Knowing without Knowledge

Knowing Without Knowledge We suppose first, then, that the wise man knows all things, as far as possible, although he has not knowledge of each of them individually; […] (Aristotle, Metaphysics, Book I (A), Part 2, 982a8) Aristotle Metaphysics. Also available online at http://files.vsociety.net/data/library/Section%201%20(A,G,M,S,Z)/Aristotle/Unknown%20Album/Metaphysics.pdf (Downloaded on 25 July, 2011). See last paragraph of page 3 of document. The title may sound paradoxical. However, the title emphasises a distinction which is often overlooked in epistemology. The result of conflating ‘knowing’ with ‘knowledge’ is a host of confusions about what is going on in epistemology. Grammatically, ‘knowledge’ is a noun whereas ‘to know’ is a verb. Nouns designate objects of some sort, hence knowledge is an object, whereas verbs designate relations. A basic initial distinction in the study of verbs is that between transitive and intransitive verbs. Transitive verbs have an object whereas intransitive verbs do not. In ‘Sachin plays cricket’, ‘plays’ is a transitive verb and in ‘Chandramukhi loves Devdas’, ‘loves’ is a transitive verb. In ‘the baby is sleeping soundly’ ‘is sleeping’ is an intransitive verb; similarly, in ‘I am running at the racecourse’ ‘am running’ is an intransitive verb. In transitive verbs it makes sense to ask plays what? or loves whom? In intransitive verbs to ask ‘is sleeping what/whom?’ or ‘am jogging what/whom?’ do not make sense. Staying strictly with the words used for designating verbs, this distinction is not categorical. In other words some verbs may be both transitive and intransitive. One may say ‘the child is playing’ without intending to specify any object of the verb ‘is playing’, though this is not likely with ‘loves’ as even in ‘Christ loves’ what is meant is that Christ loves everyone. Similarly in ‘I am running a marathon’ it does make sense to ask what am I running? However, in ‘I exist’ or the ‘universe exists’ it makes no sense to ask ‘what/whom am I existing?’ or ‘what/whom is the universe existing?’ There is probably no instance in which ‘exists’ is used to designate a verb, in which it makes sense to ask what is the object of ‘exists’. We may hence extend our original distinction into a four-fold distinction: 1. verbs that are always transitive, such as to love or loving; 2. verbs that are always intransitive, such as to exist or existing; 3. verbs which are usually transitive but may be intransitive in some uses such as to play or playing; and 4. verbs which are usually intransitive but may be intransitive in some of their uses such as ‘to run’ or ‘running’. The big 100 rupee question now is what kind of a verb is to know or knowing? Before we answer this question let us investigate why transitive verbs are called ‘transitive’: We call these verbs transitive because these verbs have the property of transitivity. What is transitivity? To transit means to pass through. Each of the verbs met, wrote and destroys in our examples has its action conveyed (carried) to the object. We might also say that the action begins with the subject (he, she, rust in our sentences) and passes through the verb to the object. This property of the verb is transitivity. Hence we call these verbs transitive. http://www.english-language-grammar-guide.com/transitive-verb.html (boldfacing and italics are original, but the underlining is mine for emphasis) The phrases ‘action conveyed (carried) to the object’, ‘action begins with the subject’ and ‘passes through the verb to the object’ clearly state that a transitive verb is a two-termed relation where the terms are the subject term and the predicate term. More simply put, a transitive verb signifies a binary relation between a subject and an object. The relation itself may be one-one, many-one, one-many or many-many. In the case of everyone’s favorite verb and relation ‘love’ it is many-many. So we can all live in a harmonious world. Now, we categorise the verb ‘to know’. Let us look at some common uses of the verb: ‘Sonia Gandhi knows Italian’, ‘I know that I am sitting here’, ‘A. R. Rehman knows music’, ‘Stephen Hawking knows that E = mc2’, ‘Sania Mirza knows how to play tennis’, ‘Baba Ramdev knows God’ and ‘my mother really knows me’. Perhaps this choice of the examples exposes my presupposition that knowing is a transitive verb as in each of these there is a the knower as subject and the known as object. Are there uses of ‘know’ which are of an intransitive verb? Let us try. We could say something like ‘Abebe knows running’ but here the verb ‘running’ is used as a noun and it is the object of the verb know. Can we meaningfully say ‘Nagarjuna knows’ just as we say ‘the baby sleeps’? Most would agree that whereas ‘the baby sleeps’ is a complete statement, ‘Nagarjuna knows’ is incomplete. When we say ‘the baby sleeps’ we need not complete it with something like ‘the baby sleeps soundly’, but we need to complete ‘Nagarjuna knows’ as ‘Nagarjuna knows nothing’ or ‘Nagarjuna knows logic’ or ‘Nagarjuna knows