Episode post here. David North does it again! Many thanks to him for typing this transcript up.
Matt Teichman:
Hello and welcome to Elucidations, a philosophy podcast
recorded at the University of Chicago. I’m Matt Teichman.
Jaime Edwards:
And I’m Jaime Edwards.
Matt Teichman:
With us today is Marko Malink, assistant professor of
philosophy at the University of Chicago, and he’s here to talk to us about modal
syllogistic. Marko Malink, welcome.
Marko Malink:
Hello, Matt. Hi, Jaime.
Matt Teichman:
The first question that’s on everybody’s mind is “what
is syllogistic?”
Marko Malink:
Aristotle’s syllogistic is basically the first system of
formal logic in the history of philosophy. It is a theory where Aristotle gives
an account of valid deductive inferences. As such, it is like logical systems
which we find in the 20th century, but there is a crucial difference between
Aristotle’s syllogistic and those logical systems. This is connected to the
kinds of sentences which Aristotle considers in his syllogistic. So in his
syllogistic, Aristotle is mainly concerned with sentences such as “Every B is an
A.” For example, “every man is an animal.” And he studies inferences of the form
“every C is a B,” “every B is an A,” therefore “every C is an A.” That’s one of
the basic inferences which Aristotle considers in his syllogistic. And he gives
an account of which of these inferences you can construct out of such
sentences. He also considers sentences of the type “some B is an A” or “no B is
an A” and he gives an account of which inferences we can construct out of these
sentences are valid, and which of these inferences aren’t valid.
Matt Teichman:
What would be an example of a valid inference and an
example of an invalid inference?
Marko Malink:
An example of a valid inference is the one I just gave:
every C is a B, every B is an A, and therefore every C is an A. A further
example would be: every C is a B, no B is an A, therefore no C is an
A. Aristotle also says which of these inferences is invalid. So for example, if
no C is a B and every B is an A, then it does not follow that no C is an A.
Matt Teichman:
So, we’re talking about different argument patterns:
these are templates for building arguments out of actual sentences without “B”-s
and “C”-s. Where we plug stuff in wherever we have a “B,” “A,” or “C”, as in:
“every giraffe is a mammal,” “every mammal breathes,” therefore “every giraffe
breathes”, or something like that. We abstract away from these argument patterns
that have actual sentences in them to get to the more abstract forms, so that we
can say: anything that has this form will be a good argument.
Marko Malink:
That’s right. One of the big differences between
Aristotle’s syllogistic and systems of logic in the 20th century concerns, as I
said, is the syntax of his sentences. For Aristotle, the basic sentences consist
of two terms and a third part which connects these two terms. For example, the
sentence “every B is an A” consists of two terms, namely B and A (“man” and
“animal” would be examples of these terms), and a third part which forms a
sentence out of these two terms. The third part would correspond to the words
“every” and “is.”
Aristotle himself actually doesn’t say “every B is an A.” He rather says things like “A belongs to all B”. And in this way of saying things, this third element would correspond to the phrase “belongs to all.” Aristotle considers these sentences, which have this tripartite syntax consisting of two terms which are of the same syntactic type and the third element, as basic sentences.
Whereas in 20th century logic, basically Fregean first order logic, the basic sentences consist of just two terms. They have a bipartite syntax consisting of a singular term and a general term. And these two terms, single and general terms, are not of the same syntactic type. No singular term can take place of a general term and no general term can take the place of a singular term. They aren’t of the same syntactic type, whereas for Aristotle, the subject and predicate in his syllogistic theory are of the same syntactic type. Every subject of a sentence can also serve, in principle, as the predicate of a sentence, and every predicate can in principle also serve as the subject of a sentence.
Matt Teichman:
To go back to the “every man is an animal” example,
Aristotle would think of that sentence as really having three parts: “every,”
“man,” and “animal.” Whereas a modern logician would just think of sentence as
having two parts. So what would the two parts be in “every man is an animal”?
Marko Malink:
In 20th century logic, “every man is an animal,” unlike
for Aristotle, is not considered a primitive sentence. Rather, it is considered
a complex sentence built up from several primitive sentences. As we said before,
the primitive sentences in 20th century logic consist of just these two types: a
singular and general term. So “man” and “animal” would be general terms, and
singular terms are, like, “Socrates” and “Matt,” for example. And Aristotle’s
sentences are represented in 20th century logic as complex sentences. This is
what Frege did in his Begriffsschrift when he set it up; he had to prove or show
how he can express Aristotle’s sentences in his own Begriffsschrift. He did this
by saying: a sentence like “every B is an A” or “every man is an animal” will be
expressed in the following way. I’ll just say for every X, where X corresponds
to singular terms, if X falls under man, then X also falls under the general
term animal. So, this whole phrase corresponds to a complex sentence in 20th
century logic (such as Frege’s Begriffsschirft). This is one of the big
differences between Aristotle’s logic and 20th century logic. For Aristotle,
these quantified sentences which he studies in his syllogistic are primitive
sentences and for 20th century logic they are complex sentences. This isn’t just
a point about syntax; it also has implications for the semantic interpretation
of these sentences.
