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Epistemic Logic |
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Introduction |
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Each state of knowledge of a person can be thought of as a possible world. So we can make up a modal logics to regularise our talk about these states. These logics replace the [] operator with the operator K which is also indexed to point to the particular person who is doing the knowing. So
Kap translates 'a knows that p'
We'll be looking at Hintikka's epistemic logic, which is is an adaptation of S4.
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S4 Knowledge |
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Let a, b, c, d, ... index the persons involved Ka, Kb, Kc, Kd, ... are the knowledge operators Pa, Pb, Pc, Pd, ... are defined as Px =: ~Kx~
Semantic Definition
Ea, Eb, Ec, Ed, ... are the epistemic accessibility relations, which are also different for each person
The S4 rules are now given the obvious modifications:
KPN ~KxX w/ ~PxX w/ ... ... Px~X w Kx~X w
PR PxX w/v ... wExv X v where v is new to this path
KR KxX w\v wExv ... X v
KTR KxX w\ ... X w
KKR KxX w\v wExv ... KxX v
and Hintikka also adds the 'Transmissibility of Knowledge Rule'
TrKR KxKyX w ... KxX w
Axiomatic Definition
To the standard PS axiom schemas and the rule of MP we add:
EN: |- A Þ |- KxA (Epistemic Necessitation)
K1: Kx(A É B) É (KxA É KxB) (Distribution) K2: (KxA É A) (Veridicality) K3: (KxA É KxKxA) (KK-Thesis or positive introspection) K4: KxKyA É KxA (Transmissibility)
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Interpreting Epistemic Logics |
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The reason I've included the axiomatic rules for the S4 version of epistemic logic is that it makes even clearer than the semantic rules some of the oddities of the logic. For example, the rule EN seems most counterintuitive - it doesn't seem to be true of any person but God. According to that rule, if X is a theorem of the logic then one knows that X. The claim that one knows all theorems is called the Strong Logical Omniscience Thesis (SLOT). Similarly odd is the rule K1 of distribution. According to this rule one knows all the logical consequences of everything one knows. This claim is the Deductive Omniscience Thesis (DOT.) There is also debate about whether K3 is believable. It may remind you of Rumsfeld's claim that there are known knowns. He thought it was uncontroversial that there are things that you know and that you know that you know. According to K3 everything you know is like that; which is to say that the category of unknown knowns is empty. Those who quibble about this generally point to instances where we suddenly realise or admit to ourselves something that we really knew all along, like not liking French films for example. But such examples look like fallacies of equivocation, because two kinds of knowledge are both being referred to using Kx. Rumsfeld also thought there were known unknowns and unknown unknowns, that is, things that you know that you don't know and things that you don't know that you don't know. There has been controversy in the modal logic crowd about whether there really are things that you don't know that you don't know.
These oddities and perplexities are addressed in a couple of different ways. Lemmon takes the logic to be concerned with the knowledge of ideal epistemic agents. (God would be one of those I'd guess; and so would the 'rational man' of rational choice theory and classical economics.) In this case the nature of the 'knowledge' is not in question. On the other hand, Hintikka prefers to make the oddities seem less odd by modifying our understanding of the 'knowledge' that the logic is about.
The Ideal Reasoner
It turns out, of course, that if we're prepared to say that epistemic logic is that which applies to the ideal knower then we might as well go the whole hog and make it an S5 logic. We add the axiom
K5: (PxA É KxPxA)
and it is provable in that system that
T1: (~A É Kx~KxA) (Platonic Principle) T2: (~KxA É Kx~KxA) (Negative introspection)
It's pretty clear that no real epistemic agent satisfies is described by these theses, although a normative one might be - depending on what the norm is supposed to judge.
Suppose we wanted to have a system that could credibly describe an epistemic agent like ourselves: what would we need to change? It's hard to think of any changes that would do the trick. As long as we have the rule of epistemic necessitation or the distribution axiom we're likely to have problems with omniscience of one form or another. One solution is to remove that rule, or to replace it with a Weak Necessitation Rule:
WN: |-PL A Þ |- KxA
According to this rule, anything which is a tautology in PL is known to X, which is still pretty unbelievable. However, if we remove all the modal axioms (including Distribution,) then we get the agent being 'logically' omniscient, but not deductively omniscient - the agent knows all the truths of classical propositional logic.
Interpreting 'Knowing'
According to Hintikka, all the oddities are much less odd if we understand KxA to mean something like 'it is knowable by x that A,' and by 'knowable' he means that its negation is indefensible, and by 'indefensible' he means that is logically inconsistent with other things known to x. If that's what the K operator means then we have no real problem with the omniscience claims - given a reasonably rational agent who is able to follow logical deductions. Of course, this means we're no longer talking about an agent 'knowing' things, rather we're looking at an agent's 'commitments.'
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| Further Reading | |
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E. J. Lemmon, G. P. Henderson (1959) 'Is There Only One Correct System of Modal Logic?' Proceedings of the Aristotelian Society: Supplementary Volumes 33 23-56
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