notwithstanding the debate between the substitutional andobjectual semantics, we do not need any formal tools to spell out thesemantics of quantifiers. All that matters is that a certainquantified statement ‘There are \(F\)s’ is implied by whatwe believe for us to be committed to \(F\)s. What does not matter iswhether the semantics of the quantifier in “There are\(F\)s” (assuming it contains a quantifier^[4]) is objectual or substitutional.

However, even if one agrees that what matters for ontologicalcommitment is whether or not what one believes implies that there are\(F\)s, for a certain kind of thing \(F\), there might still be roomfor formal tools. First of all, it isn’t clear what implieswhat. Whether or not a set of statements that express my beliefs implythat there are entities of a certain kind might not be obvious, andmight even be controversial. Formal methods can be useful indetermining what implies what. On the other hand, even though formalmethods can be useful in determining what implies what, it is notclear which formal tools are the right ones for modeling a naturalsystem of representations. It might seem that to determine which arethe right formal tools we already need to know what the implicationalrelations are between the natural representations that we attempt tomodel, at least in basic cases. This could mean that formal tools areonly of limited use for deciding controversial cases ofimplication.

But then, again, it has been argued that often it is not at all clearwhich statements really involve quantifiers at a more fundamentallevel of analysis, orlogical form. Russell (1905) famously argued that “the King of France”is a quantified expression, even though it appears to be a referringexpression on the face of it, a claim now accepted by many. AndDavidson (1967) argued, that ‘action sentences’ like“Fred buttered the toast” involve quantification overevents in the logical form, though not on the surface, a claim that ismore controversial. One might argue in light of these debates thatwhich sentences involve quantification over what can’t befinally settled until we have a formal semantics of all of our naturallanguage, and that this formal semantics will give us the ultimateanswer to what we are quantifying over. But then again, how are we totell that the formal semantics proposed is correct, if we don’tknow the inferential relations in our own language?

One further use that formal tools could have besides all the above isto make ambiguities and different ‘readings’ explicit, andto model their respective inferential behavior. For example, formaltools are especially useful to make scope ambiguities explicit, sincedifferent scope readings of one and the same natural language sentencecan be represented with different formal sentences which themselveshave no scope ambiguities. This use of formal tools is not restrictedto ontology, but applies to any debates where ambiguities can be ahindrance. It does help in ontology, though, if some of the relevantexpressions in ontological debates, like the quantifiers themselves,exhibit such different readings. Then formal tools will be most usefulto make this explicit. Whether or not quantifiers indeed do havedifferent readings is a question that will not be solved with formaltools, but if they do then these tools will be most useful inspecifying what these readings are. For a proposal of this latterkind, see Hofweber 2016. One consequence of this is a meta-ontologydifferent from Quine’s, as we will discuss below.

All this discussion in this section assumed that ontologicalcommitment is connected to a conception of ontology that concerns whatthere is. But this is not universally accepted, in particularrecently. Maybe ontology does not concern what there is, but what isfundamental, in some sense of the word. If so, then issues tied toquantifiers are not of the most central importance when it comes toontological commitment, although they would still play a role. Themain question would then be connected to fundamentality. And here,too, formal languages might play a role in determining what one iscommitted to being fundamental. We will more closely discuss the roleof formal languages in conceptions of ontology as concerning thefundamental below, in section 4.5.

4.2 Is logic neutral about what there is? (L2) meets (O2)

Logically valid inferences are those that are guaranteed to be validby their form. And above we spelled this out as follows: an inferenceis valid by its form if as long as we fix the meaning of certainspecial expressions, the logical constants, we can ignore the meaningof the other expressions in the statements involved in the inference,and we are always guaranteed that the inference is valid, no matterwhat the meaning of the other expressions is, as long as the whole ismeaningful. A logical truth can be understood as a statement whosetruth is guaranteed as long as the meanings of the logical constantsare fixed, no matter what the meaning of the other expressions is.Alternatively, a logical truth is one that is a logical consequencefrom no assumptions, i.e. an empty set of premises.

Do logical truths entail the existence of any entities, or is theirtruth independent of what exists? There are some well knownconsiderations that seem to support the view that logic should beneutral with respect to what there is. On the other hand, there arealso some well known arguments to the contrary. In this section wewill survey some of this debate.

