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Logic and Ontology
A number of important philosophical problems are problems in the overlap of
logic and ontology. Both logic and ontology are diverse fields within
philosophy, and partly because of this there is not one single philosophical
problem about the relation between logic and ontology. In this survey article we
will first discuss what different philosophical projects are carried out under
the headings of "logic" and "ontology" and then we will look at several areas
where logic and ontology overlap.
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 be able to
survey the history of the different conceptions of logic, or of ontology.
Instead we will focus on the already very diverse debate in the more or less the
20th century English speaking philosophical tradition. Even with this
restriction, ‘logic’ and ‘ontology’ are understood in different ways.
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.
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 based on them. It does not, however, cover 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. An inference is valid just in case the truth of the premises
guarantees the truth of the conclusion, or in other words that if the premises
are true then the conclusion has to be true as well, or again in other words, if
it can't be that the premises are true but the conclusion is false. An inference
being formally valid 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 Stanford Encyclopedia of Philosophy article 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 subtle 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 article on 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. It can be the consequences of my beliefs that
lead me to abandon them. Logic might tell us at least this much, though:
whatever reason I have for believing A and if A then
B, it is a reason to believe B. See (Harman 1986) for more on
the role of logic in reasoning.
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 can'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 mathematical 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 there relationship invite many hard
questions, but we should briefly look at this nonetheless.
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 is
our natural language. 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 language, through its logical and
non-logical vocabulary. 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 article 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 the
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, 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 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 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 and
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 from of judgment with the ‘syntactic’
structure 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 if 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.
As a first approximation, ontology is the study of what there is. Many
classical philosophical problems are problems in ontology, like the question
whether or not there is a god, or the problem of the existence of universals.
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 this way. For example the
problem how a universal relates to a particular that has it (assuming there are
universals and particulars), or the problem how an event like John eating a
cookie relate 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 fussy. But we have at least two parts to the overall
philosophical project of ontology: first, say what there is, what exists, what
the stuff is reality is made out off, 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. Lets 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 seems 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 commited 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 last couple of decades, partly because one meta-ontological view, the one
often associated with Quine, has been accepted as the correct one, but this
acceptance has been challenged in recent years. One motivation for the study of
meta-ontology is simply the question 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
below.
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 question it
aims to answer should be understood, and with what methodology they can be
answered.
The relationship between these four seems rather straightforward. (O4) will
have to say how the other three are supposed to be understood. 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.
The debates about logic and ontology overlap at various places. Given the
division of ontology into (O1) - (O4), and the division of logic into (L1) -
(L4) we can look at several areas of overlap. In the following we will discuss
some paradigmatic debates on the relationship between logic and ontology,
divided up by areas of overlap.
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 an
ontological question. 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 commit 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 ‘∃xFx’ is true just in case
there is an object in the domain of quantification that, when assigned as the
value of the variable ‘x’, satisfies the open formula ‘Fx’. 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 his (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 ‘∃xFx’ is true just in case there is a
term in the language that when substituted for ‘x’ in ‘Fx’ has
a true sentence as its result. Thus, ‘∃xFx’ is true just in case there
is an instance ‘Ft’ which is true, for ‘t’ a term in the
language in question, substituted for all (free) occurrences of ‘x’ in
‘Fx’. 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
Fs’ is implied by what we believe for us to be committed to
Fs. What does not matter is whether the semantics of the quantifier in
“There are Fs” (assuming it contains a quantifier[4]) is objectual or substitutional.[4]
However, even if one agrees that what matters for ontological commitment is
whether or not what one believes implies that there are Fs, 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 famously argued, in (Russell 1905), 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 widely accepted. And Davidson argued, in (Davidson
1967), 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 2000) and (Hofweber 2005). One consequence of this is a meta-ontology
different from Quine's, as we will discuss below.
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 the 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 in (Kant 1781/7), namely (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
article 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 (L2)). 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., ‘∃x x=x’. Similarly, it is a logical truth in (standard versions
of) second order logic that ‘∃ F ∀x(Fx 
¬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 exist.
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
be 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).
