## 1. Introduction



Both logic and ontology are important areas of philosophy covering
large, diverse, and active research projects. These two areas overlap
from time to time and problems or questions arise that concern both.
This survey article is intended to discuss some of these areas of
overlap. In particular, there is no single philosophical problem of
the intersection of logic and ontology. This is partly so because the
philosophical disciplines of logic and of ontology are themselves
quite diverse and there is thus the possibility of many points of
intersection. In the following we will first distinguish different
philosophical projects that are covered under the terms
'logic' and 'ontology'. We will then discuss a
selection of problems that arise in the different areas of
contact.



'Logic' and 'ontology' are big words in
philosophy, and different philosophers have used them in different
ways. Depending on what these philosophers mean by these words, and,
of course, depending on the philosopher's views, sometimes there
are striking claims to be found in the philosophical literature about
their relationship. But when Hegel, for example, uses
'logic', or better 'Logik', he means something
quite different than what is meant by the word in much of the
contemporary philosophical scene. We will not attempt to survey the
history of the different conceptions of logic and of ontology, nor the
history of the debate about their relationship. Instead this article
will look at this issue fairly top down, with an emphasis on areas of
overlap that are presently actively debated. For more historical
information, see Kneale and Kneale 1985. Nonetheless, two
historically important figures, namely Gottlob Frege and Immanuel
Kant, will make repeat appearances below.


## 2. Logic



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


### 2.1 Different conceptions of logic



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



A second discipline, also called 'logic', deals with
certain valid inferences and good reasoning connected to them. The
idea here is that there are certain patterns of valid inferences which
are both an object of study in itself as well as connected to certain
patterns of good reasoning. How this connection between inference and
reasoning is to be understood more precisely and to what extent it
obtains is controversial, and beyond the scope of this survey.
However, see Christensen 2005 for more. In any case, logic does not
capture good reasoning as a whole. That is the job of the theory of
rationality. Rather it deals with inferences whose validity can be
traced back to the formal features of the representations that are
involved in that inference, be they linguistic, mental, or other
representations. Some patterns of inference can be seen as valid by
merely looking at the form of the representations that are involved in
this inference. Such a conception of logic thus distinguishes validity
from formal validity. An inference is *valid* just in case the
truth of the premises guarantees the truth of the conclusion, or
alternatively if the premises are true then the conclusion has to be
true as well, or again alternatively, if it can't be that the
premises are true but the conclusion is false.



Validity so understood is simply a modal notion, a notion about what
has to be the case. Others might think of validity as involving a more
fine grained hyperintensional notion, but in any case, validity so
understood is not what logic is concerned with. Logic is concerned
with *formal validity*, which can be understood as follows. In
a system of representations, for example a language, it can be that
some inferences are always valid as long as the representational or
semantic features of certain parts of the representations are kept
fixed, even if we abstract from or ignore the representational
features of the other parts of the representations. So, for example,
as long as we stick to English, and we keep the meanings of certain
words like "some" and "all" fixed, certain
patterns of inference, like some of Aristotle's syllogisms, are
valid no matter what the meaning of the other words in the
 syllogism.[1]
 To call an inference formally valid is to assume that certain words
have their meaning fixed, that we are within a fixed set of
representations, and that we can ignore the meaning of the other
words. The words that are kept fixed are the logical vocabulary, or
logical constants, the others are the non-logical vocabulary. And when
an inference is formally valid then the conclusion logically follows
from the premises. This could be generalized for representations that
are not linguistic, like graphic representations, though it would
require a bit more work to do so. Logic is the study of such
inferences, and certain related concepts and topics, like formal
invalidity, proof, consistency, and so on. The central notion of logic
in this sense is the notion of logical consequence. How this notion
should be understood more precisely is presently widely debated, and a
survey of these debates can be found in the entry on
 logical consequence.



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



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



And there are other notions of 'logic' as well. One of
them is historically prominent, but not very widely represented in the
contemporary debate. We will briefly discuss it here nonetheless.
According to this conception of logic, it is the study of the most
general features of thoughts or judgments, or the form of thoughts or
judgments. Logic thus understood will for example be concerned with
the occurrence of subject and predicate structure that many judgments
exhibit, and with other such general features of judgments. It will
mostly be concerned with thoughts, and not directly with linguistic
representations, though, of course, a proponent of this conception can
claim that there is a very close connection between them. To talk
about the form of a judgment will involve a subtly different notion of
'form' than to talk about the form of a linguistic
representation. The form of a linguistic representation, basically,
was what was left once we abstract from or ignore the representational
features of everything except what we keep fixed, the logical
constants. The form of a thought, on the other hand, is often
understood as what is left over once we abstract from its content,
that is, what it is about. We will briefly pursue the question below
how these notions of form are related to each other. This conception
of logic is associated with Kant. Kant distinguished different notions
of logic (for example transcendental logic, general logic, etc.), but
we won't be able to discuss these here. See the entry on
 Immanuel Kant
 for more.



One important philosophical aspect of logic, at least in the senses
that deal with logical consequence and the forms of judgements, is its
normativity. Logic seems to give us a guide how we ought to reason,
and how we ought to draw inferences from one representation to
another. But it is not at all clear what guide it gives us, and how we
should understand more precisely what norms logic puts on our
reasoning. For example, logic does not put us under the norm "If
you believe \(A\) and you believe if \(A\) then \(B\), then you ought
to believe \(B\)." After all, it might be that I should not
believe \(A\) and if \(A\) then \(B\) in the first place. So, in
particular I shouldn't believe \(B\). A reductio ad absurdum is
a 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 lead
to a belief that \(0=1\). The consequences of my beliefs can lead me
to abandon them. Still, if I have some reasons for my beliefs then I
have at least some prima facie, but not necessarily conclusive, reason
to hold the consequences of those beliefs. Logic might thus tell us at
least 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 distinctive
normative role, and Field 2009 for a nice critical discussion of
Harman's view and an argument why logic should be tied to norms
of rationality. A survey of this an related issues can be found in the
entry on
 the normativity of logic.