If logical truth are ones whose truth is guaranteed as long as themeaning of the logical constants is kept fixed then logical truths aregood candidates for being analytic truths. Can analytic truths implythe existence of any entities? This is an old debate, often conductedusing “conceptual truths” instead of “analytictruths”. The most prominent debate of this kind is the debateabout the ontological argument for the existence of God. Manyphilosophers have maintained that there can be no conceptualcontradiction in denying the existence of particular entities, andthus there can be no proof of their existence with conceptual truthsalone. In particular, an ontological argument for the existence of Godis impossible. A famous discussion to this effect is Kant’sdiscussion of the ontological argument (Kant 1781/7, KrVA592/B620 ff). On the other hand, many other philosophers havemaintained that such an ontological argument is possible, and theyhave made a variety of different proposals how it can go. We will notdiscuss the ontological argument here, however, it is discussed indetail in different formulations in the entry onontological arguments in this encyclopedia.

Whatever one says about the possibility of proving the existence of anobject purely with conceptual truths, many philosophers havemaintained that at leastlogic has to be neutral about whatthere is. One of the reasons for this insistence is the idea thatlogic is topic neutral, or purely general. The logical truths are theones that hold no matter what the representations are about, and thusthey hold in any domain. In particular, they hold in an empty domain,one where there is nothing at all. And if that is true then logicaltruths can’t imply that anything exists. But that argument mightbe turned around by a believer in logical objects, objects whoseexistence is implied by logic alone. If it is granted that logicaltruths have to hold in any domain, then any domain has to contain thelogical objects. Thus for a believer in logical objects there can beno empty domain.

There is a close relationship between this debate and a commoncriticism that standard formal logics (in the sense of (L1))won’t be able to capture the logical truths (in the sense of(L3)). It is the debate about the status of the empty domain in thesemantics of first and second order logical systems.

It is a logical truth in (standard) first order logic that somethingexists, i.e., ‘\(\exists x\,x=x\)’. Similarly, it is alogical truth in (standard versions of) second order logic that‘\(\exists F\forall x\,(Fx \vee \neg Fx)\)’. These areexistentially quantified statements. Thus, one might argue, logic isnot neutral with respect to what there is. There are logical truthsthat state that something exists. However, it would be premature toconclude that logic is not neutral about what there is, simply becausethere are logical truths in (standard) first or second order logicwhich are existential statements. If we look more closely how it comesabout that these existential statements are logical truths in theselogical systems we see that it is only so because, by definition, amodel for (standard) first order logic has to have a non-empty domain.It is possible to allow for models with an empty domain as well (wherenothing exists), but models with an empty domain are excluded, again,by definition from the (standard) semantics in first order logic. Thus(standard) first order logic is sometimes called the logic of firstorder models with a non-empty domain. If we allow an empty domain aswell we will need different axioms or rules of inference to have asound proof system, but this can be done. Thus even though there areformal logical systems, in the sense of (L1) in which there arelogical truths that are existential statements, this does not answerthe question whether or not there are logical truths, in the sense of(L2), that are existential statements. The question rather is whichformal system, in the sense of (L1), best captures the logical truths,in the sense of (L2). So, even if we agree that a first order logicalsystem is a good formal system to represent logical inferences, shouldwe adopt the axioms and rules for models with or without an emptydomain?

A related debate is the debate about free logic. Free logics areformal systems that drop the assumption made in standard first andhigher order logic that every closed term denotes an object in thedomain of the model. Free logic allows for terms that denote nothing,and in free logic certain rules about the inferential interactionbetween quantifiers and terms have to be modified. Whether free orun-free (standard) logic is the better formal model for naturallanguage logical inference is a further question. For more discussionof logic with an empty domain see Quine 1954 and Williamson 1999.For a sound and complete proof system for logic with an empty domain,see Tennant 1990. For a survey article on free logic, see Lambert2001.

How innocent logic is with respect to ontology is also at the heart ofthe debate about the status of second order logic as logic. Quine(1970) argued that second order logic was “set theory insheep’s clothing”, and thus not properly logic at all.Quine was concerned with the questions of whether second orderquantifiers should be understood as ranging over properties or oversets of individuals. The former were considered dubious in variousways, the latter turn second order logic into set theory. Thisapproach to second order logic has been extensively criticized byvarious authors, most notably George Boolos, who in a series ofpapers, collected in part I of Boolos 1998, attempted to vindicatesecond order logic, and to propose a plural interpretation, which isdiscussed in the article onplural quantification.

