on Aristotle's Posterior Analytics 73b10-17.

“Again, in another way what holds of something because of itself in itself, and what does not hold because of itself is incidental. E.g. if there was lighting while he was walking, that was incidental: it was not because of his walking that there was lightning—that, we say, is incidental. But what holds of itself holds in itself—e.g. if something dies while it is being sacrificed, it died in the sacrifice since it died because of the sacrifice, and it was not incidental that it died while being sacrificed.”

Aristotle, Po. An. 73b10-17, (trans.  J. Barnes), emphasis in trans.

Let ‘W’ abbreviate “William is walking” and ‘L’ abbreviate “There is lighting”. Now, suppose that

W • L

It follows that

W ⊃ L

That is, from the fact that William is walking and there is lighting, it follows that if William is walking then there is lighting. But, Aristotle seems to deny this, for he says that, “it was not because of his walking that there was lighting”. Thus, Aristotle might be thought of asserting:

~(W ⊃ L)

But this would be mistake, for ~(W ⊃ L) implies

W • ~L

in which case Aristotle would be having us suppose, at the same time, that

L • ~L

Since Aristotle isn’t having us suppose a contradiction, how shall we represent what he’s saying? Here’s one option. When he says that “it was not because of his walking that there was lighting”, he isn’t denying W ⊃ L, but rather that W ⊃ L is necessary—viz.

~□(W ⊃ L)

So we might put it like this: Aristotle would agree that statements of the form P • Q imply statements of the form P ⊃ Q but not □(P ⊃ Q).

On the other hand, there are some cases in which statements of the form □(P ⊃ Q) are true. Aristotle’s example is that of a thing dying because it was sacrificed. Let ‘S’ and ‘D’ abbreviate “William is sacrificed” and  “William died”, respectively. Though Aristotle denies □(W ⊃ L), he affirms

□(S ⊃ D)

Since a statement is necessary iff its negation is impossible, □(S ⊃ D) implies

~◇~(S ⊃ D)

which, via material implication, De Morgan’s law, and double negation, implies

~◇(S • ~D)

On the other hand, since a statement is not necessary iff its negation is possible, ~□(W ⊃ L) implies

◇~(W ⊃ L)

Which, via material implication, De Morgan’s law, and double negation, implies

 ◇(W • ~L)

on epistemology as a subdiscipline of metaphysics.

My epistemology professor asks us to write down our thoughts at the end of the lecture, mostly, I think, as an efficient way for him to take attendance. One time I remember writing something like: “it seems to me that epistemology is just the metaphysics of knowledge.”

Apparently, Gustav Bergmann has similar sentiments: “Epistemology is merely the ontology of the knowing situation.” (from his “Ontological Alternatives”)

on whether modus ponens is located somewhere.

Literally, the marks that [the logician] makes on paper are just as much a part of the world as is [ordinary language]. Logically, to vary the metaphor, these marks are nowhere.

Gustav Bergmann, “Strawson’s Ontology”

Plato's Gorgias in two sentences.

Though it’s true that some of the just suffer pain and some of the wicked experience pleasure, the just always prosper and wicked never do. This is because one prospers if and only if one is just, and therefore, one fails to prosper if and only if one is wicked.

when knowledge de dicto implies knowledge de re.

Suppose that Angela believes that Santa Clause exists. One way to think of Angela’s belief is in terms of a relationship between Angela and a certain proposition, as in the following expression:  

B_a((∃x)x = Santa Clause))

i.e.—Angela believes that there is something such that it is Santa Clause.

Another way to think of Angela’s belief is in terms of a relationship between Angela and a thing, as in:  

(∃x)B_a(x = Santa Clause)

i.e. There is something such that Angela believes that it is Santa Clause.

Notice that first expression implies the existence of Angela but nothing else. The second expression implies the existence of Angela and as well as the thing which Angela believes to be Santa Clause. (Even if there is no such thing as Santa Clause, the second expression asserts that there is at least one thing which bears the relation of being believed to be Santa Clause by Angela).

Notice that this difference between these expressions disappears if the relation in question is not belief but knowledge. Consider:

K_a((∃x)x = Santa Clause))

i.e.—Angela knows that there is something such that it is Santa Clause. And

(∃x)K_a(x = Santa Clause)

i.e.—There is something such Angela knows that it is Santa Clause.

Regarding the latter expression: If there is something such that Angela knows that it is Santa Clause, then both Angela and Santa Clause exist. And the same goes for the latter expression: Angela knows that there is something such that it is Santa Clause only if Angela exists and her  belief is true, and her belief that there is something such that it is Santa Clause is true only if—you guessed it—there is something such that is Santa Clause. Generalizing, this latter point underwrites the following inference rule of epistemic logic:

          K_m((∃x)(φx))

    ∴ (∃x)(φx)

thank you, Mr. Russell.

“In the motions of mutually gravitating bodies, there is nothing that can be called a cause, and nothing that can be called an effect; there is merely a formula.”

Bertrand Russell, “On the notion of cause”

Ergo, there is no incompatibility between the “laws of physics” and God turning water to wine, raising the dead, allowing a donkey to speak, and the like.