bliss’ or something where there is an object. The verb ‘know’ seems always to require an object so it is transitive. However, let us consider ‘Momo is knowing’ or in the same stream as ‘Momo is sleeping’, ‘Momo is weeping’, ‘Momo is writing’, ‘Momo is thinking’, ‘Momo is singing’ and ‘Momo is breathing’. Sleeping, weeping, writing, thinking, singing and breathing are activities and they are also processes. In this list, ‘writing’, ‘singing’ and ‘thinking’ are usually taken to be transitive verbs as one is writing something, singing something or thinking something. However, it still makes sense to say that someone is writing in an unqualified manner without specifying the object in order to convey that the person is in the process of writing. Hence, we are treating all these as intransitive words. Each intransitive verb designates an activity which is a processe. I contend that ‘know’ in their respective uses in ‘Momo is knowing’ or ‘Mammo is knowing’ are also intransitive as they refer to the process of knowing that Momo and Mammo are going through. Momo is the little girl in Michael Ende’s novel Momo (1973) and Mammo is the little girl in the film version of Shyam Benegal’s Mammo (1994) adapted from Khalid Mohammed’s story. And the knowing as a process is both the girls going through their lives. At the beginning of Metaphysics, Aristotle claims ‘All men by nature desire to know’ (980a 21). http://files.vsociety.net/data/library/Section%201%20(A,G,M,S,Z)/Aristotle/Unknown%20Album/Metaphysics.pdf (Downloaded on 25 July, 2011). See beginning of Book I, page 2 of pdf document. He does not say what they desire to know but simply that they desire to know. However, Aristotle does seem to treat ‘know’ as a transitive verb as one must also be desiring and seeking to know something. So, there is an object of any instance of knowing. So it seems like the verb ‘know’ falls under category 1, that of verbs which are always transitive, as there seem to be no instances of the verb that are intransitive. However, in my categorisation above the classic use-mention conflation has taken place. I have conflated words that indicate verbs with verbs themselves. In other words the word ‘play’ does not always indicate the same verb so that play as a transitive verb is different from play as an intransitive verb. As far as verbs themselves are concerned it seems like there are only two types of verbs: transitive and intransitive. However, I would like to argue that there is a third type of verb that is both transitive and intransitive. So, I revise my distinction, and this time it applies only to verbs and not to words signifying verbs: 1v. transitive verbs; 2v. intransitive verbs; and 3v. verbs which are both transitive and intransitive. There may indeed by a fourth type of verbs which are neither transitive nor intransitive, but at the moment I cannot think of any examples. We can also go back now and extend the earlier typology of words that indicate verbs by adding a fifth category: 5. verbs that are always both transitive and intransitive: 1. verbs that are always transitive, such as to love or loving; 2. verbs that are always intransitive, such as to exist or existing; 3. verbs which are usually transitive but may be intransitive in some uses such as to play or playing; 4. verbs which are usually intransitive but may be intransitive in some of their uses such as ‘to run’ or ‘running’; and 5. verbs which are always both transitive and intransitive. Again, here I am persisting with the use-mention conflation but purposefully in the use of the word ‘verb’. This is a conflation that Russell continuously makes while the context always makes it clear whether we are talking about language or the world. In this list of five types I am using ‘verbs’ to refer to words that designate verbs, this is because literally a ‘verb’ is a part of speech so it is very much a word or set of words. What each word for a verb designates is a relation which we can simply call by the name of the verb. In the case of ‘know’ then, the word ‘know’ designates the verb know which is simultaneously the transitive verb know which is a two-termed relation between the knower and the known as well as the intransitive verb knowing which is the activity of knowing. More precisely the verb designated by ‘to know’ is simultaneously the relation in ‘S knows that p’ which we may signify as SKp as well as the activity of knowing that is being undertaken by S. For ‘S knows p’ to be successful or complete in any case, first, S must be involved in the activity of knowing p, and second, S must know p at the end of the process or activity. ‘knowing’ and ‘know’ in the previous sentence are clearly distinct, and my contention is that both together constitute the ordinary verb to know. So when we ordinarily say ‘S knows that p’ what we mean is that S is knowing (involved in the activity of knowing) and that at the end S knows that p. Let me put it a bit more precisely: ‘S knows that p’ = (1) S is knowinga from t1–tn and (2) S knowsr p at tn. (1) is knowing as an