Jaime Edwards:
So, you said that Frege redescribes Aristotle’s
sentences in some way and this has consequences for how we understand
Aristotle’s sentences in light of that, some of which might be misleading. Would
you say something about this?
Marko Malink:
This is true. When Frege gives his redescription of
Aristotle’s sentences by saying “every B is an A” just means, or should be
expressed as a complex sentence of the form, “for every X, if X falls under B,
then the individual X also falls under A,” this syntactic description
implies—unlike Aristotle’s description, which, on the face of it, doesn’t
imply any semantic interpretation—Frege’s redescription implies that the truth
of these sentences depends solely on the individuals which fall under general
terms.
So, the truth of these sentences like “every B is an A” or “every C is an A” just depends on which individuals fall under the general terms B, A, or C. Whenever we have two terms which have exactly the same set of individuals falling under them, we can replace one term by the other term and the sentence still remains true or false, whatever it was before. The set of individuals which fall under a general term is often referred to as the extension of this term. In this sense, Frege’s redescripton of Aristotle’s sentences implies an extensional semantics. Because, on his redescription, the truth of Aristotle’s sentences just depends on the set of individuals which fall under the general terms involved in them, i.e. just depends on the extension of the terms. In this sense, it implies an extensional semantics. And this is different from what we find in Aristotle himself because Aristotle does not really specify in any detail what the semantics of his sentences should be, and in particular, he does not explicitly state that we should give them an extensional interpretation. In fact, there is some indication that Aristotle didn’t think of his sentences, or their truth conditions, as being purely extensional in the sense just described.
Matt Teichman:
So we have two different things that “every man is an
animal” might mean, and Frege’s account of what “every man is an animal” means
is something like: you go around to each individual man and you check to see
whether he’s an animal. Is this guy an animal? Yes, he is. Is that guy an
animal? Yes, he is. And then if, after you’ve done all that, they all turn out
to be animals, then the sentence “every man is an animal” is true. Whereas it
seems like on Aristotle’s understanding of the sentence “every man is an
animal,” what it takes for that sentence to be true isn’t necessarily just about
what’s the case with a whole bunch of individuals. Maybe it’s more about
something else. Maybe the concepts man and animal, or maybe some other sort
of thing.
Marko Malink:
I think so. Unfortunately, Aristotle himself does not
really describe in any detail what the semantics of his sentences is—what
their truth conditions are. But there are some indications which seem to
suggest, at least, that he did not have in mind a purely extensional
interpretation, which rather suggests that he thought of their truth conditions
as somehow being based on his theory of predication.
So for example, one of the basic principles of Aristotle’s theory of predication is that it relies on a distinction between substances and non-substances. Terms like man or animal in some way correspond to or signify substances, whereas terms like green, walking or moving, these correspond in some way to non-substances. One of his basic principles is that substances cannot be predicated of non-substances; rather, non-substances are predicated of substances. It seems like in his syllogistic, Aristotle adheres in some way to this principle. Namely, it is striking that when we look at the concrete examples of sentences of the type “every B is an A,” which Aristotle gives in his syllogistic, we will find that very often, Aristotle gives examples where the subject term is a substance term (like man) and the predicate of the sentence is a non-substance term, like in “every man is walking.” This kind of example we find very often in Aristotle. But what we don’t find in his syllogistic are examples where the subject term is a non-substance and the predicate term is a substance. So, Aristotle does not give examples of the type “everything which is walking is a man” and we don’t even find examples of the kind “everything walking is an animal.” So, we don’t find examples of the type “everything walking is an animal”, even though, from the extensional point of view, they are obviously true, because every individual which falls under the term “walking” also falls under the term “animal.” So the extension of the first term “walking” is a subset of the extension of the second term “animal”—so from Frege’s point of view, this sentence is perfectly true.
If this were Aristotle’s account of the semantics of his sentences, we would expect Aristotle to actually use these sentences. But he doesn’t do it. Now of course, Aristotle may have had all sorts of reasons not to use these sentences. But I think there are reasons to think that the reason why he doesn’t use them is that he thinks of his sentences as somehow being tied to his more metaphysical theory of predication, which relies on the contrast between substances and non-substances—and in which there are no predications where a substance is predicated of a non-substance.
Jaime Edwards:
Would you say something more about the significance of
this result for Aristotle? Beacuse one of the things we are taught when we study
logic is that it tracks what follows formally from one thing to the next. It’s
the formal features that follow, and it doesn’t so much matter the natures of
the As or the Bs—it just matters that the relations follow logically. But what
you’re describing is that Aristotle limits his use of syllogism to a
metaphysical commitment to: certain things are substances, and certain things
are only predicates.
