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,
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
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
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.
Besson, C. 2009. Logical Knowledge and Gettier Cases. The Philosophical Quarterly 59:1-19.