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epistemology, logic, and philosophy of mind

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  • ian
    A probabilistic epistemology (PE) must not rely on epistemic probabilities to justify the system, but instead must look to what actually justifies belief.
    Message 1 of 1 , May 25, 2010
      A probabilistic epistemology (PE) must not rely on epistemic probabilities to justify the system, but instead must look to what actually justifies belief. (Pollock 1986) This can be supplemented with naturalized epistemology (NE) by using psychological findings to support justifications in beliefs that lead to the use of probabilities to make judgments.

      "At the very least, the probabilist owes us an account of probability in terms of which he wants his theory to be understood, and there is good reason for being skeptical about there being any kind of appropriate probability." (Pollock 1986)
      For the non-cognitivist conducting natural epistemology there is no need for a justification of normativity because the moral cannot be reasoned. Thus there are merely facts about reality (resting on our experiences) and the beliefs that are justified by these discoveries. But Quine is correct to say that all sentences can be revised (Dancy 1993). But does this spell defeat for necessary truths, in the logical sense of necessity? It looks blatantly obvious that it does, since no revision could be done to theorems…but that might not necessarily (in the colloquial sense) be true. To speak of what an argument entails is to speak of its necessities in the logical sense, but if we look at what we mean by `revise' we may see that necessary truths are still possible.
      A theorem can be revised in the following manner:
      {(P v Q)->R}->(Q->R)}
      It can be revised by actually describing the full potential of the relationships. I will introduce a meta-variable to signify the disjunctive antecedent in the first conditional with a single φ to denote an inclusive disjunction but also either one of the variables, thus we have:
      {(P v Q)->R}->( φ ->R)}
      Both of these theorems are Laws of Logic, such is what it means to be a theorem in formal systems of logic. They are different, yet they both represent the same fundamental truth, i.e. that if there is a disjunctive antecedent bound by a conditional, then the consequent is established with either one of the disjuncts. The revision is an improvement because it expresses the relationship more explicitly, but can it be revised again? If we look at what it means semantically, but without referents to actual objects we can see that it can indeed be revised again, because it could be said of P, Q, and R that
      { {[(P=R)->(P v R)]->Q}->(φ->Q)}
      And indeed, this could be revised further, ad infinitum as relationships can always be added to well-formed formulas given they do not alter the integrity of the structure of the theorem. Therefore Quine was mistaken to believe that theorems and necessary truths can't be revised, because he was correct to say that every sentence can be revised.
      I present the following arguments to illustrate how the revision of a theorem can bear direct relevance to weighing the epistemic foundations of arguments that occur within philosophy:
      Let, in the domain of Human Brains:
      Fx = x has property F
      Mx = there is a material aspect to x‟s mind
      T = Physicalism is tenable
      ~Nx = x has no neuronal assemblies
      ~Rx = x has no episodic memory
      C = consciousness
      Cx = x has consciousness
      (x){[(Fx->Mx)&(Mx->T)]&(~Nx->~Rx)}->(x){[( Fx)&(F=C)]->[(Cx->Mx)->T]}
      Therefore, revising it we have:
      Let, in the Domain of Human Brains:
      P = Physicalism is true
      B = the brain is the seat of mental operations
      Rx = x has episodic memory
      ~Nx = x has no neuronal assemblies
      2.(x)(~Rx <-> ~Nx)
      4.(x)~Nx /:. Ba & Pa
      5. Rp<->Np UI 2
      6.[(~Rp->~Np)&(~Np->~Rp)] Equivalence 5
      7.(~Np&~Rp)->B UI 3
      8. ~Np->~Rp simp. 6
      9. ~Np UI 4
      10. ~Rp
      11. ~Np&~Rp
      12. B, Modus Ponens 11, 7
      13. B->P contraposition 1
      14. P Modus Ponens 12, 13 5 | P a g e
      15. B & P
      But to combine them in revision once more reveals:
      Let: in the domain of human brains
      B = there is a correlated brain state with conscious experience,
      M = there is a material aspect to consciousness,
      N = there is a neuronal assembly,
      R = there is episodic memory
      P = physicalism is true.
      1. (B=>M)
      2. (M=>P)
      3. [(~R=>~N)=>B] /:. [(N=>R)=>P]
      Thus we have established a necessary truth based on a sentence that Quine believed wasn't revisable. But this argument is challenged by the following theorems:
      Cx – x is consciousness
      Hy – y is a solution to the Halting Problem
      Pzx – z is a Physical Theory of x
      Mzx – z is a mathematical model of x
      Iz – z is incomplete
      Pyx – y is a Physical Theory of x
      Mzy – z is a mathematical model of y
      Ex – x has a Physical explanation
      Gx – x is emergent
      Fzx – z can explain x
      The Law of Physical Explanation
      In the domain of consciousness:
      (x)~(ᴲy)(z){[{(Cx & Hy) & Mzx] & (~Iz->Pzx)]->(~Pzx->Iz)}
      This states that if x is consciousness and there is no solution to the Halting Problem and z is a mathematical model of x, if z is complete then there can be a Physical theory of consciousness, then if there is no Physical theory of consciousness and then the physical models must remain incomplete. What this means is simple, if we are to analyze consciousness and develop a mathematical model of it in order to derive a Physical Theory there is the chance that the physical theory will inherently be incomplete given the nature of what it is studying. This can be made more explicit.
      Physical Incompleteness Theorem
      Domain: Physical Theories
      (ᴲx)(ᴲz){[(Mzx->Iz)->(y)[(Mzx & Pyx)->(Pyx->Iy)]}
      What this states is that there is some x and some z such that if z is a mathematical model of x then z is incomplete. Then for all y, if z is a mathematical explanation of x and y is a Physical Theory of x then if y is a Physical Theory of x, y is incomplete. What this means is that some mathematical models will always be incomplete and therefore that any Physics that is based on these incomplete mathematical systems run the risk of being incomplete. The relationship between Theoretical Physics and Mathematics is close, but what must be shown is that consciousness can only be explained by Theoretical Physics, if it is to be explained by Physics, and that the mathematical model of consciousness must always be incomplete. The former is outside the scope of this paper, but the latter can be addressed.
      Mathematical Model of Cognition Incompleteness Theorem
      (x)~(ᴲy)(z){[(Cx & Gx)&[Hy &(Mzx->Iz)]]->[(Hy->(Mzx->~Fzx))->[(Iz & Mzx)->~Fzx]]}
      This therefore means that for all x, there is not some y for all z such that if x is an emergent consciousness and there is no solution to the Halting Problem and z is a mathematical model of this emergent property then the model is doomed to incompleteness, therefore since there is no solution to the Halting Problem, if z is mathematical model of x then it cannot explain x, then z is an incomplete mathematical model and it cannot ever explain consciousness.