And, of course, logic does not tell us how we ought to reason or infer
in all particular cases. Logic does not deal with the particular
cases, but only with the most generally valid forms of reasoning or
inference, ones that are valid no matter what one reasons about. In
this sense logic is often seen to be topic neutral. It applies no
matter what one is thinking or reasoning about. And this neutrality,
or complete generality of logic, together with its normativity, is
often put as "logic is about how we ought to think if we are to
think at all" or "logic is the science of the laws that we
ought to follow in our thinking no matter what we think about".
There are well known philosophical puzzles about normativity, and
these apply to logic as well if it is normative. One is why it is that
thinkers are under such norms. After all, why shouldn't I think
the way I prefer to think, without there being some norm that governs
my thinking, whether I like it or not? Why is there an
"ought" that comes with thinking as such, even if I
don't want to think that way? One idea to answer this is to
employ the notion of a 'constitutive aim of belief', the
idea that belief as such aims at something: the truth. If so then
maybe one could argue that by having beliefs I am under the norm that
I ought to have true ones. And if one holds that one of the crucial
features of logically valid inferences is that they preserve truth
then one could argue that the logical laws are norms that apply to
those who have beliefs. See Velleman 2000 for more on the aim of
belief. The normativity of logic will not be central for our
discussion 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 of
formally 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 of
logic relate to each other. The details of their relationship invite
many hard questions, but we should briefly look at this
nonetheless.


### 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. One
straightforward, though controversial view, is the following. For any
given system of representations, like sentences in a natural language,
there is one and only one set of logical constants. Thus there will be
one formal language that best models what logically valid inferences
there are among these natural representations. This formal language
will have a logical vocabulary that captures the inferential
properties of the logical constants, and that models all other
relevant features of the natural system of representation with
non-logical vocabulary. One especially important system of
representations are our natural languages. Thus (L1) is the study of
formal languages of which one is distinguished, and this one
distinguished language nicely represents the fixed and non-fixed
features of our natural languages, through its logical and non-logical
vocabulary, at least assuming that our natural languages are similar
to each other in this regard. And validity in that formal language, a
technical notion defined in the appropriate way for that formal
language, nicely models logical validity or logical consequence in our
natural language system of representations. Or so this view of the
relationship between (L1) and (L2) holds.



This view of the relationship between (L1) and (L2), however, assumes
that there is one and only one set of logical constants for each
system of representations. A contrary view holds that which
expressions 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 and
belief). This does not mean that 'believes' is a logical
constant in an absolute sense. Given other interests, other
expressions can be treated as logical. According to this conception,
different formal languages will be useful in modeling the inferences
that are formally valid given different set of 'logical
constants' or expressions whose meaning is kept fixed.



This debate thus concerns whether there is one and only one set of
logical constants for a system of representations, and if so, which
ones are the logical ones. We will not get into this debate here, but
there is quite a large literature on what logical constants are, and
how logic can be demarcated. For a general discussion and further
references, see for example Engel 1991. Some of the classic papers
in this debate include Hacking 1979, who defends a proof-theoretic
way of distinguishing logical constants from other expressions. The
leading idea here is that logical constants are those whose meaning
can be given by proof-theoretic introduction and elimination rules. On
the other hand, Mauthner 1946, van Benthem 1986, van Benthem
1989, and Tarski 1986 defend semantic ways to mark that difference.
The leading idea here is that logical notions are 'permutation
invariant'. Since logic is supposed to be completely general and
neutral with respect to what the representations are about, it should
not matter to logic if we switch around the objects that these
representations are about. So, logical notions are those that are
invariant under permutations of the domain. Van Benthem 1989 gives a
general formulation to this idea. See the entry on
 logical constants
 for more.



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



The relationship between (L2) and (L4) on the other hand raises some
questions. For one, of course, there is an issue about what it means
to say that judgments have a form, and whether they do in the relevant
sense. But one way in which this question could be understood directly
ties it to (L2). If thoughts, and thus judgments, are realized by
minds having a certain relation to mental representations, and if
these representations are themselves structured like a language, with
a "syntax" and a "semantics" (properly
understood), then the form of a judgment could be understood just like
the form of a sentence. Such a view of thoughts is commonly called the
Language of Thought hypothesis, see Fodor 1975, and if it is
correct then in the language of thought there might be logical and
non-logical vocabulary. The form of a judgment could be understood
along the lines we understood the form of a linguistic representation
when we talked about formally valid inferences. Thus the relationship
between (L2) and (L4) is rather direct. On both conceptions of logic
we deal with logical constants, the difference is that one deals with
a system of mental representations, the other with a system of
linguistic representations. Both, presumably, would deal with
corresponding sets of logical constants. Even though mental and
linguistic representations form different sets of representations,
since they are closely connected with each other, for every logical
constant in one of these sets of representations there will be another
one of the corresponding syntactic type and with the same content, or
at least a corresponding inferential role.
  





But this conception of their relationship assumes that the
"general features of judgments" or "forms of
judgment" which (L4) is concerned with deal with something like
the logical constants in the language of thought. Here the judgment as
a mental act is assumed to operate on a mental representation that
itself has syntactic structure. And the form of the judgment was
understood as the form of the representation that represents the
content of the judgment, whereby the form of the representation was
understood along the lines of (L2), involving logical constants. But
what if we can't understand "form of judgment" or
"form of thought" that way? One way this could fail is if
the language of thought hypothesis itself fails, and if mental states
do not involve representations that have something like a syntactic
form. The question then becomes, first how should we understand
'form of judgement' more precisely, and secondly, how does
logic, as the discipline concerned with forms of judgments in the
sense of (L4), relate to (L2)?



One way to answer the first question is to understand "form of
judgment" as not being concerned with the representation that
might be involved in a judgment, but rather with the content of the
judgment, i.e. with what the judgment is representing to be the case.
Contents of judgments can be seen as propositions, and these can be
understood as entities that are structured, for example Russellian
propositions. Such propositions are ordered sets whose members are
objects and properties. How such a conception of (L4) relates to (L2)
will in part depend on how one thinks of the logical constants in
Russellian propositions. If they are higher-order properties or
functions that are members of these propositions alongside other
objects and properties then presumably the logical constants have
content. But this seems to be in conflict with an understanding of
(L4) as being concerned with the form that is left once we abstract
from 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 of
the judgment, with logical constants if the latter have content.



