Modal logic, knowledge, and an old joke

Recently, I posted about necessary truth and about the logic of belief. I would like to follow up on that by discussing epistemic logic, the logic of knowledge. Knowledge is traditionally understood as justified true belief (more on that below), and we can capture the concept of knowledge using the 4 rules of S4 modal logic. These are in fact the same 4 rules that I used for for necessary truth, and the first 3 rules are the same as those I used for belief (the fourth rule adds the fact that knowledge is true).

Knowledge is specific to some person, and I am replacing the previous modal operators with  Ⓚ  which is intended to be read as “John knows” (hence the K in the circle):

• if P is any tautology, then  Ⓚ P
• if  Ⓚ P  and  Ⓚ (P ⇒ Q)  then  Ⓚ Q
• if  Ⓚ P  then  Ⓚ Ⓚ P
• if  Ⓚ P  then  P

For those who prefer words rather than symbols:

• if P is any tautology, then John knows P
• if John knows both P and (P implies Q), then John knows Q
• if John knows P, then John knows that he knows P
• if John knows P, then P is true

Epistemic logic is useful for reasoning about, among other things, electronic commerce (see this paper of mine from 2000). How does a bank know that an account-holder is authorising a given transaction? Especially if deceptive fraudsters are around? Epistemic logic can highlight which of the bank’s decisions are truly justified. For this application, the first rule (which implies knowing all of mathematics) actually works, because both the bank’s computer and the account-holder’s device can do quite sophisticated arithmetic, and hence know all the mathematical facts relevant to the transaction they are engaged in.

But let’s get back to the idea of knowledge being justified true belief. In his Theaetetus, Plato has Theaetetus suggest exactly that:

“Oh yes, I remember now, Socrates, having heard someone make the distinction, but I had forgotten it. He said that knowledge was true opinion accompanied by reason [ἔφη δὲ τὴν μὲν μετὰ λόγου], but that unreasoning true opinion was outside of the sphere of knowledge; and matters of which there is not a rational explanation are unknowable – yes, that is what he called them – and those of which there is are knowable.” (Theaetetus, 201c)

Although he also uses essentially this same definition in other dialogues, Plato goes on to show that it isn’t entirely clear what kind of “justification” or “reason” is necessary to have true knowledge. In a brief 1963 paper entitled “Is Justified True Belief Knowledge?,” the philosopher Edmund Gettier famously took issue with the whole concept of justified true belief, and provided what seemed to be counterexamples.

My personal opinion, which I have argued elsewhere, is that “justified true belief” works fine as a definition of knowledge, as long as the justification is rigorous enough to exclude beliefs which are “accidentally correct.” For analysing things like electronic commerce, a sufficient level of rigour would involve the use of epistemic logic, as described above.

One of Gettier’s supposed counterexamples involves a proposition of the form  P ∨ Q  (P or Q) such that:

• Smith believes and knows  P ⇒ (P ∨ Q)
• Smith believes P
• P is false
• Q is true, and therefore so is  P ∨ Q

From these propositions we can use doxastic logic to infer that Smith believes the true statement  P ∨ Q,  but we cannot infer (using epistemic logic) that Smith knows  P ∨ Q. A famous old joke is perhaps relevant here:

“A physicist, a philosopher, and a mathematician are travelling through Scotland by train. Through the window, they observe a black sheep in a field. ‘Aha,’ says the physicist, ‘I see that Scottish sheep are black!’ The philosopher responds, ‘No! Some Scottish sheep are black!’ The mathematician, looking shocked, replies: ‘What are you guys saying? All we know is that at least one sheep in Scotland is black on at least one side.’”

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In this post series: logic of necessary truth, logic of belief, logic of knowledge, logic of obligation

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