Language
and Ontology
Aristotle
and “Primary Substances”
Language and The Correspondence Theory of Truth
Universals and Existential
Import
Categorical Claims and
Class membership
Ancient and Medieval Categorical
Logic and Ontology
Categorical Logic and the Square of
Opposition
Add Existential Import and more
relations appear!
·
Logical Relations:
Implication
·
Logical Relations:
Contrariety
·
Logical Relations:
Sub-Contrariety
Language and Ontology:
I want to gather into a single lecture the topic of language
and ontology. What follows is a somewhat superficial overview, but I think
it introduces some important topics that we can look at in greater detail
later. Also some of these topics I cover in greater length elsewhere in
the course, but I think it useful to gather them together into one single
coherent lecture.
So “ontology” is technically speaking the “Study of Being”
coming from the Greek word “ontos.” The
compound word “ontology” combines onto- (Greek: ὄν,
on; GEN. ὄντος,
ontos, 'being' or 'that which is') and. -logia (λογία, 'logical
discourse'). But often in philosophical circles when one talks about “an
ontology” one is talking about the items that one thinks “exist,” a sort of
catalog of existing things: cats, dogs, ponies, cacti, etc. These are
things that “be” or “exist.” And so my “ontology,” assuming I believe in
cats and dogs and cacti, would include these items. My ontology currently
does not include “Santa Claus” (sorry) or unicorns, but you might change my
mind on that. Now this raises an interesting question. If unicorns
do not exist, then what do I mean when I say the sentence “Unicorns have a
single horn protruding from their foreheads.”? Indeed how can I refer to
“them” at all? More on this in a bit.
Now Aristotle famously refers to a “substance” (existing
thing) as anything that can be the object of predication. I didn't know
how difficult a concept that is for students to get a hold of these days until
I realized (relatively recently, really) that students spend very little time
learning grammar in school. That was not always the case, of
course. In fact we used to refer to secondary education schools as
“grammar schools.” When I was in school (middle school & high
school), we spent a great deal of time on grammar. That is, the parts of
speech and the structure of sentences and paragraphs etc. So I learned
about “predication” in grammar classes, long before I learned very much about
Aristotle. Nowadays I think they have dropped a lot of that from the
curriculum in middle and high schools because they spend a lot of time on
computer programming and robotics, etc. which were not anything my generation
had to worry about back in the “olde days.’
Because I think a command of grammar is very important for
understanding how language and arguments are supposed to work, and also for the ontological commitments of various
sentences, I have started including a series of lectures on diagramming
sentences in my Intro classes. (My 8th grade English teacher,
Sister Mary DePazzi would be so pleased.)
Diagramming sentences has not been much in vogue as a pedagogical device for the past thirty years or so. There are, however, many grammarians and English instructors who hold that analyzing a sentence and portraying its structure with a consistent visual scheme can be helpful—both for language beginners and for those trying to make sense of the language at any level, especially for language learners who tend to be visual-learning types.
So “predication” is act of “predicating” a predicate-noun or
predicate-property of a subject. A useful way of displaying what is going
on in a predication used to be to diagram sentences. In any diagram of a
sentence, the first four steps are always the same. First, one draws a
horizontal line.
_________________________________________________
Second, one bisects that line with a perpendicular line.
_________________________l_________________________
l
So now we have a grid with 4 quadrants, two on top and two on
the bottom.
__________________1_______l______2___________________
l
Then in the upper left corner of the grid (1), one places the
subject of the sentence.
______________Subject______l______2___________________
l
And in the upper right corner of the grid (2) one places the
verb of the sentence.
______________Subject______l______Verb________________
l
These are the first four steps for any diagram of any
sentence. Depending on how complicated the sentence is, the diagram can
also become very complicated, but the initial four steps are always the same.
So in the simple sentence “Mary danced.” I would draw a
horizontal line (step 1), a perpendicular line (step 2), in the upper left
quadrant I would put “Mary” (step 3) and in the upper right quadrant I would
put “danced” (step 4).
____________________Mary______l_____danced___________________
l
And I'd be done. Every word in the sentence has a space
on the diagram and the diagram is displaying the relationship between the words
of the sentence. In this very simple sentence, the diagram is displaying
that the subject Mary is engaged in the activity of dancing.