Another way to understand "form" as being concerned with
what the judgment is about, rather than the judgment itself, is to
think of what it is about, the world, itself as having a form. In this
sense we associate "form" neither with the representation
that is involved in the judgment, nor with the proposition which is
its content, but rather with the world that is judged about. On such a
conception the world itself has a form or basic structure. (L4) would
be concerned with this structure. How (L4) relates to (L2) is then a
somewhat tricky question. One way, again, could be that the logical
constants that (L2) is concerned with correspond to the structure of
what a representation in which they occur is about, but don't
contribute to the content of that representation. This again seems
incompatible with the logical constants themselves having content. So,
whether one associates form of judgment with the
'syntactic' structure of a representation that is involved
in the judgment, or with the content of that representation, or with
the structure of what the representation is about, the relationship
between (L4) and (L2) will in part depend on whether one thinks the
logical constants themselves contribute to content. If they do, and if
form is contrasted with content, then a close association seems
impossible. If the logical constants don't have content, then it
might be possible.



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



Thus there are many ways in which (L1), (L2), (L3), and (L4) are
connected, 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. Some
contest this formulation of what ontology is, so it's only a
first approximation. Many classical philosophical problems are
problems in ontology: the question whether or not there is a god, or
the problem of the existence of universals, etc.. These are all
problems in ontology in the sense that they deal with whether or not a
certain thing, or more broadly entity, exists. But ontology is usually
also taken to encompass problems about the most general features and
relations of the entities which do exist. There are also a number of
classic philosophical problems that are problems in ontology
understood in this way. For example, the problem of how a universal
relates to a particular that has it (assuming there are universals and
particulars), or the problem of how an event like John eating a cookie
relates to the particulars John and the cookie, and the relation of
eating, assuming there are events, particulars and relations. These
kinds of problems quickly turn into metaphysics more generally, which
is the philosophical discipline that encompasses ontology as one of
its parts. The borders here are a little fuzzy. But we have at least
two parts to the overall philosophical project of ontology, on our
preliminary understanding of it: first, say what there is, what
exists, what the stuff of reality is made out of, secondly, say what
the most general features and relations of these things are.



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



One of the troubles with ontology is that it not only isn't
clear what there is, it also isn't so clear how to settle
questions about what there is, at least not for the kinds of things
that have traditionally been of special interest to philosophers:
numbers, properties, God, etc. Ontology is thus a philosophical
discipline that encompasses besides the study of what there is and the
study of the general features of what there is also the study of what
is involved in settling questions about what there is in general,
especially for the philosophically tricky cases. How we can find out
what there is isn't an easy question to answer. It might seem
simple 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 this
question is to see if what we believe already rationally settles this
question. That is to say, given that we have certain beliefs, do these
beliefs already bring with them a rational commitment to an answer to
such questions as 'Are there numbers?' If our beliefs
bring with them a rational commitment to an answer to an ontological
question about the existence of certain entities then we can say that
we are committed to the existence of these entities. What precisely is
required for such a commitment to occur is subject to debate, a debate
we will look at momentarily. To find out what one is committed to with
a particular set of beliefs, or acceptance of a particular theory of
the world, is part of the larger discipline of ontology.



Besides it not being so clear what it is to commit yourself to an
answer to an ontological question, it also isn't so clear what
an ontological question really is, and thus what it is that ontology
is supposed to accomplish. To figure this out is the task of
meta-ontology, which strictly speaking is not part of ontology
construed narrowly, but the study of what ontology is. However, like
most philosophical disciplines, ontology more broadly construed
contains its own meta-study, and thus meta-ontology is part of
ontology, more broadly construed. Nonetheless it is helpful to
separate it out as a special part of ontology. Many of the
philosophically most fundamental questions about ontology really are
meta-ontological questions. Meta-ontology has not been too popular in
the latter parts of the 20th century, partly because one
meta-ontological view, the one often associated with Quine, had been
widely accepted as the correct one, but this acceptance has been
challenged in recent years in a variety of ways. One motivation for
the study of meta-ontology is simply the question of what question
ontology aims to answer. Take the case of numbers, for example. What
is the question that we should aim to answer in ontology if we want to
find out if there are numbers, that is, if reality contains numbers
besides whatever else it is made up from? This way of putting it
suggest an easy answer: 'Are there numbers?' But this
question 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 than
the number 8. If the latter, then there is a number which is less than
8, namely 7, and thus there is at least one number. Can ontology be
that easy? The study of meta-ontology will have to determine, amongst
others, if 'Are there numbers?' really is the question
that the discipline of ontology is supposed to answer, and more
generally, what ontology is supposed to do. We will pursue these
questions further below. As we will see, several philosophers think
that ontology is supposed to answer a different question than what
there is, but they often disagree on what that question is.



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



(O1)
 the study of ontological commitment, i.e. what we or others are
committed to,
(O2)
 the study of what there is,
(O3)
 the study of the most general features of what there is, and how
the things there are relate to each other in the metaphysically most
general ways,
(O4)
 the study of meta-ontology, i.e. saying what task it is that the
discipline of ontology should aim to accomplish, if any, how the
questions it aims to answer should be understood, and with what
methodology 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. In
particular, 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 be
only a first approximation for how to state what ontology is supposed
to do. Maybe it is supposed to answer the question what is real
instead, or what is fundamental, some other question. Whatever one
says here will also affect how one should understand (O1). We will at
first work with what is the most common way to understand (O2) and
(O1), and discuss alternatives in turn. If (O1) has the result that
the beliefs we share commit us to a certain kind of entity then this
requires us either to accept an answer to a question about what there
is in the sense of (O2) or to revise our beliefs. If we accept that
there is such an entity in (O2) then this invites questions in (O3)
about its nature and the general relations it has to other things we
also accept. On the other hand, investigations in (O3) into the nature
of entities that we are not committed to and that we have no reason to
believe exist would seem like a rather speculative project, though, of
course, 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 division
of logic into (L1)-(L4) we can consider several issues where
logic, understood a certain way, overlaps with ontology, understood a
certain way. In the following we will discuss some paradigmatic
debates 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 the
ontological question 'Are there numbers?' is, assuming
(O4) tells us this is the ontological question about numbers. One
strategy to see whether our beliefs already commit us to an answer of
this question is as follows: first, write out all those beliefs in a
public language, like English. This by itself might not seem to help
much, since if it wasn't clear what my beliefs commit me to, why
would it help to look at what acceptance of what these sentences say
commits me to? But now, secondly, write these sentences in what is
often called 'canonical notation'. Canonical notation can
be understood as a formal or semi-formal language that brings out the
true underlying structure, or 'logical form' of a natural
language sentence. In particular, such a canonical notation will make
explicit which quantifiers do occur in these sentences, what their
scope is, and the like. This is where formal languages come into the
picture. After that, and thirdly, look at the variables that are bound
by these
 quantifiers.[3]
 What values do they have to have in order for these sentences all to
be true? If the answer is that the variables have to have numbers as
their values, then you are committed to numbers. If not then you
aren't committed to numbers. The latter doesn't mean that
there are no numbers, of course, just as you being committed to them
doesn't mean that there are numbers. But if your beliefs are all
true 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 really
gain from these 'canonical notations' in determining
ontological commitment? One attempt to answer this, which partly
motivates the above way of doing things, is based on the following
consideration: We might wonder why we should think that quantifiers
are of great importance for making ontological commitments explicit.
After all, if I accept the apparently trivial mathematical fact that
there is a number between 6 and 8, does this already commit me to an
answer to the ontological question whether there are numbers out
there, as part of reality? The above strategy tries to make explicit
that and why it in fact does commit me to such an answer. This is so
since natural language quantifiers are fully captured by their formal
analogues in canonical notation, and the latter make ontological
commitments obvious because of their semantics. Such formal
quantifiers are given what is called an 'objectual
semantics'. This is to say that a particular quantified
statement \(\lsquo \exists x\,Fx\rsquo\) is true just in case there
is an object in the domain of quantification that, when assigned as
the value of the variable \(\lsquo x\rsquo\), satisfies the open formula
\(\lsquo Fx\rsquo\). This makes obvious that the truth of a
quantified statement is ontologically relevant, and in fact ideally
suited to make ontological commitment explicit, since we need entities
to assign as the values of the variables. Thus (L1) is tied to (O1).
The philosopher most closely associated with this way of determining
ontological commitment, and with the meta-ontological view on which it
is based, is Quine (in particular Quine 1948). See also van Inwagen
1998 for a presentation sympathetic to Quine.