activity or process which is the intransitive dimension of the verb to know; and (2) is the transitive dimension of the verb to know where the subject is S and the object is p. Even though in (1) ‘knowinga’ designates an activity it is an activity with the intent of knowing p, so the object is very much there, but it would not be appropriate to treat knowinga as a transitive verb because we cannot say that S is knowing p from t1–tn until S has known p at tn. The qualifications of the time constraints are usually assumed in any definition of knowledge but I have embedded them here for clarity. The process of knowinga (the subscript ‘a’ designates activity which is the intransitive verb knowing) goes on for some time which is designated by t1–tn and at the end of the process, that is, at the moment tn if S has come to knowr p (the subscript ‘r’ designates the transitive relation of the verb know); then S knows that p. This is basically what I have elsewhere called ‘knowledge as actively justified true belief’, and later by dropping the truth condition I have revised it to ‘knowledge as actively justified belief’. Now, my main contention is that what is commonly called ‘knowledge’ refers to a class of satisfied ‘S knows that p’s, that is, all the cases of knows where both the conditions (1) and (2) have been fulfilled. In other words ‘S knows that p’ is true when S has successfully performed the activity of knowing over a specified period of time and S has completed the activity with the result that S knows that p at the moment at the end of the process. The traditional definition of knowledge as we all know is as justified true belief (JTB): S knows that p =def. (i) S believes that p; (ii) S is justified in believing that p; and (iii) that p is true. We can now reformulate this definition as follows: ‘S knows that p’ at tn = (1) S is knowinga from t1–tn and (2) S knowsr p at tn =def. S believes that p at t1, t2,…., tn; S sincerely performs the activity of justification of p from t1 to tn; that p is true at tn. It is normally assumed that the seeking of knowledge begins when there already is a belief and it is also taken for granted that truth cannot be a result of a process. Hence, it is justification where the process is contained. So, S goes through a process from from t1 to tn at the end of which she/he can be said to be justified in believing that p. Rather than give a separate recursive definition of each of (1) and (2) I have given a combined definiens with the three conditions (i), (ii) and (iii). This may be a technical problem in the way we usually understand recursive definitions. In the traditional definition the three conditions (i) S believe that p, (ii) S is justified in believing that p and (iii) that p is true; each of (i), (ii) and (iii) in the definiens are necessary conditions for the definiendum ‘S knows that p’. They are also jointly sufficient for the definiendum. Furthermore, each of (i), (ii) and (iii) are logically independent, that is, (a) ‘S believes that p’ does not imply that ‘S is justified in believing p’ so that S may well believe p without being justified in believing p; (b) ‘S believes that p’ does not imply ‘that p is true’ so that S may believe p without that p being true; (c) ‘S is justified in believing p’ does not imply ‘S believes that p’ so that S may be justified in believing p without believing that p; (d) ‘S is justified in believing that p’ does not imply ‘that p is true’ so that S may be justified in believing that p without p being true; (e) ‘that p is true’ does not imply ‘S believes that p’ so that p may well be true without S’s believing that p; and (f) ‘that p is true’ does not imply ‘S is justified in believing that p’ so that p may well be true without S’s being justified in believing that p. In first defining ‘S knows that p’ as (1) S is knowinga from t1–tn and (2) S knowsr p at tn, (1) and (2) must be logically independent, that is, S’s knowing from t1–tn does not imply ‘S knowsr that p at tn’, and ‘S knowsr that p at tn’ does not imply ‘S is knowinga from t1–tn’. This autonomy is preserved. Is the autonomy preserved with the extended definiens? (a) ‘S believes that p at t1, t2,…., tn’ does not imply ‘S sincerely performs the activity of justification of p from t1 to tn’ because S may well believe p at every moment in the process and yet not sincerely perform the activity of justifying p during the same period; (b) ‘S believes that p at t1, t2,…., tn’ does not imply ‘that p is true’ because obviously the truth of p is independent of S’s believing p at any time; (c) ‘S sincerely performs the activity of justification of p from t1 to tn’ does not imply ‘S believes that p at t1, t2,…., tn’ because S may sincerely perform the justification activity of p without at any moment actually believing p or coming to believe p; (d) ‘S sincerely performs the activity of justification from t1 to tn’ does not imply ‘that p is true’ as the truth of a proposition is also independent of the justification for it and justification does not guarantee truth. Many