A particularly important and pressing case of the ontologicalimplications of logic are logicist programs in the philosophy ofmathematics, in particular Frege’s conception of logical objectsand his philosophy of arithmetic. Frege and neo-Fregeans following himbelieve that arithmetic is logic (plus definitions) and that numbersare objects whose existence is implied by arithmetic. Thus inparticular, logic implies the existence of certain objects, andnumbers are among them. Frege’s position has been criticized asbeing untenable since logic has to be neutral about what there is.Thus mathematics, or even a part thereof, can’t be both logicand about objects. The inconsistency of Frege’s originalformulation of his position sometimes has been taken to show this, butsince consistent formulations of Frege’s philosophy ofarithmetic have surfaced this last point is moot. Frege’sargument for numbers as objects and arithmetic as logic is probablythe best known argument for logic implying the existence of entities.It has been very carefully investigated in recent years, but whetheror not it succeeds is controversial. Followers of Frege defend it asthe solution to major problems in the philosophy of mathematics; theircritics find the argument flawed or even just a cheap trick that isobviously going nowhere. We will not discuss the details here, but adetailed presentation of the argument can be found in the entry onFrege’s theorem and foundations of arithmetic as well as Rosen 1993, which gives a clear and readablepresentation of the main argument of Wright (1983), which in turn ispartially responsible for a revival of Fregean ideas along theselines. Frege’s own version is in his classicGrundlagen(1884). A discussion of recent attempts to revive Frege can be foundin Hale and Wright 2001, Boolos 1998 and Fine 2002. A discussionof Frege’s and Kant’s conceptions of logic is inMacFarlane 2002 which also contains many historical references.

4.3 Formal ontology. (L1) meets (O2) and (O3)

Formal ontologies are theories that attempt to give precisemathematical formulations of the properties and relations of certainentities. Such theories usually propose axioms about these entities inquestion, spelled out in some formal language based on some system offormal logic. Formal ontology can been seen as coming in three kinds,depending on their philosophical ambition. Let’s call themrepresentational, descriptive, and systematic. We will in this sectionbriefly discuss what philosophers, and others, have hoped to do withsuch formal ontologies.

A formal ontology is a mathematical theory of certain entities,formulated in a formal, artificial language, which in turn is based onsome logical system like first order logic, or some form of the lambdacalculus, or the like. Such a formal ontology will specify axiomsabout what entities of this kind there are, what their relations amongeach other are, and so on. Formal ontologies could also only haveaxioms that state how the things the theory is about, whatever theymay be, relate to each other, but no axioms that state that certainthings exist. For example, a formal ontology of events won’t saywhich events there are. That is an empirical question. But it mightsay under what operations events are closed under, and what structureall the events there are exhibit. Similarly for formal ontologies ofthe part-whole relation, and others. See Simons 1987 for a wellknown book on various formal versions of mereology, the study of partsand wholes.

Formal ontologies can be useful in a variety of different ways. Onecontemporary use is as a framework to represent information in anespecially useful way. Information represented in a particular formalontology can be more easily accessible to automated informationprocessing, and how best to do this is an active area of research incomputer science. The use of the formal ontology here isrepresentational. It is a framework to represent information, and assuch it can be representationally successful whether or not the formaltheory used in fact truly describes a domain of entities. So, a formalontology of states of affairs, lets say, can be most useful torepresent information that might otherwise be represented in plainEnglish, and this can be so whether or not there indeed are any statesof affairs in the world. Such uses of formal ontologies are thusrepresentational.

A different philosophical use of a formal ontology is one that aims tobe descriptive. A descriptive formal ontology aims to correctlydescribe a certain domain of entities, say sets, or numbers, asopposed to all things there are. Take common conceptions of set theoryas one example. Many people take set theory to aim at correctlydescribing a domain of entities, the pure sets. This is, of course, acontroversial claim in the philosophy of set theory, but if it iscorrect then set theory could be seen as a descriptive formal ontologyof pure sets. It would imply that among incompatible formal theoriesof sets only one could be correct. If set theory were merelyrepresentational then both of the incompatible theories could beequally useful as representational tools, though probably fordifferent representational tasks.