      What can revision tell us now about these contradictory, necessary truths? This is an epistemic challenge, and there are two sides to this issue given natural and probabilistic epistemology: the status of being necessary truths, the status of the truths being claimed relative to science and probability.
      Before anything can be established, the definition of probability must be stated; probability is the quantification of the likelihood for an event to occur, or proposition to be true. An event is a state that reality may take, and a proposition is a combination of ideas. Ideas themselves are formed by beliefs and create knowledge wherein knowledge is the database used within reason found to be true. Epistemic probability is the likelihood of an epistemic proposition to be true and can be quantified through empirical investigation. Thus our justification for knowledge is the probability that the event will occur or the proposition is likely true given the evidence. This probabilistic epistemic basis necessitates empirical verification given the need for numerical values of probability, which can only be gleaned through induction, a posteriori. This weakens the system, but for a corollary system in logic to be developed from this epistemic basis one may find that the truths discovered have a high probability of being true. This can't be known a priori, given that a number of one event must occur multiple times, or all information must be incorporated into the proof. Since induction loses strength the more a conclusion is validated by experimentation and no proof can ever contain all relevant premises, the situation looks bleak for any logic based upon the epistemic basis put forth in these preceding thoughts.
      A combination of scientific investigation with the proof of epistemic truths justified analytically, a posteriori would give basis to the probability of their truth given what must be known in order for an epistemic truth to be justified. Some claim that epistemology is concerned only with synthetic, a priori truths but Quine argues that there is no true distinction between synthetic and analytic because all truths are analytic and a posteriori (this holds some weight, but needs formalization – without begging the question). Thus, the traditional methods of epistemology are mistaken, and can be subsumed under empirical investigation. (Dancy 1993) If epistemology is to take on a purely descriptive status it could lose its application to normative claims (Baergen 1995); however, few natural epistemologists take it this far. Natural epistemology may make normative claims in that a given function of an organism can be said to be `good' or `bad' if a standard can be established based on knowledge of how the organism is to function. Psychology at its base deals with taxons and arbitrary distinctions, but the conglomeration of categories that results does in fact produce a system of normativity within the descriptions the field discovers; however, the epistemology I am attempting to establish is non-cognitivist to a degree, making normativity impossible to establish through any means of reason.
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