Marko Malink:
Ok. Well, the question which you just asked is a
difficult question, and part of the difficulty is that it’s not prima facie
clear what it means to say that logic is formal. There are senses such as the
one you just described in which, given what I said, Aristotle’s syllogistic
would not be formal, because in some way it is sensitive to the distinction
between substance terms and non-substance terms. Whereas one might expect that
a purely formal logic should not be sensitive to such metaphysical
distinctions.
But there are also senses in which Aristotle s logic, as I see it, is still formal, even if it is sensitive to the contrast between substance terms. As I see it, this distinction is built into the truth conditions of his sentences. So any pattern of inference which is valid will still be valid, whichever term we put into the placeholders A, B, and C. It’s just that in some cases which we, given Frege’s account of these sentences, might expect to be true, they in fact aren’t true. For example, a sentence like “everything walking is an animal.” We would expect this example to be true, maybe, but given what I said, if this is true, these sentences aren’t in fact true, but still, any pattern of inference in which such a sentence occurs either as a premise or as a conclusion is still valid. So, in this sense Aristotle’s logic is still formal. It’s not that his arguments are valid if we put in substance terms in certain places. They’re still valid whatever terms we choose and, in this sense, it’s still formal. It’s just that the truth conditions of these sentences don’t correspond exactly to what we might expect them to be from the Fregean point of view.
Matt Teichman:
Ok, so we’ve got this distinction between substance
terms and non-substance terms. The substance terms would be e.g. man,
whale, giraffe, that kind of stuff—and the non-substance terms,
intuitively, are properties that something might happen to have by coincidence
but not a property that something has because of the kind of thing that it is.
Then these non-substance terms would be e.g. being green and walking—these
are properties you might happen to have but don’t define what you are. So what
we seem to be concluding from the fact that Aristotle only says things like
“every man is walking” but not “every walking thing is a man” is that Aristotle
thought: saying “every A is a B” isn’t just saying something about whether all
As happen to be Bs, but something more than that. Something like being a B is
part of what it is to be an A, maybe.
Marko Malink:
I think this is not exactly the conclusion I want to draw
because Aristotle does use examples of the kind “every man is walking” or even
“every man is white.” And in these cases it’s obvious that walking is not part
of what it is to be a man. So he does accept purely accidental predications,
it’s rather that the truth conditions are restricted in such a way that they
don’t depend solely on the extension of the terms involved, i.e. on the set of
individuals falling under the terms involved. In this sense, the conclusion I
want to draw is that the semantics of Aristotle’s syllogistic is a
non-extensional semantics.
Jaime Edwards:
Could you say something about the significance of
Aristotle’s use of substance terms in the way that you’ve described?
Marko Malink:
This is also a difficult question, because as long as we
confine ourselves to what is called the assertoric syllogistic—which is the
syllogistic that is concerned with non-modal statements, like “every A is a B”
or “every C is a B”—as long as we are concerned with these non-modal
sentences, it is actually quite hard to see what the significance is of (what I
claim to be) these non-extensional features of Aristotle’s sentences.
After Aristotle has developed his assertoric, non-modal syllogistic, Aristotle also develops an even more complex system of modal syllogistic, where he is concerned with modally qualified sentences. So he not just considers sentences of the type “every B is an A”, but he also considers sentences of the type “every B necessarily is an A” or “every B possibly is an A” or “some Bs possibly are an A” or “no A necessarily is an A.” And so on. So he considers such modally quantified sentences, and he goes on to study inferences which consist of these modally qualified sentences. And when we turn to the modal syllogistic, these non-extensional features that depend on the distinction between substance terms and non-substance terms become more significant.
For example, one of Aristotle’s main claims in the modal syllogistic is that the following pattern of inference is valid: namely, if every C in fact is a B, and if every B necessarily is an A, then it follows from this that every C necessarily is an A. So although the one premise “every C is a B” is just an assertoric non-modal premise, the conclusion of this pattern of inference is a necessity proposition. So we can infer “every C necessarily is an A” from the true sentence “every C is in fact a B” and the modally qualified premise “every B necessarily is an A.”
Now, this is one of Aristotle’s main claims in the modal syllogistic, but this claim that this inference is valid was already denied by some of Aristotle’s pupils, in particular by Theophrastus. Theophrastus said: no, Aristotle, this pattern of inference can’t be valid, because I have a counterexample to it. And one of Theophrastus’ counterexamples is as follows: everything walking is a man, every man necessarily is an animal—therefore, if Aristotle’s inference were valid, we could infer that everything walking necessarily is an animal. By this counterexample, Theophrastus took himself to prove that Aristotle’s pattern of inference is invalid. But of course, what Theophrastus is doing in this counterexample is that he used a premise of the form “everything walking is a man”—because under some circumstances, it might happen that every individual who is walking turns out to be man, as Matt described before. So if the truth conditions even of his non-modal sentences were purely extensional—depending only on the set of individuals who fall under the terms—then this premise which Theophrastus uses in his counterexample could be true, in at least some contexts, and we would have very good counterexamples to some inferences which Aristotle claims to be valid in the modal syllogistic.