Finally, formal ontologies have been proposed as systematic theoriesof what there is, with some restrictions. Such systematic theorieshope to give one formal theory for all there is, or at least a goodpart of it. Hardly anyone would claim that there can be a simpleformal theory that correctly states what concrete physical objectsthere are. There does not seem to be a simple principle thatdetermines whether there are an even or odd number of mice at aparticular time. But maybe this apparent randomness only holds forconcrete physical objects. It might not hold for abstract objects,which according to many exist not contingently, but necessarily if atall. Maybe a systematic, simple formal theory is possible of allabstract objects. Such a systematic formal ontology will most commonlyhave one kind of entities which are the primary subject of the theory,and a variety of different notions of reduction that specify how other(abstract) objects really are entities of this special kind. A simpleview of this kind would be one according to which all abstract objectsare sets, and numbers, properties, etc. are really special kinds ofsets. However, more sophisticated versions of systematic formalontologies have been developed. An ambitious systematic formalontology can be found in Zalta 1983 and Zalta 1999 [2022] (see theOther Internet Resources).

Representational formal ontologies, somewhat paradoxically, areindependent of any strictly ontological issues. Their success orfailure is independent of what there is. Descriptive formal ontologiesare just like representational ones, except with the ambition ofdescribing a domain of entities. Systematic formal ontologies gofurther in not only describing one domain, but in relating allentities (of a certain kind) to each other, often with particularnotions of reduction. These theories seem to be the most ambitious.Their motivation comes from an attempt to find a simple and systematictheory of all, say, abstract entities, and they can rely on theparadigm of aiming for simplicity in the physical sciences as a guide.They, just like descriptive theories, will have to have as theirstarting point a reasonable degree of certainty that we indeed areontologically committed to the entities they aim to capture. Withoutthat these enterprises seem to have little attraction. But even if thelatter philosophical ambitions fail, a formal ontology can still be amost useful representational tool.

4.4 Carnap’s rejection of ontology. (L1) meets (O4) and (the end of?) (O2)

One interesting view about the relationship between formal languages,ontology, and meta-ontology is the one developed by Carnap in thefirst half of the 20th century, and which is one of the startingpoints of the contemporary debate in ontology, leading to thewell-known exchange between Carnap and Quine, to be discussed below.According to Carnap one crucial project in philosophy is to developframeworks that can be used by scientists to formulatetheories of the world. Such frameworks are formal languages that havea clearly defined relationship to experience or empirical evidence aspart of their semantics. For Carnap it was a matter of usefulness andpracticality which one of these frameworks will be selected by thescientists to formulate their theories in, and there is no one correctframework that truly mirrors the world as it is in itself. Theadoption of one framework rather than another is thus a practicalquestion.

Carnap distinguished two kinds of questions that can be asked aboutwhat there is. One are the so-called ‘internal questions’,questions like ‘Are there infinitely many prime numbers?’These questions make sense once a framework that contains talk aboutnumbers has been adopted. Such questions vary in degree of difficulty.Some are very hard, like ‘Are there infinitely many twin primenumbers?’, some are of medium difficulty, like ‘Are thereinfinitely many prime numbers?’, some are easy like ‘Arethere prime numbers?’, and some are completely trivial, like‘Are there numbers?’. Internal questions are thusquestions that can be asked once a framework that allows talk aboutcertain things has been adopted, and general internal questions, like‘Are there numbers?’ are completely trivial since once theframework of talk about numbers has been adopted the question if thereare any is settled within that framework.

But since the internal general questions are completely trivial theycan’t be what the philosophers and metaphysicians are after whenthey ask the ontological question ‘Are there numbers?’ Thephilosophers aim to ask a difficult and deep question, not a trivialone. What the philosophers aim to ask, according to Carnap, is not aquestion internal to the framework, but external to it. They aim toask whether the framework correctly corresponds to reality, whether ornot therereally are numbers. However, the words used in thequestion ‘Are there numbers?’ only have meaning within theframework of talk about numbers, and thus if they are meaningful atall they form an internal question, with a trivial answer. Theexternal questions that the metaphysician tries to ask aremeaningless. Ontology, the philosophical discipline that tries toanswer hard questions about what therereally is is based ona mistake. The question it tries to answer are meaningless questions,and this enterprise should be abandoned. The words ‘Are therenumbers?’ thus can be used in two ways: as an internal question,in which case the answer is trivially ‘yes’, but this hasnothing to do with metaphysics or ontology, or as an externalquestion, which is the one the philosophers are trying to ask, butwhich is meaningless. Philosophers should thus not be concerned with(O2), which is a discipline that tries to answer meaninglessquestions, but with (L1), which is a discipline that, in part,develops frameworks for science to use to formulate and answer realquestions. Carnap’s ideas about ontology and meta-ontology aredeveloped in a classic essay (Carnap 1956b). A nice summary ofCarnap’s views can be found in his intellectual autobiography(Carnap 1963).