epistemologists, starting with Plato himself in the Theaetetus (the Bible of epistemology), claim that real knowledge is when one knows that one knows. Why is that? Since truth is a condition that is outside the knower, it is not determined by the knower, so that it is possible for S to know something accidentally or to be absolutely convinced that she knows p but since p turns out to be false she does not know p. Knowing that one knows will supposedly take care of this problem because one would know whether or not one really knows. Does this happen? Let us look at this by applying the traditional definition in two stages: S knows that ‘S knows that p’ =def. (i) S believes that S knows that p; (ii) S is justified in believing that S knows that p; and (iii) that S knows that p is true. Now, condition (iii) is exactly the first order knowledge of p, that is ‘S knows that p’, the definition of which has the third condition of p being true, which is outside the individual. Hence, the element of luck has not been eliminated. Traditionally, there have been two opposing views on what is called the KK principle. The KK principle simply states that if one knows that p then it is implied that one knows that one knows that p: S knows that p → S knows that S knows that p Some, like Hintikka, accept this principle motivated by the modal axiom in S5 that if p is necessary then it is necessary that p is necessary: p → p Others, like Chisholm, reject the KK principle while they accept the modal axiom. Chisholm claims that it is obvious that ‘S knows that p’ implies that ‘S knows that p’ is true, which satisfies condition (iii) Also, though perhaps not so obvious, ‘S knows that p’ implies that ‘S is justified in believing that S knows that p’. However, Chisholm does not think that ‘S knows that p’ implies that ‘S believes that S knows that p’. So, in order to get the implication we need the additional condition that S believes that S knows that p. In other words: S knows that p and S believes that S knows that p → S knows that S knows that p This means that on the traditional definition one may know something without believing that one knows. This is indeed a solution to a Socratic paradox. If Socrates claims that he does not know anything, then he knows at least one thing, that is, that he does not know anything else except this very proposition. We can now say that though Socrates is committed to knowing this one proposition, he does not believe that he knows. Socrates’ own activity for which he was famous/notorious in Athens was to expose the ignorance of others, that is, to make alleged experts realize that they did not know what they claimed to know? Was he exposing that though these experts may have had knowledge they did not know that they had knowledge? If we carefully go through some of the dialogues we see that in his activity what Socrates really does is to show that whereas experts believe that they know something or a lot of things on closer examination they don’t really know even if they believe that they know. Since one of the two necessary and sufficient conditions for knowing that one knows, here the main condition of first order knowing is not satisfied, then obviously the second order knowing does not take place. Similarly we can resolve the sceptics self referential paradox as well. A common objection to scepticism is that if there is nothing that can be known then what about this very proposition that nothing can be known, is it known or not known? If it is known then it is not true that nothing can be known and hence scepticism is false; if, on the other hand, the sceptic cannot know that she/he cannot know anything, then her/his statement about scepticism may or may not be true. Now, the sceptic can say that he believes that he knows but he may not actually know the proposition that states scepticism, or alternatively, s/he may say that s/he knows that s/he does not know but s/he does not believe this. So, we have two varieties of scepticism. Let us look more closely at Chisholm’s implication more colosely by first spelling out the first order and second order definitions of knowing and knowing that one knows more explicitly: S knows that p =def. S believes that p, S is justified in believing that p, and that p is true. S knows that ‘S knows that p’ =def. S believes that S believes that p, S believes that S is justified in believing that p, S believes that p is true, S is justified in believing that S believes that p, S is justified in believing that S is justified in believing that p, S is justified in believing that p is true; and that S believes that p is true, that S is justified in believing that p is true, and that p is true. Now, when S knows that p then conditions (g), (h) and (i) are satisfied because that is exactly the definition of first order knowledge. Condition (f) is the same as condition (h) hence it is also implied by the first order knowing of p. (e) is not so obviously implied by S knows that p. How is the second order justification of first order justification