The above account of ontological commitment has been criticized from a
variety of different angles. One criticism focuses on the semantics
that is given for quantifiers in the formal language that is used as
the canonical notation of the natural language representations of the
contents of beliefs. The above, objectual semantics is not the only
one that can be given to quantifiers. One widely discussed alternative
is the so-called 'substitutional semantics'. According to
it we do not assign entities as values of variables. Rather, a
particular quantified statement \(\lsquo \exists x\,Fx\rsquo\) is
true just in case there is a term in the language that when
substituted for \(\lsquo x\rsquo\) in \(\lsquo Fx\rsquo\) has a true
sentence as its result. Thus, \(\lsquo \exists x\,Fx\rsquo\) is true
just 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 quantifiers
has often been used to argue that there are ontologically innocent
uses of quantifiers, and that what quantified statements we accept
does not directly reveal ontological commitment. Gottlieb (1980)
provides more details on substitutional quantification, and an attempt
to use it in the philosophy of mathematics. Earlier work was done by
Ruth Marcus, and is reprinted in Marcus 1993.



Another objection to the above account of determining ontological
commitment goes further and questions the use of a canonical notation,
and of formal tools in general. It states that if the ontological
question about numbers simply is the question 'Are there
numbers?' then all that matters for ontological commitment is
whether or not what we accept implies 'There are numbers'.
In particular, it is irrelevant what the semantics for quantifiers in
a formal language is, in particular, whether it is objectual or
substitutional. What ontological commitment comes down to can be
determined at the level of ordinary English. Formal tools are of no,
or at best limited, importance. Ontological commitment can thus
according to this line of thought be formulated simply as follows: you
are committed to numbers if what you believe implies that there are
numbers. Notwithstanding the debate between the substitutional and
objectual semantics, we do not need any formal tools to spell out the
semantics of quantifiers. All that matters is that a certain
quantified statement 'There are \(F\)s' is implied by what
we believe for us to be committed to \(F\)s. What does not matter is
whether 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 ontological
commitment is whether or not what one believes implies that there are
\(F\)s, for a certain kind of thing \(F\), there might still be room
for formal tools. First of all, it isn't clear what implies
what. Whether or not a set of statements that express my beliefs imply
that there are entities of a certain kind might not be obvious, and
might even be controversial. Formal methods can be useful in
determining what implies what. On the other hand, even though formal
methods can be useful in determining what implies what, it is not
clear which formal tools are the right ones for modeling a natural
system of representations. It might seem that to determine which are
the right formal tools we already need to know what the implicational
relations are between the natural representations that we attempt to
model, at least in basic cases. This could mean that formal tools are
only of limited use for deciding controversial cases of
implication.



But then, again, it has been argued that often it is not at all clear
which statements really involve quantifiers at a more fundamental
level of analysis, or
 logical form.
 Russell (1905) famously argued that "the King of France"
is a quantified expression, even though it appears to be a referring
expression on the face of it, a claim now accepted by many. And
Davidson (1967) argued, that 'action sentences' like
"Fred buttered the toast" involve quantification over
events in the logical form, though not on the surface, a claim that is
more controversial. One might argue in light of these debates that
which sentences involve quantification over what can't be
finally settled until we have a formal semantics of all of our natural
language, and that this formal semantics will give us the ultimate
answer to what we are quantifying over. But then again, how are we to
tell that the formal semantics proposed is correct, if we don't
know the inferential relations in our own language?



One further use that formal tools could have besides all the above is
to make ambiguities and different 'readings' explicit, and
to model their respective inferential behavior. For example, formal
tools are especially useful to make scope ambiguities explicit, since
different scope readings of one and the same natural language sentence
can be represented with different formal sentences which themselves
have no scope ambiguities. This use of formal tools is not restricted
to ontology, but applies to any debates where ambiguities can be a
hindrance. It does help in ontology, though, if some of the relevant
expressions in ontological debates, like the quantifiers themselves,
exhibit such different readings. Then formal tools will be most useful
to make this explicit. Whether or not quantifiers indeed do have
different readings is a question that will not be solved with formal
tools, but if they do then these tools will be most useful in
specifying what these readings are. For a proposal of this latter
kind, see Hofweber 2016. One consequence of this is a meta-ontology
different from Quine's, as we will discuss below.



