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Gettier is widely considered to have falsified justified true belief accounts of knowledge. In this paper, I provide clear proofs that show that the subjects of Gettier’s cases cannot believe what Gettier claims they believe. To do this, I clarify how we are to characterize belief using propositions and the tools of truth functional logic. Furthermore, no alternative characterization would seem to be forthcoming. Because of this, Gettier’s cases entirely fail against justified true belief accounts of knowledge, despite what has been widely believed for over a half a century.
Gettier is widely considered to have provided counterexamples to Justified True Belief (JTB) accounts of knowledge, those accounts that claim, or similarly claim, that a subject S knows a proposition P if:
(i) P is true
(ii) S believes P
(iii) S is justified in believing P
Such an account of knowledge, as Gettier himself notes, has origins going at least back to Plato, and, as he also notes, is similarly expressed by Ayer and Chisholm. Gettier provides two cases against JTB accounts of knowledge. Case II relies on the subject forming a disjunctive belief (i.e. of the form: P or Q) and Case I relies on the subject forming an existential belief (i.e. of the form: there is at least one x such that x is/does something .…). The main conclusion of this paper that I will argue is that the specific beliefs that Gettier requires the subjects to believe in order for his cases to work as counterexamples to JTB theories cannot be held by those subjects. Indeed, his subjects would explicitly reject them. With this, JTB theories have been wrongfully maligned as defeated in the half century and two decades since Gettier first proposed his challenge. JTB theories emerge from Gettier’s shadow. We will get to the specifics of Gettier’s cases after we examine the central consideration of this paper: how to logically characterize beliefs. I will develop a view of characterizing beliefs according to the truth conditions of the propositions in question. The acceptance, rejection, or lack thereof of such propositions is to be the basis of the characterization of belief using such propositions.
Characterizing Beliefs: Propositionally, Truth Functionally, Logically
I take it that if Sam believes it is raining, then it is entirely pedestrian to translate such a state of affairs as Sam believes R, where R is the proposition1 that it is raining. Moreover, a proposition is either true or false.2
I find it convenient, additionally, to make use of a slightly different phrase from “S believes P”, which is meant to be synonymous with it, but is more clear about what is believed about P. When we take it that S believes P, we take it that S accepts that P is true. Here, ‘believes’ and ‘accepts’ are wholly synonymous, but ‘accepts’ contrasts nicely with ‘rejects’, which we will also make convenient use of; furthermore, the addition of ‘is true’ is meaningful: if S accepts rather that P is false, we would not say that S believes P. Indeed, we should rather say that S rejects P. And so, rejecting P just amounts to accepting that P is false. This nicely allows us to speak of believing P or rejecting P while using the minimalistic term ‘accept’ (particularly since we are already making use of the notion of truth in the theory of knowledge we are examining). And so, in what follows, unless specified otherwise, “S accepts P” will mean the same as “S accepts P as true”, and “S rejects P” will mean the same as “S accepts P as false”.3 Note that for any proposition P, Sam need not accept or reject it. He may not have any such stance about it at all. He might be open to accepting that it is raining, for example, but has not yet seen any indication or gotten any information about it one way or another. Thus, he neither accepts nor rejects that it is raining. Given that Sam is a normal functioning adult, we should readily conclude that, given more information (perhaps he looks out the window and checks), Sam may readily come to accept as true that it is raining, or that it is not raining, as the case may be. We might say that Sam is open about there being rain.
The next level of sophistication for characterizing what Sam believes includes adding in the logical operators. That Sam believes that it is raining and that he is at home may readily be translated as “Sam believes R&H”, where R is the proposition that it is raining and H is the proposition that he is at home, and ‘&’ is the logical operator ‘and’. A logical operator just combines two truth functional items (i.e. things that can be either true or false—propositions, conditions, or ideas, as we mentioned) into one truth functional item, which will be true or false according to the truth values of its parts. Different logical operators follow different rules about what the larger truth value is given the truth values of its parts. For our example, the truth value of a proposition with the logical connective ‘&’ (and) is true if both parts are true, otherwise it is false. Intuitively this makes perfect sense: Sam’s belief that it is raining and he is at home is true if it is raining and Sam is at home, and is false otherwise. If just one part of his belief, either that it is raining or that he is at home, is false, then his entire belief is false.
Suppose that we are considering whether or not to characterize Sam’s beliefs with a conjunction. What we would need to consider is whether or not Sam accepts each conjunct. Moreover, if Sam rejects a conjunct, then this would be proof positive that characterizing Sam’s belief using such a conjunction is incorrect. Indeed, Sam even rejects the conjunction. That is, even if he accepts the other conjunct as true, still, he is rejecting one conjunct, and so is rejecting the conjunction. For our case, if Sam accepts the truth of R&H, then he accepts R as true and he accepts H as true. In such a case, characterizing Sam’s state as Sam believes R&H is entirely appropriate. If it turned out that Sam did not accept the truth of either R or H, however, it would be incorrect to characterize Sam’s state as Sam believes R&H. If Sam rejects that he is at home, because he is at the hardware store and he is not insane, say, then he no longer believes R&H.