implied by S knows that p? Surely I could have adequate justification required for first order knowing without having justification of the first order justification. There is a normal requirement of theories like reliabilism that if I am justified in believing that p then my justification itself is reliable or justified otherwise I am nor really justified I believing that p. In other words, the process of justifying that p guarantees that this process itself is justified. So, condition (2) of first order knowing entails condition (e) of second order knowing. Condition (d) is again obviously implied by first order knowing. If I have a belief p as condition (1) states, then I possess this mental state of a belief ; and my very possession of this mental state is a self-justification that I am justified in believing that p. Hence, (1) implies (d). (1) also implies (c). Rather (c) is the spelling out of (1) as when one believes any proposition she/he believes that that proposition is true. This is the kind of property of believing that Ayer would argue shows that all knowledge is propositional. In other words If one were to say something like ‘I know God’ and that this begins with my belief in God, then what does ‘I believe God’ or ‘I believe in God’ mean? This may be a molecular conjunction proposition of the type ‘I believe that God exists’ and ‘I believe that God is all good’ and ‘I believe that God is all powerful’ and ‘I believe that God is omniscient’ and ‘I believe that God is everywhere’ (there could be further conjuncts which may not be easy for an atheist like me to think up of). So believing in God and knowing God has been reduced to propositional belief and propositional knowledge. Does (1) also imply (a)? Since belief is a mental state if I have a belief then surely I must believe that I have this belief. One may argue that though this should be the case it may not be so at all times. We may have people who believe something without believing that they believe it. One quick way to respond to this is to say that there are no second order beliefs or that second order beliefs are redundant as believing p implies that one believes that one believes that p. One may give a rebuttal to this by analogy. Suppose someone’s arm is amputated. S/he can see that her/his arm is not there on her/his body; however she/he still feels that arm as part of her/his body and hence does not believe that the arm is no longer part of her/his body, or perhaps believes both that the arm is and is not a part of her/his body. Since one can and does occasionally hold contradictory beliefs there is no problem with this scenario. Should we then say that (1) does not imply (a)? Let us look at a concrete example: I believe that tomorrow is Saturday. Does this imply that I believe that I believe that tomorrow is Saturday. If beliefs are mental states then the first order mental states would imply second order mental states which would imply third order mental states, and so on, an infinite regress gets generated. However, since this is a benign and not a vicious infinite regress, there is no reason to reject second order mental states. So, it could well be that I possess the first order mental state of the belief that tomorrow is Saturday but I do not possess the second order mental state of belief that I have the belief that tomorrow is Saturday. I may even have a second order mental state that I do not believe that tomorrow is Saturday, which conflicts with my first order mental state of tomorrow being Saturday. I think if we look closely enough at ordinary cases of believing this often happens. So, let us at least leave it as an open question as to whether or not (a) is implied by (1). The case of (1) implying (b) is even more precarious, and this is the exact point where Chisholm differs from Hintikka. My believing that tomorrow is Saturday does not imply that I believe that I am justified in believing that tomorrow is Saturday. So clearly (1) does not imply (b). Perhaps (b) is implied by (2). That is, my justification for believing that tomorrow is Saturday implies that I believe that I am justified in believing that tomorrow is Saturday. Please, just stare at this, and you will see that this implication is just not there. Suppose Shanti is a an excellent student in her geometry class. When asked to prove the Pythagorian theorem she provides a flawless proof and scores an 25 out of 25 for that question. Since being able to provide a proof is a sufficient justification for a mathematical proposition, Shanti surely is justified in believing the conclusion of the Pythagorian theorem. However, she may actually believe that she is not justified in believing the Pythagorian theorem. Even though she can produce the proof on demand and understands all the steps, she may still feel that she does not really understand the proof and for her to believe that she is justified she needs to get greater understanding. Furthermore, since belief is a psychological state, it just turns out for whatever reason that she just does not