All this discussion in this section assumed that ontological
commitment is connected to a conception of ontology that concerns what
there is. But this is not universally accepted, in particular
recently. Maybe ontology does not concern what there is, but what is
fundamental, in some sense of the word. If so, then issues tied to
quantifiers are not of the most central importance when it comes to
ontological commitment, although they would still play a role. The
main question would then be connected to fundamentality. And here,
too, formal languages might play a role in determining what one is
committed to being fundamental. We will more closely discuss the role
of formal languages in conceptions of ontology as concerning the
fundamental 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 valid
by their form. And above we spelled this out as follows: an inference
is valid by its form if as long as we fix the meaning of certain
special expressions, the logical constants, we can ignore the meaning
of the other expressions in the statements involved in the inference,
and we are always guaranteed that the inference is valid, no matter
what the meaning of the other expressions is, as long as the whole is
meaningful. A logical truth can be understood as a statement whose
truth is guaranteed as long as the meanings of the logical constants
are fixed, no matter what the meaning of the other expressions is.
Alternatively, a logical truth is one that is a logical consequence
from no assumptions, i.e. an empty set of premises.



Do logical truths entail the existence of any entities, or is their
truth independent of what exists? There are some well known
considerations that seem to support the view that logic should be
neutral with respect to what there is. On the other hand, there are
also some well known arguments to the contrary. In this section we
will survey some of this debate.



If logical truth are ones whose truth is guaranteed as long as the
meaning of the logical constants is kept fixed then logical truths are
good candidates for being analytic truths. Can analytic truths imply
the existence of any entities? This is an old debate, often conducted
using "conceptual truths" instead of "analytic
truths". The most prominent debate of this kind is the debate
about the ontological argument for the existence of God. Many
philosophers have maintained that there can be no conceptual
contradiction in denying the existence of particular entities, and
thus there can be no proof of their existence with conceptual truths
alone. In particular, an ontological argument for the existence of God
is impossible. A famous discussion to this effect is Kant's
discussion of the ontological argument (Kant 1781/7, KrV
A592/B620 ff). On the other hand, many other philosophers have
maintained that such an ontological argument is possible, and they
have made a variety of different proposals how it can go. We will not
discuss the ontological argument here, however, it is discussed in
detail in different formulations in the entry on
 ontological arguments
 in this encyclopedia.



Whatever one says about the possibility of proving the existence of an
object purely with conceptual truths, many philosophers have
maintained that at least *logic* has to be neutral about what
there is. One of the reasons for this insistence is the idea that
logic is topic neutral, or purely general. The logical truths are the
ones that hold no matter what the representations are about, and thus
they 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 logical
truths can't imply that anything exists. But that argument might
be turned around by a believer in logical objects, objects whose
existence is implied by logic alone. If it is granted that logical
truths have to hold in any domain, then any domain has to contain the
logical objects. Thus for a believer in logical objects there can be
no empty domain.



There is a close relationship between this debate and a common
criticism 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 the
semantics of first and second order logical systems.



It is a logical truth in (standard) first order logic that something
exists, i.e., '\(\exists x\,x=x\)'. Similarly, it is a
logical truth in (standard versions of) second order logic that
'\(\exists F\forall x\,(Fx \vee \neg Fx)\)'. These are
existentially quantified statements. Thus, one might argue, logic is
not neutral with respect to what there is. There are logical truths
that state that something exists. However, it would be premature to
conclude that logic is not neutral about what there is, simply because
there are logical truths in (standard) first or second order logic
which are existential statements. If we look more closely how it comes
about that these existential statements are logical truths in these
logical systems we see that it is only so because, by definition, a
model 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 (where
nothing 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 first
order models with a non-empty domain. If we allow an empty domain as
well we will need different axioms or rules of inference to have a
sound proof system, but this can be done. Thus even though there are
formal logical systems, in the sense of (L1) in which there are
logical truths that are existential statements, this does not answer
the question whether or not there are logical truths, in the sense of
(L2), that are existential statements. The question rather is which
formal 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 logical
system is a good formal system to represent logical inferences, should
we adopt the axioms and rules for models with or without an empty
domain?



A related debate is the debate about free logic. Free logics are
formal systems that drop the assumption made in standard first and
higher order logic that every closed term denotes an object in the
domain of the model. Free logic allows for terms that denote nothing,
and in free logic certain rules about the inferential interaction
between quantifiers and terms have to be modified. Whether free or
un-free (standard) logic is the better formal model for natural
language logical inference is a further question. For more discussion
of 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 Lambert
2001.



How innocent logic is with respect to ontology is also at the heart of
the debate about the status of second order logic as logic. Quine
(1970) argued that second order logic was "set theory in
sheep's clothing", and thus not properly logic at all.
Quine was concerned with the questions of whether second order
quantifiers should be understood as ranging over properties or over
sets of individuals. The former were considered dubious in various
ways, the latter turn second order logic into set theory. This
approach to second order logic has been extensively criticized by
various authors, most notably George Boolos, who in a series of
papers, collected in part I of Boolos 1998, attempted to vindicate
second order logic, and to propose a plural interpretation, which is
discussed in the article on
 plural quantification.



A particularly important and pressing case of the ontological
implications of logic are logicist programs in the philosophy of
mathematics, in particular Frege's conception of logical objects
and his philosophy of arithmetic. Frege and neo-Fregeans following him
believe that arithmetic is logic (plus definitions) and that numbers
are objects whose existence is implied by arithmetic. Thus in
particular, logic implies the existence of certain objects, and
numbers are among them. Frege's position has been criticized as
being untenable since logic has to be neutral about what there is.
Thus mathematics, or even a part thereof, can't be both logic
and about objects. The inconsistency of Frege's original
formulation of his position sometimes has been taken to show this, but
since consistent formulations of Frege's philosophy of
arithmetic have surfaced this last point is moot. Frege's
argument for numbers as objects and arithmetic as logic is probably
the best known argument for logic implying the existence of entities.
It has been very carefully investigated in recent years, but whether
or not it succeeds is controversial. Followers of Frege defend it as
the solution to major problems in the philosophy of mathematics; their
critics find the argument flawed or even just a cheap trick that is
obviously going nowhere. We will not discuss the details here, but a
detailed presentation of the argument can be found in the entry on
 Frege's theorem and foundations of arithmetic
 as well as Rosen 1993, which gives a clear and readable
presentation of the main argument of Wright (1983), which in turn is
partially responsible for a revival of Fregean ideas along these
lines. Frege's own version is in his classic *Grundlagen*
(1884). A discussion of recent attempts to revive Frege can be found
in Hale and Wright 2001, Boolos 1998 and Fine 2002. A discussion
of Frege's and Kant's conceptions of logic is in
MacFarlane 2002 which also contains many historical references.