And similar reasoning will apply to all of the logical operators. More generally, if Sam rejects a truth assignment that makes a sentence containing logical operators true, then Sam does not accept the truth of the very same sentence containing those logical operators. Let me show this with more logical operators, from which all of the others may be derived. Beliefs involving other logical connectives may seem trickier on the surface, but they follow the same underlying pattern. Let’s say that Sam believes that nearby rain always necessitates nearby clouds. That is, if it is ever raining nearby, then there are clouds nearby. A straightforward way to characterize Sam’s belief on this matter is: if R then C (Symbolically: R→C, where C is additionally the proposition that there are clouds nearby). This characterization suits our purpose because Sam does not have to believe that it is raining in order to believe that given rain, there are clouds. Nor does he have to believe it is not raining. This part is irrelevant. What he cannot accept as true, however, is that it is raining and that there are no clouds. This would be a clear counterexample to his belief, which explicitly is that nearby rain always has nearby clouds.
Conditionals are logically equivalent to disjunctions in the following way: P→Q is logically equivalent to ~PvQ (where P and Q stand for names of arbitrary propositions). Being logically equivalent, it is unsurprising that what we take Sam to accept or not accept lines up perfectly between them: he may or may not believe it is raining (again, perhaps he believes neither if he hasn’t looked outside), and, if he does, then he also believes there are clouds. And the counterexample to either the conditional characterization or the disjunctive characterization of Sam’s belief is that he believes that it is raining and that there are no clouds.
Let us say Sam looks outside, sees rain, and so now accepts that it is raining. Being a sane adult, he also rejects that it is not raining. Due to this, it would be woefully incomplete to characterize Sam’s beliefs as we did when he was open to these possibilities, as ~RvC. This is because ~RvC has truth assignments that make ~RvC true, but which Sam now explicitly rejects: namely the ones where it is true that it is not raining. This point seems obvious, but is the first pitfall of Gettier (Case II), so let me explain more. If we characterize Sam’s belief as ~RvC simpliciter and give this information to an outsider (who also knows how to read predicate logic), then what would one conclude about Sam’s beliefs in this case? Let’s say the outsider has just the following information:
Report on Sam’s Belief About Rain and Clouds
R = “It is raining (nearby).”
C = “There are clouds (nearby).”
Given just this information, one has enough just to say that Sam rejects that there is rain without clouds. Besides that, Sam can be open to believing any other possibilities: he can believe that it is raining, and he can believe that it is not raining, and he can believe that there are clouds, and he can believe that there are no clouds (provided he does not also believe it is raining, as we said). This obviously gets Sam’s current beliefs wrong—Sam rejects that it is not raining, and Sam rejects that there are no clouds. To leave Sam’s beliefs at ~RvC is just to mischaracterize what he believes.4 Sam explicitly rejects truth assignments of the atomic sentences that would make the full sentence ~RvC true; rejecting some of its truth preserving assignments, Sam therefore does not believe just ~RvC. This is not an apt characterization of what he believes.5
Let us next consider how to characterize Sam’s beliefs when he believes something that may be translated using the existential quantifier. Let’s say that Sam believes there is some person who owes him money. He has a vague recollection of going to dinner with his friends and having a few drinks. He picked up one of his friend’s tabs, and now he needs money, but who was it? Someone owes him money. Thus, we may characterize his belief with the help of quantification: there is some x such that x owes Sam money. Now, in first order logic and similar systems, there is a domain of discourse, or the set of the things that are under discussion, and which get named by the constants (which are usually single lower-case letters) of the language. Let’s stipulate that the domain of discourse is simply Sam’s five friends who accompanied him last night. How existential sentences are true is that there is at least one constant that names an object of the domain of discourse and that, when substituted with all instances of the variable of the existential quantifier (above, this is the part of the phrase that is ‘There is some x such that’) in the sentence (e.g., x), and the existential quantifier itself removed, results in a true sentence. Finally, such a substitution furthermore results in a true sentence when any predicate of the sentence with that substituted constant is assigned by a truth assignment in such a way that the entire sentence turns out true, according to any further quantifiers or logical connectives. Conversely, such a sentence is false when there is no such constant that results in a true sentence when substituting the variable as just described.6
For example, if one substitution (or more) of the variable x with a constant that names one of Sam’s friends in the sentence “x owes Sam money” results in a true sentence, then the proposition “someone owes Sam money” is true. Furthermore, such a proposition aptly characterizes Sam’s belief. If any truth assignment for the predicate ‘x owes Sam money’ makes the existential sentence true, then of course Sam is open to this. Perhaps they all owe him money, because he picked up the tab that included them all. Let’s say Sam calls his friends and finds out that Redacted owes him money from last night. To make things simple, Redacted also confirms no one else owes him money. What is Sam’s belief on the matter? Say that a report is created regarding Sam’s beliefs about who owes him money from last night. It reads:
Report on Sam’s Belief About Who Owes Him Money From Last Night
Sam believes: ∃x Osx
The domain of discourse is Sam’s five friends from last night,
‘Osϕ’ is the predicate ‘ϕ owes Sam money’, where ϕ stands for the name of some term of our language,
’s’ names Sam, and
∃x is the existential quantifier, more colloquially: “There is an x such that . . . .”