believe that she is justified in believing the conclusion of the Pythagorian theorem. Let us turn to a famous historical case. Fermat said that he had a proof for his last theorem, that is the famous Fermat’s last theorem. But before he could write down this proof, he died. Now let us speculate that he actually had the proof in his head but had not written it down. Once he wrote it down, he would then have to check it for errors before he would be convinced that this is the correct proof. Since he had not gone through this process we could conjecture that even though he was very confident that he had the proof, he may well have had the belief that he had the proof for Fermat’s last theorem. This may seem unlikely but is definitely possible. Again, beliefs are a matter of psychology and at a certain moment one suddenly for whatever reason acquires a mental state of having a certain belief. One may have very strict Cartesian psychological requirements for oneself that one will only believe that which is absolutely beyond doubt. But even then one may end up not believing as that mental state of belief just does not happen. Descartes never says that we can suspend our beliefs, but insists that our beliefs will always be there. What we suspend is our knowledge of these beliefs and apply the criterion of clarity and distinctness for knowledge not for belief. So, (2) does not imply (b). What about (1) and (2) together? Do they in conjunction imply (b)? When I have justification for the belief that tomorrow is Saturday and I also believe that tomorrow is Saturday then is it not implied that I believe that I am justified in believing that tomorrow is Saturday? Again, I would request that we stare at this for a while and we would reach the answer ‘no’. If one is an externalist regarding justification the counterexample will come easy. One may have adequate externalist justification for tomorrow being Saturday, yet one may not believe that one has an externalist justification for tomorrow being Saturday, since belief is an internal matter both for externalists and internalists. It gets a litter tougher for internalism. Internalists like Bonjour claim that the justification must be available to the knower upon reflection. Now, if I have internalist justification for believing that tomorrow is Saturday and I possess the mental state of the belief that tomorrow is Saturday, do I not also have the belief that I have internalist justification for believing that tomorrow is Saturday? The answer is still ‘no’. Fermat’s proof was internalist as the proof was available only to him and he claimed that it was really available to him on reflection and surely he believed in the conclusion. Yet he may not yet have formed the belief that he had the internalist justification as the proof. Shanti’s proof of the Pythagorian theorem that she produced flawlessly on the exam may be available to her on reflection on demand and she may indeed believe the conclusion of the Pythagorian theorem; yet, she may not believe that her internal justification is sufficient for the conclusion of the Pythagorian theorem. We must keep the psychological distinction between a belief and internalist justification of a belief. Believing that one is internally justified in believing that tomorrow is Saturday is a belief, a mental state; and believing that tomorrow is Saturday is also a belief, a distinct mental state then the one just stated. So that one may have one belief without having the other one. So, (1) does not imply (b). Being justified in believing that tomorrow is Saturday does not imply belief either by itself or conjointly with the belief that tomorrow is Saturday. To think that an implication holds here would be to commit what Ryle calls a category mistake. Within the definition of knowing we put believing in a propositional form: ‘S believes that p’. Now, it is traditionally agreed that ‘S knows that p’ implies that ‘S believes that p’. This is challenged by those contemporary epistemologists who claim that knowledge is not a species of belief. However, ‘S believes that p’ does not imply ‘S believes that q’ where p is distinct from q, whatever p and q may be. Nor does ‘S believes that p’ in conjunction ‘S is justified in believing that p’ entail any proposition of the form ‘S believes that S is justified in believing that p’. Within the logic of beliefs we cannot even claim that if S believes that p and S believes that q then it implies that S believes that p and q, though in all likelihood S would believe p and q if s/he believed each of p and q. So, the implication we are looking for is just not there. What about (1), (2) and (3) conjointly, do they imply (b)? Truth is an external condition, so introducing the proposition that tomorrow is Saturday is true into the conjunction of I believe that tomorrow is Saturday and I am justified in believing that tomorrow is Saturday does not add anything that would assure the belief that I am justified in believing that tomorrow is Saturday. To use a metaphor, when we conjoined (1) and (2) we