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



Formal ontologies are theories that attempt to give precise
mathematical formulations of the properties and relations of certain
entities. Such theories usually propose axioms about these entities in
question, spelled out in some formal language based on some system of
formal logic. Formal ontology can been seen as coming in three kinds,
depending on their philosophical ambition. Let's call them
representational, descriptive, and systematic. We will in this section
briefly discuss what philosophers, and others, have hoped to do with
such formal ontologies.



A formal ontology is a mathematical theory of certain entities,
formulated in a formal, artificial language, which in turn is based on
some logical system like first order logic, or some form of the lambda
calculus, or the like. Such a formal ontology will specify axioms
about what entities of this kind there are, what their relations among
each other are, and so on. Formal ontologies could also only have
axioms that state how the things the theory is about, whatever they
may be, relate to each other, but no axioms that state that certain
things exist. For example, a formal ontology of events won't say
which events there are. That is an empirical question. But it might
say under what operations events are closed under, and what structure
all the events there are exhibit. Similarly for formal ontologies of
the part-whole relation, and others. See Simons 1987 for a well
known book on various formal versions of mereology, the study of parts
and wholes.



Formal ontologies can be useful in a variety of different ways. One
contemporary use is as a framework to represent information in an
especially useful way. Information represented in a particular formal
ontology can be more easily accessible to automated information
processing, and how best to do this is an active area of research in
computer science. The use of the formal ontology here is
representational. It is a framework to represent information, and as
such it can be representationally successful whether or not the formal
theory used in fact truly describes a domain of entities. So, a formal
ontology of states of affairs, lets say, can be most useful to
represent information that might otherwise be represented in plain
English, and this can be so whether or not there indeed are any states
of affairs in the world. Such uses of formal ontologies are thus
representational.



A different philosophical use of a formal ontology is one that aims to
be descriptive. A descriptive formal ontology aims to correctly
describe a certain domain of entities, say sets, or numbers, as
opposed to all things there are. Take common conceptions of set theory
as one example. Many people take set theory to aim at correctly
describing a domain of entities, the pure sets. This is, of course, a
controversial claim in the philosophy of set theory, but if it is
correct then set theory could be seen as a descriptive formal ontology
of pure sets. It would imply that among incompatible formal theories
of sets only one could be correct. If set theory were merely
representational then both of the incompatible theories could be
equally useful as representational tools, though probably for
different representational tasks.



Finally, formal ontologies have been proposed as systematic theories
of what there is, with some restrictions. Such systematic theories
hope to give one formal theory for all there is, or at least a good
part of it. Hardly anyone would claim that there can be a simple
formal theory that correctly states what concrete physical objects
there are. There does not seem to be a simple principle that
determines whether there are an even or odd number of mice at a
particular time. But maybe this apparent randomness only holds for
concrete physical objects. It might not hold for abstract objects,
which according to many exist not contingently, but necessarily if at
all. Maybe a systematic, simple formal theory is possible of all
abstract objects. Such a systematic formal ontology will most commonly
have 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 simple
view of this kind would be one according to which all abstract objects
are sets, and numbers, properties, etc. are really special kinds of
sets. However, more sophisticated versions of systematic formal
ontologies have been developed. An ambitious systematic formal
ontology can be found in Zalta 1983 and Zalta 1999 [2022] (see the
Other Internet Resources).



Representational formal ontologies, somewhat paradoxically, are
independent of any strictly ontological issues. Their success or
failure is independent of what there is. Descriptive formal ontologies
are just like representational ones, except with the ambition of
describing a domain of entities. Systematic formal ontologies go
further in not only describing one domain, but in relating all
entities (of a certain kind) to each other, often with particular
notions of reduction. These theories seem to be the most ambitious.
Their motivation comes from an attempt to find a simple and systematic
theory of all, say, abstract entities, and they can rely on the
paradigm of aiming for simplicity in the physical sciences as a guide.
They, just like descriptive theories, will have to have as their
starting point a reasonable degree of certainty that we indeed are
ontologically committed to the entities they aim to capture. Without
that these enterprises seem to have little attraction. But even if the
latter philosophical ambitions fail, a formal ontology can still be a
most 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 the
first half of the 20th century, and which is one of the starting
points of the contemporary debate in ontology, leading to the
well-known exchange between Carnap and Quine, to be discussed below.
According to Carnap one crucial project in philosophy is to develop
*frameworks* that can be used by scientists to formulate
theories of the world. Such frameworks are formal languages that have
a clearly defined relationship to experience or empirical evidence as
part of their semantics. For Carnap it was a matter of usefulness and
practicality which one of these frameworks will be selected by the
scientists to formulate their theories in, and there is no one correct
framework that truly mirrors the world as it is in itself. The
adoption of one framework rather than another is thus a practical
question.



Carnap distinguished two kinds of questions that can be asked about
what 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 about
numbers has been adopted. Such questions vary in degree of difficulty.
Some are very hard, like 'Are there infinitely many twin prime
numbers?', some are of medium difficulty, like 'Are there
infinitely many prime numbers?', some are easy like 'Are
there prime numbers?', and some are completely trivial, like
'Are there numbers?'. Internal questions are thus
questions that can be asked once a framework that allows talk about
certain things has been adopted, and general internal questions, like
'Are there numbers?' are completely trivial since once the
framework of talk about numbers has been adopted the question if there
are any is settled within that framework.