Now if I said “Mary danced slowly.”
well again the first four steps are the same horizontal line, bisecting line,
place the subject and place the verb. But I have a word left over:
“slowly.” I have to find a place for that word
on the diagram. When diagramming sentences, modifying words usually end
up going underneath the word they modify. So in this sentence “slowly”
modifies how Mary is dancing. It is an adverb and so it
would go underneath “danced” on our diagram.
____________________Mary______l_____danced___________________
So far so good.
If I said “The professor is very
smart.” the first four steps are the same.
Horizontal line, perpendicular line, the subject of the sentence “professor”
left hand side, the verb of sentence “is” on the right-hand side.
But now I have three words left over: “the” “smart” and
“very.”
“The” is easy enough. That's a definite article and it
is modifying “professor” so it goes underneath that
word on the diagram.
Where does “smart” go? Well “smart” in this case is a
predicate adjective. In other words I am predicating smart
of “professor.” I'm saying that this is a quality that the professor has.
When diagramming a predicate adjective one places a slightly
slanted line on the horizontal line and then the predicate adjective after that
and so now I have diagramed, “The professor is smart.”
___________Professor________l_____is____/smart___________
l
But I still have one word left over, “very.” Where does
that go? Well in this case it is modifying smart. It is telling you
the degree of smartness professor has. The professor is smart/
very. And so, it would go underneath smart.
___________Professor________l_____is____/smart___________
l
But the value of the diagram is that it points out that I am
predicating “smart” of “professor” so smartness is a
property of the professor.
Aristotle and “Primary Substances”
For the ancient Greek philosopher Aristotle, this indicates
the professor is a “primary substance” whereas “smartness” is NOT a primary
substance. Rather it is a secondary substance. “Smartness” is just
a quality of the subject of predication, what is being predicated of
“professor.” Again, there's a sense in which “smartness” can't exist on
its own, so it is not “primary” here; smartness can only exist as a property of
existing things such as “professors” and “students” etc. That’s why “smartness”
is a secondary substance. Its existence
is dependent upon primary subsrtances.
The bigger point for this lecture is to notice that Aristotle
is taking his ontology (catalogue of existing things) from language. In
other words, when we find true sentences and we parse them into subjects and
predicates (and predicate nouns and predicate adjectives etc.) we start to
define our ontology and our ontological commitments. That is, we start to
define our catalog of things we think exist and mark out our “ontological
commitments.”.
I think we can say that Plato is doing pretty much the same thing
as well, taking his ontological commitments as an interpretation of what must
be the case when certain sentences are true. Notice, when I say, “There is
something that all cats have in common.” and that this sentence is true, I seem
to be making an ontological commitment. There is, or there
exists, something (some thing) that all cats
have in common. What is that thing? Well, Plato would claim the
answer to that question is “Cat Form.” “Cat Form” is what all and only
cats have in common which makes them cats (and not, say, dogs). Aristotle
too would say that cat form or cat essence is what it is that all cats have in
common by virtue of which they are cats. And for Aristotle just as for
Plato, cat essence was thought to me a real, mind independent reality, existing
in the world, which we can come to know.[1]
But notice further the curious reversal of ontological
dependency between the views of Plato and Aristotle on this point. For Plato
Cat Form is ontologically prior to cat particulars. That
is, Plaot claims that there could not actually be
individual cats if there did not exist Cat Form, but there could exist Cat
Form without any existing individual cats. Particulars are
ontologically dependent upon Forms according to Plato, but Forms were
ontologically independent, and thus ontologically prior to, their particular
instantiations according to Plato. For Aristotle, it is exactly the
opposite. For Aristotle there could not exist cat essence or cat form if there
were not actual, particularly existing, cats. For Aristotle actual individual particular cats are the primary substances. Cat form is a secondary substance that cannot
exist on its own (like smartness or redness). It can only exist as a
property of individual existing things (i.e. real cats). Were there no
individual cats, there would not be a cat form. So, cat form is a ontologically
dependent on the particulars according to Aristotle, and the particular cats on
ontologically prior to the form.
Language and The Correspondence Theory
of Truth
Now this roughly corresponds to what is called the “correspondence theory of truth.”
On this view, sentences are true when they correspond to or accurately
“picture” the world.