were trying to find out whether we could produce an apple from an apple and an orange; and when we combine (1), (2) and (3), then are trying to find out whether an apple, orange and a horse can produce an apple, which is even more absurd. So, after this long exercise we must come to the conclusion that only beliefs may imply beliefs, or rather, propositions of the type ‘S believes p’ can only be implied by propositions of the same type ‘S believes that p’. But we have just said that even this is not true as no proposition ‘S believes that p’ implies ‘S believes that q’ when p is distinct from q. Even ‘S believes that p at t1’ does not imply that ‘S believes that p at t2’ as a person’s belief may change for moment to moment as the Buddhists and Hume tell us. Furthermore, and this may not even be acceptable to the Buddhists or Hume, even ‘S believes that p at t1’ does not imply ‘S believes that p at t1’ for two reasons. First, there is a quantitative reason of not being able to pin down t1 as time is continuous and not really moments as each moment consists of further moments. Second, there is a reason from the logic of beliefs. Within the logic of beliefs, a person may well believe p as well as not p at the same moment. Again, this is unlikely or highly improbable but possible. One example of this is from the history of physics in which Jules Henri Poincaré (1854–1912) held two contradictory principles at the same time. This is not an explicit holding of the believing p and believing not p at the same time but it can easily be reduced to this as in Poincaré’s words if the one principle is true the other is false and vice versa. But if believing p implies believing p then S cannot believe not p. However, one may object that we are reaching an absurdity which goes beyond logic and the notion of ‘imply’. Let us then return to the propositional statements of beliefs. ‘S believes that p’ implies ‘S believes that p’, because any proposition implies itself, and there is no way to get around it. Similarly ‘S believes that not p’ implies that ‘S believes that not p’. And both these implications do not prevent the possibility that ‘S believes p and not p’ because neither ‘S believes p’ nor ‘S believes not p’ imply that S cannot believe both p and not p. S may not well, no matter how much bad faith it is, believe in the law of non-contradiction. One implication that definitely holds is that ‘S knows that p’ implies that ‘S believes that p’ (except for those who deny that knowledge is a species of belief), but this is simply because ‘S knows that p’ (the definiendum) is equivalent to (the definiens) (1) S believes that p, (2) S is justified in believing that p and (3) p is true. Any conjunction of (1), (2) and (3) implies (1). This is simply analytic. So we conclude that at least (b) is not implied by any of or any combination of (1), (2) and (3). This means that knowing p does not imply knowing that one knows p. But with the addition, as Chisholm suggests, of (a) that one believes that one believes that p and (b) one believes that one is justified in believing that p, one’s knowing that p would entail knowing that one knows that p. Now, let us lay out the definition of knowing that one knows to our drawn out definition of ‘know’: We get a two staged definition as follows: I. S knows that S knows that p = (1) S is knowinga from from tn+1–tn+j and (2) S knowsr at tn+j that S knows that p at tn. II. S knows that S knows that p =def. (i) S believes at tn+1, at tn+2,…., at tn+j, that S is knowinga that S t1; (ii) S is justified in believing that p at tn; and (iii) that p is true at tn. S knows that ‘S knows that p’ =def. (i) S believes that S knows that p; (ii) S is justified in believing that S knows that p; (iii) ‘S knows that p’ is true; and (iv) S is justified in believing that S is justified in believing that p. Knowing that one knows is an activity, hence we may reformulate our definition as follows: S knows that ‘S knows that p’ =def. (i) S believes that S knows that p; (ii) S is justified in believing that S knows that p; (iii) ‘S knows that p’ is true; and (iv) S is justified in believing that S is justified in believing that p. and later in the same Book I he says: ‘[…] that which is desirable on its own account and for the sake of knowing it is more of the nature of wisdom than that which is desirable on account of its results, […]’ (982b 14–16) Aristotle Metaphysics, Book I (A), Part 2. Also available at http://files.vsociety.net/data/library/Section%201%20(A,G,M,S,Z)/Aristotle/Unknown%20Album/Metaphysics.pdf (Downloaded on 25 July, 2011). See top of page 4 of pdf document.. References Aristotle, Metaphysics. Translated by W. C. Ross. in Jonathan Barnes (ed.), Complete Works of Aristotle. Also Available online at http://files.vsociety.net/data/library/Section%201%20(A,G,M,S,Z)/Aristotle/Unknown%20Album/Metaphysics.pdf (Downloaded on 25 July, 2011). English-Grammar-Language-Guide. 2011. Available at http://www.english-language-grammar-guide.com/transitive-verb.html (Downloaded on 27 July 2011).