Carnap’s rejection of ontology, and metaphysics more generally,has been widely criticized from a number of different angles. Onecommon criticism is that it relies on a too simplistic conception ofnatural language that ties it too closely to science or to evidenceand verification. In particular, Carnap’s more general rejectionof metaphysics used a verificationist conception of meaning, which iswidely seen as too simplistic. Carnap’s rejection of ontologyhas been criticized most prominently by Quine, and the debate betweenCarnap and Quine on ontology is a classic in this field. Quinerejected Carnap’s conception that when scientists are faced withdata that don’t fit their theory they have two choices. Firstthey could change the theory, but stay in the same framework.Secondly, they could move to a different framework, and formulate anew theory within that framework. These two moves for Carnap aresubstantially different. Quine would want to see them as fundamentallysimilar. In particular, Quine rejects the idea that there could betruths which are the trivial internal statements, like “Thereare numbers”, whose truth is a given once the framework ofnumbers has been adopted. Thus some such internal statements would beanalytic truths, and Quine is well known for thinking that thedistinction between analytic and synthetic truths is untenable. ThusCarnap’s distinction between internal and external questions isto be rejected alongside with the rejection of the distinction betweenanalytic and synthetic truths. On the other hand, Quine and Carnapagree that ontology in the traditional philosophical sense is to berejected. Traditionally ontology has often, but not always, been anarmchair, a priori, investigation into the fundamental building blocksof reality. As such it is completely separated from science. Quine(1951) rejects this approach to ontology since he holds that therecan’t be such an investigation into reality that is completelyseparate and prior to the rest of inquiry. See Yablo 1998 for moreon the debate between Quine and Carnap, which contains many referencesto the relevant passages. The view on ontological commitment discussedin section 4.1., which is usually attributed to Quine, was developedas a reaction to Carnap’s position discussed in thissection. Simply put, Quine’s view is that to see what we arecommitted to we have to see what our best overall theory of the worldquantifies over. In particular, we look at our best overall scientifictheory of the world, which contains physics and the rest.

Carnap’s arguments for the rejection of ontology are presentlywidely rejected. However, several philosophers have recently attemptedto revive some parts or others of Carnap’s ideas. For example,Stephen Yablo has argued that an internal-external distinction couldbe understood along the lines of the fictional-literal distinction.And he argues (Yablo 1998) that since there is no fact about thisdistinction, ontology, in the sense of (O2), rests on a mistake and isto be rejected, as Carnap did. On the other hand, Thomas Hofweber hasargued that an internal-external distinction with many of the featuresthat Carnap wanted can be defended on the basis of facts about naturallanguage, but that such a distinction will not lead to a rejection ofontology, in the sense of (O2). See Hofweber 2016. Hilary Putnam(1987) has developed a view that revives some of the pragmatic aspectsof Carnap’s position. See Sosa 1993 for a critical discussion ofPutnam’s view, and Sosa 1999 for a related, positiveproposal. Robert Kraut (2016) has defended an expressivist reading ofthe internal-external distinction, and with it some Carnapianconsequences for ontology. And most of all, Eli Hirsch and AmieThomasson have defended different versions of approaches to ontologythat capture a good part of the spirit of Carnap’s view. See inparticular Hirsch 2011 and Thomasson 2015. For various views about theeffects of Carnap on the contemporary debate in ontology, see Blattiand Lapointe 2016.