When I say the sentence “My car is in the parking lot.” is true,
I am saying the sentence “My car is in the parking lot.” is true if and only if
there exists such a thing as “my car” and it is in the “parking lot.” So,
the truth of the sentence commits me to a certain ontological “state of
affairs” and a catalog of existing things (cars, parking lots, etc.).
Now even if this is a useful enough way of understanding some
sentences and what we mean when we say that the sentences are true, clearly it
does not work for all sentences. Some sentences are not so simple and this straightforward analysis does not
work. When I say “An American is stung by a bee
every 10 seconds.” well, the joke is to respond “Wow, that's one very unlucky
American.” or “Boy, that's one vindictive bee!”
But of course, this is a joke because the speaker is not
claiming there “exists” such a thing as a single American who is stung
repeatedly, or a bee that stings again and again every 10 seconds.
In this case, when I say the sentence “An American is stung by a bee every 10
seconds.” is true, I am NOT committed to that simple ontology, but rather to a
logical construction. I am saying something more like “When you take the
total number of seconds (in a year) and divide is by the total number of
American bee stings, you get about one sting for every 10 seconds.
For instance were I to say “The average plumber has two and a half children.”
________Plumber_______l__has____L__children____
l
I do NOT mean to say that there exists such a thing as
plumber/ the/ average who has children/ 2.5. What I'm truly saying is
that when one takes the total number of plumbers, the total number of plumbers’
children and divides the total number of plumbers children by the total number
of plumbers one comes up with something like two and a half children per
plumber.
So here is an example where the simple picture theory of
language and ontological predication/ commitment is inadequate and
misleading.
Now that may seem obvious enough here, but what if I were to say “Mary danced a waltz.”?
___________Mary________l__danced ___L__waltz_____
l
Well, here we have the subject “Mary.” The verb is
“danced” and it has a “direct object,” waltz.
This seems to commit me to two existing things: Mary and waltz.
But is that really true? Is
the only way that sentence could be true that there exists
these two different things? Must there exist such a thing as “a waltz”
that Mary “danced?” Some might say yes, but notice, I am saying more or
less the same thing if I simply say “Mary
waltzed.” But here, in this second sentence, which is more
or less equivalent to the first, there is no commitment to the existence
of a thing beyond Mary. Mary “is” and here the way she “is” is
waltzing. In other words, Mary danced in such a way as called waltzing.
Were I to just say, “Mary danced
slowly.” I'm simply telling you how Mary danced and I making no
ontological commitments beyond Mary. Now I am saying that Mary danced
“waltz-edly.” I'm saying how Mary danced, but
I'm not committed to the existence of another thing nothing, beyond Mary and
the way she is (is behaving). To suggest otherwise is to make an
unwarranted ontological commitment.
Some have suggested that this is a useful way of dealing with
a problem from the philosophy of mind we encounter when we talk about
qualia. Qualia are, allegedly, elements of our subjective experience such
as a red afterimage is see after a flashbulb goes off,
let’s say. Or the sweet sensation I experience when I put honey on my
tongue. The afterimage or the sensation seem to be real things that I am
ontologically committed to despite the fact that they
seem to have no physical location. Does qualia
commit us to a non-physicalist account of mind?
Consider the sentence “Jane saw a red afterimage.” What
ontological commitments does this sentence make if it is true?
______________Jane_______l__saw___L__afterimage_____
l
When I say “Jane saw a red
afterimage after the flash bulb went off.” I seem to be committed to the
existence of a “thing” called an “afterimage” which does not correspond to any thing that exists in space, but
is the direct object of Jane’s experience. In other words, Jane
experienced a “thing” and it is the direct object of
her experience (the thing that she saw) that I call an afterimage, in this case
a red one. So we might diagram the sentence this way
______________Jane_______l__saw___L__afterimage_____
l
But notice, that may not be the only way of dealing with the
truth of the sentence “Jane saw a red afterimage.” Some have suggested a
useful way of dealing with this is an adverbial account of
experience. That is, Jane saw redly/afterimagely.
In other words, I am not committed to the existence of a thing called it
afterimage, but rather a way that Jane experienced. Just as Mary who
waltzed is behaving in a certain way, Jane who experiences a red afterimage is
simply experiencing in a particular way. I'm not committed to anything
beyond Jane and how she is experiencing in terms of my ontology.