Assume that the one who reads it knows first order logic. What would one conclude about Sam’s belief regarding this matter? Easily enough, as we just covered, one would only gather that Sam believes that one of his friends (or possibly some combination of them) owes him money. If it turned out one or the other or some combination of them owed him money, Sam is perfectly open to accepting this as far as we have been told.
However, Sam believes no such thing currently, after calling and speaking with Redacted. Indeed he believes just one truth assignment of the predicate Osϕ, the one which returns true just for the substitution of ϕ with the constant that names Redacted in the original sentence, and false for any other substitution. Since Sam explicitly now rejects most truth assignments of Osϕ that would make the original existential sentence true, characterizing Sam’s belief as just that existential sentence is incorrect. Let us now treat these results with Gettier’s cases more specifically.
Let us now look at Gettier’s cases and supplement them with a more formal proof with the same result as what has been covered. Let us also examine Case II first, since it makes use of disjunctive beliefs whereas Case I makes use of existential beliefs, following the order we used above. The main reason for this is just that predicate logic (the logic of just the logical connectives) is simpler than first order logic (predicate logic and the quantifiers).
To brief, Smith believes that Jones owns a ford. Smith is also justified in having that belief. Furthermore, he deduces that the truth of Jones owning a Ford implies the proposition:
(a) Either Jones owns a Ford, or Brown is in Barcelona.
He picked that latter disjunct quite at random and has no idea where Brown is. Unknown to Smith, Brown is in Barcelona. Also, Jones does not own a Ford. Jones instead uses a rental, against all the previous times he has owned the Ford he drives. Thus, Smith clearly does not know (a), but, Gettier claims, Smith believes (a), Smith is justified in holding (a), and (a) is true. Thus, JTB theories are incorrect because they do not provide sufficient conditions for knowledge. Obviously, we should grant that (a) is true; furthermore, if it were that Smith could believe (a) then he would be justified, since he just follows a logical entailment. However, Smith clearly does not believe just (a). In particular, Smith rejects certain truth assignments that would make (a) true: namely the ones where “Jones owns a Ford” is false. Indeed, Smith can deduce his acceptance of the truth of the negation that Jones owns a Ford and Brown is in Barcelona:
F (Jones owns a Ford)
~F&B (Suppose Jones doesn’t own a Ford, and that Brown is in Barcelona)
~F (Conjunction Elimination on the above supposition)
~~F (Double Negation on F)
⊥ (Absurdity: contradiction of ~F and ~~F)
~(~F&B) (Therefore, negate the supposition that led to the absurdity)
In order to claim that Smith believes (a), Gettier must have some alternative method for characterizing Smith’s beliefs from the one that I have provided, but he makes no mention of it, and there is no alternative that I can tell that would be forthcoming. Without such an account, we have no way of justifying Gettier’s claim that Smith believes (a). But besides this, it would seem perfectly clear that Smith could not believe (a). In particular, Smith rejects truth assignments of (a) that would allow it to be true. This is not to say that Smith misses his deduction: to characterize Smith’s belief as (a) is, besides being incorrect, just incomplete. Smith doesn’t believe (a). He believes, more specifically, F&(a). Gettier’s claim about Smith’s belief here makes the mistake of effectively giving Smith amnesia about the belief he uses to derive (a) at all. If we suppose that Smith indeed does have amnesia about his starting belief, then Smith loses any justification he has for holding it. Gettier has therefore completely failed to provide a counterexample to JTB theories with Case II.
Case I is dealt with similarly as Case II. It may even be left as an exercise to the reader. Smith would not believe what Gettier claims he does, because Smith explicitly rejects a particular truth assignment that leads to that belief being true.
In brief, there is a job that Smith and Jones are applying for and Smith believes that Jones is the man who will get the job, and Jones has ten coins in his pocket. Smith is justified in this belief because he heard about Jones getting the job from the boss, and he grabbed some coins, counted them, and gave them to Jones, who put it in his pocket. Smith did this because Jones said he needed change (This is a slight modification, but I found Gettier’s version suspect—who counts the coins of another person’s pocket!?). From the proposition that Jones is the man who will get the job, and Jones has ten coins in his pocket, Smith deduces the following proposition:
(b) The man who will get the job has ten coins in his pocket.