4.5 The fundamental language. (L1) meets (O4) and (the new beginning of?) (O2)

Although ontology is often understood as the discipline that tries tofind out what there is, or what exists, this is rejected by many inthe contemporary debate. These philosophers think that the job ofontology is something different, and there is disagreement among themwhat it is more precisely. Among the proposed options are the projectsof finding out what is real, or what is fundamental, or what theprimary substances are, or what reality is like in itself, orsomething like this. Proponents of these approaches often find thequestions about what there is too inconsequential and trivial to takethem to be the questions for ontology. Whether there are numbers, say,is trivially answered in the affirmative, but whether numbers arereal, or whether they are fundamental, or primary substances, etc., isthe hard and ontological question. See Fine (2009) and Schaffer (2009)for two approaches along these lines. But such approaches have theirown problems. For example, it is not clear whether the questionwhether numbers are real is any different than the question whethernumbers exist. If one were to ask whether the Loch Ness monster isreal, it would naturally be understood as just the same question aswhether the Loch Ness monster exists. If it is supposed to be adifferent question, is this due to simple stipulation, or can we makethe difference intelligible? Similarly, it is not clear whether thenotion of what is fundamental can carry the intended metaphysicalweight. After all, there is a perfectly clear sense in which primenumbers are more fundamental in arithmetic than even numbers, but thisisn’t to hold the metaphysical priority of prime numbers overother numbers, but simply to hold that they are mathematically specialamong the numbers. Thus to ask whether numbers are fundamental is noteasily seen as a metaphysical alternative to the approach to ontologythat asks whether numbers exist. See Hofweber (2009; 2006, chapter 13)for a critical discussion of some approaches to ontology that rely onnotions of reality or fundamentality. Whether such approaches toontology are correct is a controversial topic in the debate aboutontology which we will not focus on here. However, this approach givesrise to a special connection between logic and ontology which we willdiscuss in the following.

The relation between the different approaches to ontology mentionedjust above is unclear. Is something that is part of reality as it isin itself something which is fundamental, or which is real in therelevant sense? Although it is unclear how these different approachesrelate to each other, all of them have the potential for allowing forthat our ordinary description of the world in terms of mid-sizeobjects, mathematics, morality, and so on, is literally true, while atthe same time these truths leave it open what the world, so to speak,deep down, really, and ultimately is like. To use one way ofarticulating this, even though there are tables, numbers, and values,reality in itself might contain none of them. Reality in itself mightcontain no objects at all, and nothing normative. Or it might. Theordinary description of the world, on this conception, leaves itlargely open what reality in itself is like. To find that out is thejob of metaphysics, in particular ontology. We might, given ourcognitive setup, be forced to think of the world as one of objects,say. But that might merely reflect how reality is for us. How it is initself is left open.

Whether the distinction between reality as it is for us and as it isin itself can be made sense of is an open question, in particular ifit is not simply the distinction between reality as it appears to us,and as it really is. This distinction would not allow for the optionthat our ordinary description of reality is true, while the questionhow reality is in itself is left open by this. If our ordinarydescription were true then this would mean that how reality appears tous is how it in fact is. But if this distinction can be made sense ofas intended then it gives rise to a problem about how to characterizereality as it is in itself, and this gives rise to a role for logic inthe sense of (L1).

If we are forced to think of the world in terms of objects because ofour cognitive makeup then it would be no surprise that our naturallanguage forces us to describe the world in terms of objects. Andarguably some of the central features of natural languages do exactlythat. It represents information in terms of subject and predicate,where the subject paradigmatically picks out an object and thepredicate paradigmatically attributes a property to it. If this iscorrect about natural language then it seems that natural language isutterly unsuitable to describe reality as it is in itself if thelatter does not contain any objects at all. But then, how are we todescribe reality as it is in itself?

Some philosophers have proposed that natural language might beunsuitable for the purposes of ontology. It might be unsuitable sinceit carries with it too much baggage from our particular conceptualscheme. See Burgess 2005 for a discussion. Or it might be unsuitablesince various expressions in it are not precise enough, too contextsensitive, or in other ways not ideally suited for the philosophicalproject. These philosophers propose instead to find a new, bettersuited language. Such a language likely will be a major departure fromnatural language and instead will be a formal, artificial language.This to be found language is often called ‘ontologese’(Dorr 2005, Sider 2009, Sider 2011), or ‘the fundamentallanguage’. The task thus is to find the fundamental language, alanguage in the sense of (L1), to properly carry out ontology, in thenew and revised sense of (O2): the project of finding out what realityfundamentally, or in itself, etc., is like. For a critical discussionof the proposal that we should be asking the questions of ontology inontologese, see Thomasson 2015 (chapter 10).