But since the internal general questions are completely trivial they
can't be what the philosophers and metaphysicians are after when
they ask the ontological question 'Are there numbers?' The
philosophers aim to ask a difficult and deep question, not a trivial
one. What the philosophers aim to ask, according to Carnap, is not a
question internal to the framework, but external to it. They aim to
ask whether the framework correctly corresponds to reality, whether or
not there *really* are numbers. However, the words used in the
question 'Are there numbers?' only have meaning within the
framework of talk about numbers, and thus if they are meaningful at
all they form an internal question, with a trivial answer. The
external questions that the metaphysician tries to ask are
meaningless. Ontology, the philosophical discipline that tries to
answer hard questions about what there *really* is is based on
a mistake. The question it tries to answer are meaningless questions,
and this enterprise should be abandoned. The words 'Are there
numbers?' thus can be used in two ways: as an internal question,
in which case the answer is trivially 'yes', but this has
nothing to do with metaphysics or ontology, or as an external
question, which is the one the philosophers are trying to ask, but
which is meaningless. Philosophers should thus not be concerned with
(O2), which is a discipline that tries to answer meaningless
questions, but with (L1), which is a discipline that, in part,
develops frameworks for science to use to formulate and answer real
questions. Carnap's ideas about ontology and meta-ontology are
developed in a classic essay (Carnap 1956b). A nice summary of
Carnap'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. One
common criticism is that it relies on a too simplistic conception of
natural language that ties it too closely to science or to evidence
and verification. In particular, Carnap's more general rejection
of metaphysics used a verificationist conception of meaning, which is
widely seen as too simplistic. Carnap's rejection of ontology
has been criticized most prominently by Quine, and the debate between
Carnap and Quine on ontology is a classic in this field. Quine
rejected Carnap's conception that when scientists are faced with
data that don't fit their theory they have two choices. First
they could change the theory, but stay in the same framework.
Secondly, they could move to a different framework, and formulate a
new theory within that framework. These two moves for Carnap are
substantially different. Quine would want to see them as fundamentally
similar. In particular, Quine rejects the idea that there could be
truths which are the trivial internal statements, like "There
are numbers", whose truth is a given once the framework of
numbers has been adopted. Thus some such internal statements would be
analytic truths, and Quine is well known for thinking that the
distinction between analytic and synthetic truths is untenable. Thus
Carnap's distinction between internal and external questions is
to be rejected alongside with the rejection of the distinction between
analytic and synthetic truths. On the other hand, Quine and Carnap
agree that ontology in the traditional philosophical sense is to be
rejected. Traditionally ontology has often, but not always, been an
armchair, a priori, investigation into the fundamental building blocks
of reality. As such it is completely separated from science. Quine
(1951) rejects this approach to ontology since he holds that there
can't be such an investigation into reality that is completely
separate and prior to the rest of inquiry. See Yablo 1998 for more
on the debate between Quine and Carnap, which contains many references
to the relevant passages. The view on ontological commitment discussed
in section 4.1., which is usually attributed to Quine, was developed
as a reaction to Carnap's position discussed in this
section. Simply put, Quine's view is that to see what we are
committed to we have to see what our best overall theory of the world
quantifies over. In particular, we look at our best overall scientific
theory of the world, which contains physics and the rest.



Carnap's arguments for the rejection of ontology are presently
widely rejected. However, several philosophers have recently attempted
to revive some parts or others of Carnap's ideas. For example,
Stephen Yablo has argued that an internal-external distinction could
be understood along the lines of the fictional-literal distinction.
And he argues (Yablo 1998) that since there is no fact about this
distinction, ontology, in the sense of (O2), rests on a mistake and is
to be rejected, as Carnap did. On the other hand, Thomas Hofweber has
argued that an internal-external distinction with many of the features
that Carnap wanted can be defended on the basis of facts about natural
language, but that such a distinction will not lead to a rejection of
ontology, in the sense of (O2). See Hofweber 2016. Hilary Putnam
(1987) has developed a view that revives some of the pragmatic aspects
of Carnap's position. See Sosa 1993 for a critical discussion of
Putnam's view, and Sosa 1999 for a related, positive
proposal. Robert Kraut (2016) has defended an expressivist reading of
the internal-external distinction, and with it some Carnapian
consequences for ontology. And most of all, Eli Hirsch and Amie
Thomasson have defended different versions of approaches to ontology
that capture a good part of the spirit of Carnap's view. See in
particular Hirsch 2011 and Thomasson 2015. For various views about the
effects of Carnap on the contemporary debate in ontology, see Blatti
and 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 to
find out what there is, or what exists, this is rejected by many in
the contemporary debate. These philosophers think that the job of
ontology is something different, and there is disagreement among them
what it is more precisely. Among the proposed options are the projects
of finding out what is real, or what is fundamental, or what the
primary substances are, or what reality is like in itself, or
something like this. Proponents of these approaches often find the
questions about what there is too inconsequential and trivial to take
them to be the questions for ontology. Whether there are numbers, say,
is trivially answered in the affirmative, but whether numbers are
real, or whether they are fundamental, or primary substances, etc., is
the hard and ontological question. See Fine (2009) and Schaffer (2009)
for two approaches along these lines. But such approaches have their
own problems. For example, it is not clear whether the question
whether numbers are real is any different than the question whether
numbers exist. If one were to ask whether the Loch Ness monster is
real, it would naturally be understood as just the same question as
whether the Loch Ness monster exists. If it is supposed to be a
different question, is this due to simple stipulation, or can we make
the difference intelligible? Similarly, it is not clear whether the
notion of what is fundamental can carry the intended metaphysical
weight. After all, there is a perfectly clear sense in which prime
numbers are more fundamental in arithmetic than even numbers, but this
isn't to hold the metaphysical priority of prime numbers over
other numbers, but simply to hold that they are mathematically special
among the numbers. Thus to ask whether numbers are fundamental is not
easily seen as a metaphysical alternative to the approach to ontology
that asks whether numbers exist. See Hofweber (2009; 2006, chapter 13)
for a critical discussion of some approaches to ontology that rely on
notions of reality or fundamentality. Whether such approaches to
ontology are correct is a controversial topic in the debate about
ontology which we will not focus on here. However, this approach gives
rise to a special connection between logic and ontology which we will
discuss in the following.



The relation between the different approaches to ontology mentioned
just above is unclear. Is something that is part of reality as it is
in itself something which is fundamental, or which is real in the
relevant sense? Although it is unclear how these different approaches
relate to each other, all of them have the potential for allowing for
that our ordinary description of the world in terms of mid-size
objects, mathematics, morality, and so on, is literally true, while at
the same time these truths leave it open what the world, so to speak,
deep down, really, and ultimately is like. To use one way of
articulating this, even though there are tables, numbers, and values,
reality in itself might contain none of them. Reality in itself might
contain no objects at all, and nothing normative. Or it might. The
ordinary description of the world, on this conception, leaves it
largely open what reality in itself is like. To find that out is the
job of metaphysics, in particular ontology. We might, given our
cognitive 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 in
itself is left open.



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



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



Some philosophers have proposed that natural language might be
unsuitable for the purposes of ontology. It might be unsuitable since
it carries with it too much baggage from our particular conceptual
scheme. See Burgess 2005 for a discussion. Or it might be unsuitable
since various expressions in it are not precise enough, too context
sensitive, or in other ways not ideally suited for the philosophical
project. These philosophers propose instead to find a new, better
suited language. Such a language likely will be a major departure from
natural 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 fundamental
language'. The task thus is to find the fundamental language, a
language in the sense of (L1), to properly carry out ontology, in the
new and revised sense of (O2): the project of finding out what reality
fundamentally, or in itself, etc., is like. For a critical discussion
of the proposal that we should be asking the questions of ontology in
ontologese, see Thomasson 2015 (chapter 10).