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 Stanford Encyclopedia of Philosophy article Frege's philosophy 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 (Frege 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.
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. 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. 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, in 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 commitment 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.
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 the scientists to formulate
theories of the word. 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. Or so Carnap's project. Carnap's ideas about ontology and
meta-ontology are developed in his classic essay (Carnap 1956). 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 a 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. Such truths 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 been an armchair, a priori investigation into the
fundamental building blocks of reality. As such it is completely separated from
science. Quine 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 his (Quine 1951). 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 word,
which contains physics and the rest.
Carnap's arguments for the rejection of ontology are presently widely
rejected. However, some philosophers have recently attempted to revive some
parts or others of Carnap's ideas. Yablo has argued that an internal-external
distinction could be understood along the lines of the fictional-literal
distinction. And, he argues in (Yablo 1998), 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, Hofweber has argued that an
internal-external distinction with many of the features that Carnap wanted can
be defended using facts about natural language, but that such a distinction will
not lead to a rejection of ontology, in the sense of (O2). See (Hofweber 2000)
and (Hofweber 2005). Putnam, for example in (Putnam 1987), has developed a view
that revives some of the pragmatic aspects of Carnap's view. See (Sosa 1993) for
a critical discussion of Putnam's view, and (Sosa 1999) for a related, positive
proposal.
One way to understand logic is as the study of the most general forms of
thought or judgment, what we called (L4). 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 the same structure in thought as
well as in reality. And similarly for other structural features. Does this
matching between thought and the world ask for a substantial philosophical
explanation? Is it a deep philosophical puzzle?
There might be a number of philosophical problems in the neighborhood of this
issue. One is epistemic, connected to the question whether or not logic is a
priori, or rather a general but still empirical discipline. We will not address
this question here. Another is to ask for an explanation of the structural
similarity between the general features of thought and the general features of
reality. We will focus on this question. And we will take the subject-predicate
structure of though, and the object-property structure of reality as our
example.
If there is an explanation of this similarity to be given it seems it could
go in one of two ways: either the structure of thought explains the structure of
reality, or the other way round. An explanation of the latter kind, where the
structure of reality explains that of thought, could go as follows: the world
has a certain basic structure, being constituted by objects which have
properties, which other objects can have as well. To properly represent a world
like this the creatures from which we evolved had to develop minds that mirror
this structure. Those who developed a different kind of mind died out. Therefore
we have a mind whose thoughts have a structure which mirrors the structure of
the world.
This kind of an explanation is a nice try, but it is rather speculative.
After all, why should we think that only the right kind of structure of thoughts
helps with survival? If reality really contains no objects and properties, but
say only a large amount of stuff, how would that shorten our survival? Of
course, if there are no objects then all our thoughts about objects would
presumably be false, and thus we face the question whether truth is beneficial
for survival, or something else might do just as well.
On the other hand, one could argue that the structure of thought explains the
structure of the world, which would most likely be an idealist route of sorts.
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 structure of our minds. 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
structure of thought explains the structure of reality.
But maybe there is not much to explain here. Maybe reality does not have
anything like a structure that mirrors the structure 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. 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 truth 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 structure 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 truth see (Künne 2003).
Whether or not there is substantial metaphysical puzzle about the
correspondence of the structure thoughts and the structure of reality will
itself depend on certain controversial philosophical topics. And if there is, it
might be quite trivial to solve or a tough nut to crack.
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 many others. The references and links below provide more in depth
discussions of these topics.
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Language’ in Linguistics and Philosophy, 4: 159-219.
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University Press.
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modal logic, Chicago: University of Chicago Press. 2nd edition.
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203-221.
- -----. 1963. ‘Intellectual autobiography’, in Schilpp 1963, pp. 3-84.
- Davidson, D. 1967. ‘The Logical Form of Action Sentences’, in Davidson
1980.
- -----. 1980. Essays on Actions and Events, Oxford: Oxford
University Press.
- Engel, P. 1991. The Norm of Truth: an introduction to the philosophy
of logic, Toronto: University of Toronto Press.