But this idea of a connection between (L1) and (O2) is notunproblematic. First there is a problem about making this approach to(O2) more precise. How to understand the notion of ‘reality initself’ is not at all clear, as is well known. It can’tjust mean: reality as it would be if we weren’t in it. On thisunderstanding it would simply be the world as it is except with nohumans in it, which would in many of its grander features be just asit in fact is. But then what does it mean? Similar, but different,worries apply to those who rely on notions like‘fundamental’, ‘substance’, and the like. Wewon’t pursue this issue here, though. Second, there is a seriousworry about how the formal language which is supposed to be thefundamental language is to be understood. In particular, is itsupposed to be merely an auxiliary tool, or an essential one? Thisquestion is tied to the motivation for a formal fundamental languagein the first place. If it is merely to overcome ambiguities,imperfections, and context sensitivities, then it most likely willmerely be an auxiliary, but not essential tool. After all, withinnatural language we have many means available to get rid ofambiguities, imperfections and context sensitivities. Scopeambiguities can be often quite easily be overcome with scope markers.For example, the ambiguities in ‘\(A\) and \(B\) or \(C\)’can be overcome as: ‘either \(A\) and \(B\) or \(C\)’ onthe one hand, and ‘\(A\) and either \(B\) or \(C\)’, onthe other. Other imprecisions can often, and maybe always, be overcomein some form or other. Formal languages are useful and oftenconvenient for precisification, but they don’t seem to beessential for it.

On other hand, the formal fundamental language might be taken to beessential for overcoming shortcomings or inherent features of ournatural language as the one alluded to above. If the subject-predicatestructure of our natural languages brings with it a object-propertyway of representing the world, and if that way of representing theworld is unsuitable for representing how reality is in itself, then acompletely different language might be required, and not simply beuseful, to describe fundamental reality. Alternatively, if the formallanguage is needed to articulatereal existence, as we mightbe tempted to put it, something which we can’t express inEnglish or other natural languages, then, too, it would be essentialfor the project of ontology. But if the formal language is needed todo something that our natural language can’t do, then what dothe sentences in the formal language mean? Since they do something ournatural language can’t do, we won’t be able to translatetheir meanings into our natural language. If we could then our naturallanguage would be able to say what these sentences say, which byassumption it can’t do. But then what do sentences in thefundamental language mean? If we can’t say or think what thesesentences say, what is the point for us to use it to try to describereality as it is in itself with them? Can we even make sense of theproject of finding out which sentences in such a language are correct?And why should we care, given that we can’t understand whatthese sentences mean?

A sample debate related to the issues discussed in this section is thedebate about whether it might be that reality in itself does notcontain any objects. See, for example, Hawthorne and Cortens 1995,Burgess 2005, and Turner 2011. Here the use of a variable andquantifier free language like predicate functor logic as thefundamental language is a recurring theme.

Formal languages might be called upon to help overcome an inherentflaw built into our natural languages, as discussed above, or theymight be called upon to overcome a limitation of our naturallanguages. One such limitation might be an expressive one. Forexample, it is unclear whether our natural languages contain trulyhigher-order quantifiers, ones that interact with, say, predicates orsentences directly, as opposed to just terms. If our natural languagesare limited in this way, then it might be tempting to carry outmetaphysics and ontology in a higher-order formal languages, whichdoes not suffer such a limitation. And since such formal languages canbe characterized precisely, that gives rise to the possibility ofgiving precise proofs of various statements of metaphysicalsignificance. This project is pursued under the label higher-ordermetaphysics by several contemporary philosophers in some from orother, see, for example, Williamson 2003 and Dorr 2016. Forcriticism, see Hofweber 2022. This approach can be seen asgeneralizing formal ontology in the sense of section 4.3 to formalmetaphysics, and is related to some of the approaches to formalontology discussed above.

4.6 The form of thought and the structure of reality. (L4) meets (O3)

One way to understand logic is as the study of the most general formsof thought or judgment, what we called (L4). And one way to understandontology is as the study of the most general features of what thereis, our (O3). Now, there is a striking similarity between the mostgeneral forms of thought and the most general features of what thereis. Take one example. Many thoughts have a subject of which theypredicate something. What there is contains individuals that haveproperties. It seems that there is a kind of a correspondence betweenthought and reality: the form of the thought corresponds to thestructure of a fact in the world. And similarly for other forms andstructures. Does this matching between thought and the world ask for asubstantial philosophical explanation? Is it a deep philosophicalpuzzle?

To take the simplest example, the form of our subject-predicatethoughts corresponds perfectly to the structure of object-propertyfacts. If there is an explanation of this correspondence to be givenit seems it could go in one of three ways: either the form of thoughtexplains the structure of reality (a form of idealism), or the otherway round (a form of realism), or maybe there is a common explanationof why there is a correspondence between them, for example on a formof theism where God guarantees a match.