Another way of dealing with the truth of sentences is to
reject the correspondence theory of truth entirely and embrace what is referred
to sometimes as the pragmatic theory of truth. Here we reject the idea
that true sentences accurately picture the world or make any ontological
commitments at all, but rather that true sentences are merely sentences that
are useful ways of speaking. This follows from a sort of Kantian view of
knowledge as mental construction(s) of experience or pragmatism more directly.
Recall that Immanuel Kant claims knowledge does not consist in grasping things
in themselves. Indeed, we can never hope to grasp things in themselves,
but only understand the world in terms of human concepts and experience as
shaped by mind. So knowledge consists in describing the world as it
presents itself for human experience.
Pragmatists go beyond the normal idea that the usefulness of
a theory is a reliable test of its truthfulness to the bolder
claim that all it means to call a theory true is that it is a useful way of
speaking. Some have defended, for instance, past life regression therapy
on these grounds. When I say John experienced a past life where he
died by drowning and this is why he is afraid of water, I do not mean to say
there exists such a thing as a “past life.” I mean that speaking this way
is a useful way of treating his fear of water. Or when I say the
personality is comprised of three parts: the Id, the ego
and the superego, I'm not committed to the existence of these three things as
ontological entities, but rather this is a useful model for classifying human
experiences and treating mental illness.
Perhaps more boldly, to say that there are protons, neutrons
and electrons does not commit us to the existence of mind independent
entities. On this view, one is merely claiming that proton talk, neutron
talk and electron talk, are useful ways of speaking to predict future
experience or categorize past experience
etc. But note further, “useful” is one of the relational
terms. To say “X is useful” will always need to be cashed out “useful
for… “ Useful will always be relative to some problem set. And of
course, sentences and the theories in which they occur that are useful (i.e.)
true relative to problem set X may NOT be useful (i.e. true) relative to
problem set Y. On this more extreme view, it is not that Newton’s physics
was wrong, it is that our problem set has changed since then.
Universals and Existential
Import
Imagine we're having a conversation and I said to you, “None of my children have graduated college.” And you reply, “Really? I didn't know you had any children.” And I responded, ‘You're quite right. I don't have any children.” At a minimum, you would think that was odd. You might even think that I had been deceptive by saying that none of my children have graduated college. It would be even weirder if I had said “All of my children have graduated college.” and then went on to say that I had no children. In these cases, you are assuming that the subject class I am referring to in my initial statement, the class of “my children,” is a nonempty set. In other words, when I say “None of my children have graduated college.’ You are assuming that I am actually asserting two separate things:
1. “None of my children have graduated college.”
And
2. “I have some children (e.g. at least one child).”
In so doing, you are interpreting the sentence with what philosophers and logicians refer to as “existential import.”
Now, if I said, “No one who is 11 feet tall could sit comfortably in a Miata.” you might think that this sentence is true despite the fact that you may or may not believe there exists any human being who is 11 feet tall. (I don't think there are, but I don't know for sure.) Nevertheless, the sentence here does NOT seem to carry with it existential import. Likewise the sentence “All unicorns have horns.” Is widely accepted as true, even by people who believe that there exists no such things as unicorns. Again, the sentence does not seem here to carry existential import.
How do we know the difference?
Well in ordinary language, I don't know that there is a hard and fast rule. This reveals a difference between what we might call “conversational implication” and “logical implication.” As a matter of conversations we treat some universal claims as carrying existential important and other as not and as I say, I don’t think there is a syntactic or even a sematic rule to guide us here. However, as a matter of standard formal interpretation, contemporary logicians understand universal claims such as “All S are P.” and “No S are P.” in what's called the hypothetical mode of interpretation and not the existential mode. In other words, according to standard formal logic interpretation, the statement “All S are P.” should be interpreted as claiming “If anything is an S, then it's a P.” Likewise, the claim “No S are P.” we interpret that as saying “If anything is an S, then it is not a P.” The question as to whether or not anything actually is an S is left open. The truth of universal statements does not commit is to the existence of members of the subject class.
As I say, this is the standard interpretation for logicians and philosophers for universal claims such as these. But in normal conversation, that rule does not always apply. This is another example of ontology following from language use. We might infer from the truth of a statement that something exists (ontological commitment) even when that is not what has been explicitly stated.