- Everett, A. and Hofweber, T. (eds.) 2000. Empty Names, Fiction, and
the Puzzles of Non-Existence, Stanford: CSLI Publications.
- Fine, K. 2002. The Limits of Abstraction, Oxford: Oxford
Univeristy Press.
- Frege, G. 1884. Die Grundlagen der Arithmetic: eine
logisch-philosophische Untersuchung zum Begriff der Zahl, Breslau: w.
Koebner; translated by J. L. Austin as The Foundations of Arithmetic: A
Logic-Mathematical Enquiry into the Concept of Number, Oxford: Blackwell,
second revised edition, 1974.
- Goble, L. 2001. Philosophical Logic, Oxford: Blackwell
Publishers.
- Gottlieb, D. 1980. Ontological Economy: substitutional quantification
and mathematics, Oxford: Oxford University Press.
- Haack, S. 1978. Philosophy of Logics, Cambridge: Cambridge
University Press.
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No 6.
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Oxford University Press.
- Harman, G. 1986. Change in View, Cambridge, MA: MIT Press.
- Hofweber, T. 2000. ‘Quantification and non-existent objects’, in Everett
and Hofweber 2000, pp. 249-274.
- -----. 2005. ‘A puzzle about ontology’, Nous, vol. 39:2. pp.
256-283;
- Kant, I. 1781/7. Kritik der reinen Vernunft, various translations
as Critique of Pure Reason.
- Künne, W. 2003. Conceptions of Truth, Oxford: Oxford University
Press.
- Lambert, K. 2001. ‘Free logics’, in Goble 2001, pp. 258-279.
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Philosophical Review, 111: 25-65.
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Invariant Theory’, American Journal of Mathematics, 68.
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University Press.
- Putnam, H. 1987. The Many Faces of Realism, La Salle: Open Court.
- Quine, W.V. 1948. ‘On what there is’, Review of Metaphysics;
reprinted in Quine 1980.
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Review; reprinted in Quine 1980.
- -----. 1954. ‘Quantification and the Empty Domain’, Journal of
Symbolic Logic, 19: 177-179.
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University Press.
- -----. 1980. From a Logical Point of View, Cambridge, MA: Harvard
University Press. 2nd edition.
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Press.
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Topics, 21: 149-86.
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Open Court
- Sosa, E. 1993. ‘Putnam's pragmatic realism’, Journal of
Philosophy, 90: 605-26.
- Tarski, A. 1986. ‘What are Logical Notions?’, History and Philosophy
of Logic, 7.
- Tennant, N. 1990. Natural Logic, Edinburgh: Edinburgh University
Press. 2nd edition.
- van Benthem, J. 1986. Essays in Logical Semantics, Dordrecht: D.
Reidel.
- -----. 1989. ‘Logical Constants across Varying Types’, Notre Dame
Journal of Formal Logic, 30/3.
- van Inwagen, P. 1998. ‘Meta-ontology’, Erkenntnis, 48: 233-250;
reprinted in van Inwagen 2001.
- -----. 2001. Ontology, Identity and Modality Cambridge University
Press
- Velleman, D. 2000. ‘On the Aim of Belief’, Chapter 11 of The
Possibility of Practical Reason, Oxford: Oxford University Press.
- Williamson, T. 1999. ‘A note on truth, satisfaction and the empty domain’,
Analysis, 59: 3-8.
- Wright, C. 1983. Frege's Conception of Numbers as Objects,
Aberdeen: Aberdeen University Press.
- Yablo, S. 1998. ‘Does ontology rest on a mistake?’, Proceedings of the
Aristotelean Society, 72: 229-61.
- Zalta, E. N. 1983. Abstract Objects: an Introduction to Axiomatic
Metaphysics, Dordrecht: D. Reidel.
Carnap, Rudolf | Frege, Gottlob | Frege, Gottlob: logic,
theorem, and foundations for arithmetic | logic: free | logical
consequence | logical
constants | ontological
arguments | ontology and ontological commitment
Acknowledgements
Thanks to two anonymous referees for their suggestions, in particular to one
of them for two sets of detailed and helpful comments. Thanks also to Rafael
Laboissiere for reporting several typos.
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