But this idea of a connection between (L1) and (O2) is not
unproblematic. First there is a problem about making this approach to
(O2) more precise. How to understand the notion of 'reality in
itself' is not at all clear, as is well known. It can't
just mean: reality as it would be if we weren't in it. On this
understanding it would simply be the world as it is except with no
humans in it, which would in many of its grander features be just as
it 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. We
won't pursue this issue here, though. Second, there is a serious
worry about how the formal language which is supposed to be the
fundamental language is to be understood. In particular, is it
supposed to be merely an auxiliary tool, or an essential one? This
question is tied to the motivation for a formal fundamental language
in the first place. If it is merely to overcome ambiguities,
imperfections, and context sensitivities, then it most likely will
merely be an auxiliary, but not essential tool. After all, within
natural language we have many means available to get rid of
ambiguities, imperfections and context sensitivities. Scope
ambiguities 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 \(C\)' on
the one hand, and '\(A\) and either \(B\) or \(C\)', on
the other. Other imprecisions can often, and maybe always, be overcome
in some form or other. Formal languages are useful and often
convenient for precisification, but they don't seem to be
essential for it.



On other hand, the formal fundamental language might be taken to be
essential for overcoming shortcomings or inherent features of our
natural language as the one alluded to above. If the subject-predicate
structure of our natural languages brings with it a object-property
way of representing the world, and if that way of representing the
world is unsuitable for representing how reality is in itself, then a
completely different language might be required, and not simply be
useful, to describe fundamental reality. Alternatively, if the formal
language is needed to articulate *real existence*, as we might
be tempted to put it, something which we can't express in
English or other natural languages, then, too, it would be essential
for the project of ontology. But if the formal language is needed to
do something that our natural language can't do, then what do
the sentences in the formal language mean? Since they do something our
natural language can't do, we won't be able to translate
their meanings into our natural language. If we could then our natural
language would be able to say what these sentences say, which by
assumption it can't do. But then what do sentences in the
fundamental language mean? If we can't say or think what these
sentences say, what is the point for us to use it to try to describe
reality as it is in itself with them? Can we even make sense of the
project of finding out which sentences in such a language are correct?
And why should we care, given that we can't understand what
these sentences mean?



A sample debate related to the issues discussed in this section is the
debate about whether it might be that reality in itself does not
contain any objects. See, for example, Hawthorne and Cortens 1995,
Burgess 2005, and Turner 2011. Here the use of a variable and
quantifier free language like predicate functor logic as the
fundamental language is a recurring theme.



Formal languages might be called upon to help overcome an inherent
flaw built into our natural languages, as discussed above, or they
might be called upon to overcome a limitation of our natural
languages. One such limitation might be an expressive one. For
example, it is unclear whether our natural languages contain truly
higher-order quantifiers, ones that interact with, say, predicates or
sentences directly, as opposed to just terms. If our natural languages
are limited in this way, then it might be tempting to carry out
metaphysics and ontology in a higher-order formal languages, which
does not suffer such a limitation. And since such formal languages can
be characterized precisely, that gives rise to the possibility of
giving precise proofs of various statements of metaphysical
significance. This project is pursued under the label higher-order
metaphysics by several contemporary philosophers in some from or
other, see, for example, Williamson 2003 and Dorr 2016. For
criticism, see Hofweber 2022. This approach can be seen as
generalizing formal ontology in the sense of section 4.3 to formal
metaphysics, and is related to some of the approaches to formal
ontology 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 forms
of thought or judgment, what we called (L4). And one way to understand
ontology is as the study of the most general features of what there
is, our (O3). Now, there is a striking similarity between the most
general forms of thought and the most general features of what there
is. Take one example. Many thoughts have a subject of which they
predicate something. What there is contains individuals that have
properties. It seems that there is a kind of a correspondence between
thought and reality: the form of the thought corresponds to the
structure of a fact in the world. And similarly for other forms and
structures. Does this matching between thought and the world ask for a
substantial philosophical explanation? Is it a deep philosophical
puzzle?



To take the simplest example, the form of our subject-predicate
thoughts corresponds perfectly to the structure of object-property
facts. If there is an explanation of this correspondence to be given
it seems it could go in one of three ways: either the form of thought
explains the structure of reality (a form of idealism), or the other
way round (a form of realism), or maybe there is a common explanation
of why there is a correspondence between them, for example on a form
of theism where God guarantees a match.



At first it might seem clear that we should try to give an explanation
of the second kind: the structure of the facts explains the forms of
our thoughts that represent these facts. And an idea for such an
explanation suggests itself. Our minds developed in a world full of
objects having properties. If we had a separate simple representation
for these different facts then this would be highly inefficient. After
all, it is often the same object that has different properties and
figures in different facts, and it is often the same property that is
had by different objects. So, it makes sense to split up our
representations of the objects and of the properties into different
parts, and to put them back together in different combinations in the
representation of a fact. And thus it makes sense that our minds
developed to represent object-property facts with subject-predicate
representations. Therefore we have a mind whose thoughts have a form
which mirrors the structure of the facts that make up the world.



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



To explain the connection differently one could endorse the opposite
order of explanatory priority, and argue that the form of thought
explains the structure of the world. This would most likely lead to an
idealist position of sorts. It would hold that the general features of
our minds explain some of the most general features of reality. The
most famous way to do something like this is Kant's in the
*Critique of Pure Reason* (Kant 1781/7). We won't be able
to discuss it here in any detail. This strategy for explaining the
similarity has the problem of explaining how there can be a world that
exists independently of us, and will continue to exist after we have
died, but nonetheless the structure of this world is explained by the
forms of our thoughts. Maybe this route could only be taken if one
denies that the world exists independently of us, or maybe one could
make this tension go away. In addition one would have to say how the
form of thought explains the structure of reality. For one attempt to
do this, see Hofweber 2019, for another one, see Gaskin 2020.



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



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



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


## 5. Conclusion



With the many conceptions of logic and the many different
philosophical projects under the heading of ontology, there are many
problems that are in the intersection of these areas. We have touched
on several above, but there are also others. Although there is no
single problem about the relationship between logic and ontology,
there are many interesting connections between them, some closely
connected to central philosophical questions. The references and links
below are intended to provide a more in depth discussions of these
topics.