Categorical Claims and
Class membership:
To be clear, even philosophers and logicians have not always agreed with this now standard interpretation of universal categorical claims. Historically, for philosophers and logicians the standard way on interpreting true universal statements was to assume they carry with them “existential import,” that is, that the subject class of these claims is not empty. Why did they adopt this convention? In part, it might because they thought there wasn’t much value in trying to reason about imaginary realities. But there was a more philosophically/metaphysically grounded justification as well.
Categorical claims are claims about the relationships between two distinct categories. for instance if I said “All cats are mammals.” I'm talking about two categories of things: the category of cats and the category of mammals. But beyond that, I'm saying that anything which is a member of the category of cats is also a member of the category of mammals. Notice, if I said that “No turtles are mammals.” I'm asserting the opposite logical relationship, that is no member of the subject class is also a member of the predicate class.
In standard form categorical logic there are only four types of categorical claims: A-claims are affirmative universal claims form “All S are P.”. E-claims are negative universal claims of the form “No S are P.” The remaining two claims are not universal, but rather particular. The I-claim is an affirmative particular claim of the form “Some S are P.” (where “some” is understood to me at least one existing thing). The O-claim is a negative particular of the form “Some S are not P.” (again where “some is understood to mean at least one existing thing).
So note that the I-claim and the O-claim to not carry with them “existential import” because they quite explicitly claim “There exists at least one thing that is an S that is also a P.” (I-claim), that is there exists at least one member of the subject class that is a member of the predicate class, or alternatively “There exists at least one thing that is an S that is not a P.” (O-claim). Both are stating ontological commitments ANS stating a relation between the two classes.
Ancient and Medieval Categorical
Logic and Ontology
One similarity between the metaphysics of Plato and Aristotle is that they were both real realists with respect to universals. That is, both Plato and Aristotle believe that words like “cat,” “dog,” “gold,” name real universals which exist, independent of our mental concepts of them. These universals are not merely “ideas” we dream up with our language or social practices, but rather are mind-independent objective realities. Where Plato and Aristotle differed is that Plato thought these universals were immaterial forms which existed in a transcendent realm. Aristotle, by contrast, held that these universals existed, but only in their instantiations. That is, the universal essence of “cat nature” exists as a mind-independent reality, but only in individual cats. Were there no cats, there could not be “cat nature” according to Aristotle. So for Aristotle, there are no uninstantiated universals. All classes named by a (real) universals must have membership. Further, if the subject class of a statement is an empty set (That is, the empty set.), one cannot truly or falsely predicate anything about it. This is because there is no member (no existing thing) corresponding to the assertion and so there is nothing which could make the statement true or false.
The relevance of this to our current discussion is this. Aristotle would claim that no universal claim is true or false which has as its subject the name of an (the) empty set. So, for Aristotle, the sentence “All unicorns are horned animals.” could only be true if there were at least one unicorn. The relevance of this to logic is that universal claims were presumed to be claims about non-empty sets on the Aristotelian and Medieval logical systems. While contemporarily logicians adopt the hypothetical view and interpret the sentence “All unicorns are horned animals.” as equivalent to “If there is any such thing as a Unicorn, then that thing would be a horned animal.”. Aristotelian and medieval logicians would not. For contemporary logicians, universal claims do not carry with them “existential import,” but for Medieval logicians they do. But this means on the hypothetical/ modern interpretation, my claim “None of my children graduated college.” and my claim “All of my children graduated college,” could both be “true” at the same time if, in fact, I have no children. What fact could you point to to prove me a liar?
Categorical Logic and the Square of
Opposition
If you have taken introduction to logic, you may have covered something called Categorical Logic/ Aristotelian Syllogistics and within that segment of the course, what is called the “Square of Opposition.” In the square of opposition, the four standard form categorical claims are arranged so that they form a square. The top two corners represent the universal claims with the Affirmative Universal (A-claim) “All S are P.” going in the upper left corner of the square and the Negative Universal (E-claim) “No S are P.” going in the upper right corner. In the lower left corner goes the Affirmative Particular (I-claim) “Some S are P and in the lower right corner goes the Negative Particular (O-claim) Some S are not P.