However, unknown to Smith, he himself will get the job; also unknown to Smith, he himself has ten coins in his pocket. Thus, Smith clearly does not know (b), but, Gettier claims, Smith believes (b), Smith is justified in holding (b), and (b) is true. Thus, JTB theories are incorrect because they do not provide sufficient conditions for knowledge. Obviously, we should grant that (b) is true; furthermore, if it were that Smith could believe (b) then he would be justified, since he just follows a logical entailment. However, Smith clearly does not believe (b). In particular, Smith rejects certain truth assignments that would make (b) true: namely those truth assignments where any constant that names anything other than Jones in the domain of discourse and that is substituted for the existential variable used by the existential quantifier and the quantifier removed is true. Indeed Smith can explicitly deduce his acceptance of the truth of the negation that someone else, say Smith himself, gets the job and has ten coins in his pocket:
Where ‘j’ names Jones, and Jϕ and Tϕ are the predicates concerning getting the job and having ten coins in a pocket:
(Jj & Tj) (Jones gets the job and has ten coins in his pocket)
∃x (Jx & Tx) (Existential Elimination from previous line)
(Js & Ts) (Suppose this is true to derive an absurdity)
~(j = s) (Obvious additional belief Smith has, that he is not identical to Jones)
∀x (Jx) → [∀y(Jy) → x = y] (Additional belief that Smith has, made explicit, that just one person will get the job)
Jj & Tj → (Js & Ts → (j = s)) (Universal Instantiation on previous line, where j replaces x and s replaces y)
Js & Ts → (j = s) (Modus Ponens on previous line)
(j = s) (Modus Ponens on previous line)
⊥ (Absurdity: Contradiction of ~(j = s) and (j = s) above)
~(Js & Ts) (Therefore, negate the supposition that lead to the absurdity)
In order to claim that Smith believes (b), Gettier must have some alternative method for characterizing Smith’s beliefs from the one that I have provided, but he makes no mention of it, and there is no alternative that I can tell that would be forthcoming. Without such an account, we have no way of justifying Gettier’s claim that Smith believes (b). But besides this, it would seem perfectly clear that Smith just could not believe (b), since he rejects all except for one of the truth assignments of (b) that would allow it to be true. This is not to say that Smith misses his deductions: to characterize Smith’s belief as (b) is, besides being incorrect, just incomplete. Smith doesn’t believe (b). He believes, more specifically, (Jj & Tj)&(b). Gettier’s claim about Smith’s belief makes the mistake of effectively giving Smith amnesia about the belief he uses to derive (b) at all. If we suppose that Smith indeed does have amnesia about his starting belief, then Smith loses any justification he has for holding it. Gettier therefore has completely failed to provide a counterexample to JTB theories with Case I.
Gettier has been taken by philosophers of the last half century and longer to have provided counterexamples to JTB theories of knowledge. I believe I have shown thoroughly and conclusively that such an assessment is entirely false. JTB theories have no cases against them as Gettier presented.
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Or perhaps the condition or the idea—it makes no difference for our purposes. I will continue using the term ‘proposition’, but these others could be substituted to the same effect ↩︎
Similarly ’true’ might be meant pragmatically, deflationarily, according to Aristotle’s correspondence theory, or according to some other plausible view. For our purposes, these specifics should not matter. ↩︎
Again, I am being explicit in my use of the terms ‘accept’ and ‘reject’ in this way for convenience in what follows. One should be able to substitute ‘accepts’ with ‘believes’ and ‘rejects’ with ‘disbelieves’ for the same result in what follows. ↩︎
An easy remedy, of course, is just to add more logical operators until there is no truth assignment concerning Sam’s beliefs about rain and clouds that Sam rejects. R & (~RvC) works fine enough. In general accomplishing this amounts to merely making explicit what Sam believes right now—that clouds always accompany rain, Yes, but also that it is raining. ↩︎
A logically inclined reader may here wonder whether or not Sam can believe in a basic logical truths, such as, say, the law of non-contradiction (or one of its many equivalencies): ~(R&~R). Yes he can: If he is open to its raining or not raining, then we may characterize his belief (or at least this relevant part) just as ~(R&~R). And if he has a stance one way or another concerning proposition R then just add the stance with ‘&’, e.g., Sam believes ~((R&~R) & R). ↩︎
There are of course varying levels of sophistication in determining how first order sentences are true, but they are all logically equivalent. For more, the interested reader may seek out Computability and Logic, 3rd edition, chapter 9, or Language Proof and Logic, 2nd Ed. chapter 12. Alternatively, there is a strong chance that this information may be found in any other textbook on First Order logic that is in wide circulation and published in the last 40 years. ↩︎