Gettier cases and logic
[This is an extract from the
penultimate draft of ‘Logical Knowledge and Gettier
Cases’ by Corine Besson.]
Suppose that Nate wants to learn the
rules of logic. And suppose further that he does not know some logical
vocabulary, for instance he does not know the word ‘if’ and does not have the
concept of material implication. (If there are worries about how Nate could reason
at all without knowing the word ‘if’ or having the concept of material
implication, we can for instance suppose that, oddly, he got by with
disjunctive syllogisms.) Brenda, who is a renowned expert in logic, agrees to
teach him. Given that Nate does not know the word ‘if’, she intends to teach
him first the rules for ‘if’ – that is to say the rules of Conditional Proof
and Modus Ponens. (I assume here that ‘if’ is truth-functional, which is a
standard thing to do in discussion concerning the nature of knowledge of the
basic rules of logic.) As Brenda and Nate are lost on a desert island, the
teaching is done orally; and to teach him, she plans on telling him the two
rules and perhaps give him a few of their instances.
She starts with the rule of
Conditional Proof (CP), which says that:
(CP) From an inference that Q from the
assumption that P, one can infer that if P,
then Q.
The teaching goes well with (CP),
which Nate understands just fine. But then, when she turns to the rule of Modus
Ponens, Brenda gets tired and irritated. She decides to trick Nate and sets
about teaching him a fallacy. She will not teach him the correct rule of Modus
Ponens (MP), which says that:
(MP) From if P, then Q, and P, one may
infer that Q.
An instance of (MP) is for example:
(MPday) From
if it is day, then it is light, and it is day, one may infer that it is
light.
Rather than (MP), Brenda decides to
teach Nate the incorrect rule known as ‘the fallacy of asserting the
consequent’ (AC), which says that:
(AC) From if P, then Q, and Q, one may
infer that P.
An instance of (AC) is:
(ACday) From
if it is day, then it is light, and it is light, one may infer that it is
day.
Now, suppose that by sheer
coincidence, just at the time at which Brenda is supposed to utter (MP), or one
of its instances, there is a whirlwind of sorts between her mouth and Nate’s
ear, which they both fail to detect. The effect of this whirlwind is that
although Brenda utters the rule (AC), Nate hears the rule (MP). That is to say,
although she utters the second occurrence of Q as the second conjoint of the
main antecedent, and the second occurrence of P as the main consequent, as she
gives samples of the incorrect rule (AC), Nate systematically hears them the
other way round; i.e., he hears the second occurrence of P as the second
conjoint of the main antecedent, and the second occurrence of Q as the main
consequent, and thus he hears samples of the correct rule (MP). For instance, although
she utters the second occurrence of ‘it is light’ before uttering the second occurrence
‘it is day’, in trying to convey (ACday), Nate hears
‘it is day’ before hearing ‘it is light’ and thus hears (MPday).
Intuitively, Nate acquires the
justified true belief that from if P, then Q and P, one may infer that Q – that
is to say, he acquires a justified true belief of rule (MP). But he does not
know it. For the process through which he acquired this belief was not
reliable. Nate was just lucky that the atmospheric conditions happened to be
capricious in a favourable way at the time Brenda tried to deceive him; if this
had not happened, he would not have acquired a justified true belief. If that
is so, this example shows that one can have a justified true belief of a basic
rule of logic and fail to know it, i.e. that knowledge of a basic rule of logic
can be gettiered.
Reference
Besson, C. 2009. Logical
Knowledge and Gettier Cases. The Philosophical Quarterly 59:1-19.