It is in cases like these, in the modal syllogistic, I think, that the claim that even the semantics of Aristotle’s assertoric (non-modal) sentences is not just purely extensional becomes significant, because it can help us to rule out certain counterexamples to inferences which Aristotle takes to be valid. And thereby, to understand why Aristotle took these inferences to be valid—these modal inferences in his modal syllogistic. The answer, coming back to your question, is that really, it’s not easy to see why my main claim should be true in the assertoric syllogistic. Because of course, the Fregean reconstruction yields a semantics for the assertoric syllogistic which is more or less in accordance with Aristotle’s claims in the assertoric syllogistic (except for this little, although well-known problem of existential import). So this is the only problem which Frege, or the Fregean way of thinking about Aristotle’s sentences, has when we confine ourselves to the assertoric syllogistic. And there are several ways of solving this problem of existential import.
So as long as we confine ourselves to assertoric syllogistic, it really looks as if Frege’s way of understanding Aristotle’s sentences is more or less okay, except for this problem of existential import. But when we turn to modal syllogistic, then it becomes more obvious that the truth conditions even of the assertoric sentences can’t be just extensional.
Matt Teichman:
Do you think this idea of Aristotle’s that claims like
“every A is a B” depend for their meaning on more than just whether each
individual A happens to be a B—do you think it has any consequences for the
way we think about possibility, or necessity, or any of these ideas?
Marko Malink:
I think it can have some such consequences. One of them
may be that once we specify the semantics of Aristotle’s assertoric and also
modal propositions in terms of his theory of predications—that is, is if we
base their truth conditions on Aristotle’s theory of predication, which in turn
is based on the distinction between substance terms and non-substance terms, and
which is also based on a distinction between essential predication and purely
accidental predication—once we do this, we might find a semantic
interpretation not only of his assertoric syllogistic but also of his modal
syllogistic, based only on this theory of predication, and the distinction of
essential predication versus accidental predication.
And if we succeed in having such an interpretation of Aristotle’s modal syllogistic, then in effect what we have is a semantics for a certain modal logic, namely Aristotle’s modal logic (i.e. Aristotle’s’ modal syllogistic), which does not appeal to anything like possible worlds semantics, but which just appeals to Aristotle’s theory of predication, based on these distinctions which we mentioned. So, this could help us to see how at least in Aristotle’s case, we can do modal logic without appealing to possible worlds, even if this modal logic is very different from the kind of modal logic which we find in the 20th century. Aristotle’s modal syllogistic obviously is very different from systems of modal logic which we find in the 20th century—but still, his modal syllogistic, on this interpretation, could help us to see how we can do modal logic without using the framework of possible worlds semantics.
Matt Teichman:
One very influential idea in contemporary logic has been
that whenever you say such and such is necessarily the case, what you’re saying
is that in every possible world, such and such is the case. And this way of
thinking about what it is for something to be necessarily the case has been
really influential and widespread. But maybe one of the lessons we can draw
from Aristotle’s modal syllogistic is that there’s other ways of defining what
it is for something to be possibly the case versus necessarily the
case—besides in terms of what’s the case in alternate worlds other than the
actual world that we live in now.
Marko Malink:
That’s right, and again, his theory of predication can
help us to do this. For example, in the
Topics, he develops a theory
of predication which is based on relations such as being a genus of, being a
differentia of, being a definition of, being an accident of, and so on. Some of
these relations of predication Aristotle regards as essential predication, where
the predicate specifies either the whole essence or part of the essence of the
subject. So in a predication like “man is an animal,” obviously “animal” stands
for a genus of the species “man.” So in this sense it is an essential
predication. Or to use one of Aristotle’s stock examples, by saying that man is
a biped, Aristotle regards biped as a specific differentia of the species
man, and in this sense, an essential feature of the species man. We could
then try to tie the semantics of Aristotle’s modal propositions—i.e. of
propositions of the form “every B necessarily is A”—we could try to tie the
truth conditions of these sentences to the relations of essential predication.
We can then use the claims which Aristotle makes in the
Topics (and in several other
writings) about the relation of essential predication to justify the claims
which Aristotle makes in his modal syllogistic.
Matt Teichman:
Marko Malink, thank you very much for joining us.
Marko Malink:
Thanks for having me here.
Elucidations isn't set up for blog comments currently, but if you have any thoughts or questions, please feel free to reach out on Twitter!