At first it might seem clear that we should try to give an explanationof the second kind: the structure of the facts explains the forms ofour thoughts that represent these facts. And an idea for such anexplanation suggests itself. Our minds developed in a world full ofobjects having properties. If we had a separate simple representationfor these different facts then this would be highly inefficient. Afterall, it is often the same object that has different properties andfigures in different facts, and it is often the same property that ishad by different objects. So, it makes sense to split up ourrepresentations of the objects and of the properties into differentparts, and to put them back together in different combinations in therepresentation of a fact. And thus it makes sense that our mindsdeveloped to represent object-property facts with subject-predicaterepresentations. Therefore we have a mind whose thoughts have a formwhich mirrors the structure of the facts that make up the world.

This kind of an explanation is a nice try, and plausible, but it israther speculative. That our minds really developed this way in lightof those pressures is a question that is not easy to answer from thearmchair. Maybe the facts do have a different structure, but our formsare close enough for practical purposes, i.e. for survival andflourishing. And maybe the correspondence does obtain, but not forthis largely evolutionary reason, but for a different, more direct andmore philosophical or metaphysical reason.

To explain the connection differently one could endorse the oppositeorder of explanatory priority, and argue that the form of thoughtexplains the structure of the world. This would most likely lead to anidealist position of sorts. It would hold that the general features ofour minds explain some of the most general features of reality. Themost famous way to do something like this is Kant’s in theCritique of Pure Reason (Kant 1781/7). We won’t be ableto discuss it here in any detail. This strategy for explaining thesimilarity has the problem of explaining how there can be a world thatexists independently of us, and will continue to exist after we havedied, but nonetheless the structure of this world is explained by theforms of our thoughts. Maybe this route could only be taken if onedenies that the world exists independently of us, or maybe one couldmake this tension go away. In addition one would have to say how theform of thought explains the structure of reality. For one attempt todo this, see Hofweber 2019, for another one, see Gaskin 2020.

But maybe there is not much to explain here. Maybe reality does nothave anything like a structure that mirrors the form of our thoughts,at least not understood a certain way. One might hold that the truthof the thought “John smokes” does not require a worldsplit up into objects and properties, it only requires a smoking John.And all that is required for that is a world that contains John, butnot also another thing, the property of smoking. Thus a structuralmatch would be less demanding, only requiring a match between objectsand object directed thought, but no further match. Such a view wouldbe broadly nominalistic about properties, and it is rathercontroversial.

Another way in which there might be nothing to explain is connected tophilosophical debates about truth. If a correspondence theory of truthis correct, and if thus for a sentence to be true it has to correspondto the world in a way that mirrors the structure and matches parts ofthe sentence properly with parts of the world, then the form of a truesentence would have to be mirrored in the world. But if, on the otherextreme, a coherence theory of truth is correct then the truth of asentence does not require a structural correspondence to the world,but merely a coherence with other sentences. For more on all aspectsof truth see Künne 2003.

Whether or not there is a substantial metaphysical puzzle about thecorrespondence of the form of thoughts and the structure of realitywill itself depend on certain controversial philosophical topics. Andif there is a puzzle here, it might be a trivial one, or it might bequite deep. And as usual in these parts of philosophy, how substantiala question is is itself a hard question.

5. Conclusion

With the many conceptions of logic and the many differentphilosophical projects under the heading of ontology, there are manyproblems that are in the intersection of these areas. We have touchedon several above, but there are also others. Although there is nosingle problem about the relationship between logic and ontology,there are many interesting connections between them, some closelyconnected to central philosophical questions. The references and linksbelow are intended to provide a more in depth discussions of thesetopics.

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Frege, Gottlob,Grundlagen der Arithmetik (in German) (PDF), the original of what is translated as theFoundations of Arithmetic.
Zalta, E., 1999 [2022],Principia Logico-Metaphysica, manuscript of Edward N. Zalta’s systematic formal ontology.

Acknowledgments

Thanks to various anonymous referees for their helpful suggestions onearlier versions of this article. Thanks also to Jamin Asay, RafaelLaboissiere, Ricardo Pereira, Adam Golding, Gary Davis, programadoor,Barnaby Dromgool, Chris Meister, and especially James Cole forreporting several errors, typos, or omissions.

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