|
AFFIRMATIVE |
NEGATIVE |
|
|
UNIVERSAL |
(#1) ALL As are Bs |
(#2) NO As is Bs |
|
PARTICULAR |
(#3) SOME As are Bs |
(#4) SOME As are not Bs |
Logical Relations: Contradiction
When arranged in this fashion, we can see that various logical relations hold between the claims. Note for instance that A-claims and O-claims state exactly the opposite thing and therefore cannot have the same truth-value. If A-claim is true, then the corresponding O-claim must be false and if O-claim is true, then the corresponding A-claim must be false. And vice versa. In other words, these two claims cannot both be true at the same time nor can they both be false at the same time. So if I know the truth-value of one (be it T or F), I immediately know what the truth-value of the other. This is the logical relation of “contradiction,” and it holds between A and O claims and it also holds between E and I claims.
Add Existential Import and
more relations appear!
If one makes the assumption of existential import for universal claims, that is, if one assumes that the subject class is not empty, a number of other logical relationship reveal themselves.
Note that if the A-claim is true, (i.e. Anything that is an S is also a P and there is at least one S.) then the I-claim (i.e. There is at least one S that is a P.) MUST be true and if the I-claim is false, then the A-claim MUST be false. This is the relation of “implication.” Here A implies I. In the same fashion, we find on the other side of the square that the E-claim implies the O-claim.
Logical Relations: Contrariety
Between the A-claim and the E-claim, the relationship of contrariety occurs. Two claims are contraries if they cannot both be true at the same time, but they CAN both be false at the same time. So, it cannot be the case that “All my cats are blue-eyed cats.” AND “None of my cats are blue-eyed cats. (again, assuming I have cats). But, it could be the case that both of these claims are false. Note, if I had some green-eyed cats and some blue-eyed cats the A-claim would be false and the E-claim would be false at the same time.
Logical Relations:
Sub-Contrariety
Between the I-claim and the O-claim, a converse relationship occurs called sub-contrariety. Two claims are sub-contraries when they CAN both be true at the same time, but then CANNOT both be false at the same time. Again, assuming I have cats, it can be true that “Some of my cats are blue-eyed.” and also true that “Some of my cats are NOT blue-eyed.” at the same time. However, it cannot be false that “Some of my cats are blue-eyed.” and also, at the same time, false that “Some of my cats are NOT blue-eyed.” Either they are blue-eyed or they are not. Assuming we are talking about a non-empty sets of cats, they either have blue eyes or they don't. In the first case, the I-claim is true, and in the second case, the O-claim is true. But, at least one of them has to be true. (Again, this is assuming we're talking about non-empty sets, i.e. assuming existential import.) Of course, it I don’t have any cats, that is it “My cats” name an (the) empty set, then both the claims “Some of my cats are blue-eyed.” and the claim “Some of my cats are NOT blue-eyed.” are false. In neither case would there exist “at least one” member of the subject class.
Now Aristotle and the medieval logicians that follow him adopted this interpretation of universal claims (i.e. with existential import). Again, they just thought it was silly to talk about non-existing things and so long as we're talking about existing things these logical relationships hold between these sets of existing objects. However, modern and contemporary logicians have adopted the hypothetical interpretation of universals. So when I say, “All my cats have blue eyes.” I'm merely saying that if something is “my cat” then it has blue eyes. This does not commit me to the existence of actual cats. Thus I am not assuming existential import in my interpretation of the claim. It's not part of the logic of the claim. So I can say the sentence “All unicorns are horned things.” is true despite the fact that there exists no such thing as unicorns.
The point of these remarks for our purpose is here is to
look at the ontological commitments which arise from certain statements.
If we are to interpret universal statement “Every person born in the United
States is a US citizen.” or “No one who failed the final exam will pass this
class.” with existential import, we are assuming that the classes were talking
about are not empty. But is that really what is being asserted or are we
assuming this without warrant? And even if it's not an assumption that
formal logicians make when conducting logical exercises in logic classes, it is
an assumption that forms (or perhaps misinforms) the background of many
conversations? So there's often a conversational implication of
existential import and we need to be aware of that. I might claim for instance
to be selling a medical remedy which is such that every person who successfully
completed the prescribed medical regime was cured of cancer. Wow.
That sounds very promising doesn't it? However,
we might ask, wait a minute. He didn't actually
explicitly say anyone had successfully completed the
prescribed medical regime. So my sentence may be “true” despite the fact that I have cured no one. How should the
law regard such claim do you think?