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Since termination at unjustified beliefs leaves everything based on those beliefs unjustified, and non-linear justification - where certain parts justify others in a coherent way - is innately circular no matter how complex, the only plausible intuitive candidate for termination is foundational (i.e. basic) beliefs which are somehow justified by appeal to something which does not itself require justification (else the regress would continue). This answer to the regress problem is known as internalist foundationalism.
The pertinent question for the internalist foundationalist is how to justify these basic beliefs in such a way that they remain basic beliefs. Self-justification begs the question, and self-evidence is inapplicable since true empirical beliefs are contingently true, not necessarily true. The only intuitive possibility left is that they are justified by appeal to experience, which then brings forth the question of how to get from experience to beliefs.
The only plausible suggestion seems to be that a belief about a given experience is justified by an act of direct acquaintance with the experiential content. For example, when a person has the experience of seeing a tree and a belief that they are seeing a tree, they also have a direct acquaintance with, or one might say awareness of, the experience and that it is of seeing a tree and not of seeing a lake. It is this acquaintance with, or awareness of, the very character of the experience that is purported to justify the belief (and itself require no justification).
Unfortunately, a dilemma immediately seems to arise from this conception of direct acquaintance stopping the regress by justifying basic beliefs. It appears that the acquaintance in question must either be an assertive or judgmental cognitive act that either conceptualizes the experience (that it was of a tree, rather than a lake) or does not. In other words, it is either propositional (by that is meant that it makes a statement, perhaps not explicitly, but at least in conceptual terms, which is either true or false - which claims that the experience was a certain way distinct from other possibilities) or it is not propositional.
If the former, the direct acquaintance seems to indeed be able to provide sufficient justification for the belief that the experience was a certain way (of a tree) and not otherwise - for the acquaintance contains evidence for the truth of the belief. The problem here is that since the direct acquaintance contains an assertion of things being one way and not another, this assertion seems to require justification. This justification cannot be provided by self-evidence (for it is of a contingent claim) nor appeal simply to the experience itself (for that would break down into externalism, which has unsolvable problems of its own), so there seems to be no way to justify the direct acquaintance without appealing again to something which itself requires justification (and thus restarts the regress).
If, however, direct acquaintance is not propositional (assertive of the experience being one way rather than another), there seems to be no requirement to justify it - for what reason could there possibly be to justify something that makes no claims of the true/false sort? Unfortunately, if direct acquaintance is such, it seems to provide no justification for a belief (which is propositional) about an experience, because there is nothing in the acquaintance that matches the belief's assertive aspect (that the experience was of one thing and not another).
Therefore, whether direct acquaintance is propositional (as defined above) or not, it fails to meet the requirements of a justification for basic beliefs (that is, it either cannot justify the basic beliefs, or it requires justification itself which renders the beliefs it justifies as non-basic). This dilemma presents a serious and basic problem for the internalist foundationalist account, for it seems that the general idea of the dilemma applies to anything which is suggested as a justification for basic beliefs (it either makes some sort of assertion - literal, conceptual or otherwise - that an experience was one way rather than another, or it does not make such an assertion).
Laurence BonJour offers a response that supposedly avoids the problem and rescues internalist foundationalism by going "through the horns" of the dilemma, so to speak. In the direct acquaintance view considered above, the acquaintance, that is to say the awareness, was viewed as an apperceptive, or metaphysically distinct, thing which had as its content something like the proposition that a belief X is being held. This second-order awareness (where the first-order would be the belief itself, with the content that X is so) appears to be no more than an occurrent version of the second-order meta-belief which is a belief with the content that it is believed X is so; and for this reason, the second order awareness is unable to justify the meta-belief (for that would be circular).
BonJour claims that looking at such a second-order, apperceptive awareness is a mistake, and that there is another way to approach the problem. He states that an intrinsic and unavoidable part of having an occurrent belief is a conscious awareness of the propositional (that is to say assertive that things are one way rather than another) content and character of the belief. His view is that this awareness is not metaphysically distinct from the first-order state, but rather constitutive of it - it is part of what makes the first-order state what it is rather than another state. For BonJour, an occurrent belief or thought (first-order state) is a conscious state on its own (not based on any higher-level states), where what the believer/thinker is conscious of is the propositional content and character of the belief.
The way BonJour thinks this avoids the dilemma is that the awareness of the (propositional) content and character of the first-order state will justify the meta-belief about the first-order state (that is, the meta-belief that one is believing X). Thus it is proposed that the meta-belief is the basic or foundational belief upon which an epistemological system will be built, though the awareness constitutive of the first-order state is in fact the regress-stopper. The key element for BonJour is that here, where generally the dilemma would propose that awareness needs to be justified, it makes sense to instead go through the horns of the dilemma by stating that the awareness does not need to be, and in fact cannot be, justified.
The reason given for this is that the constitutive awareness is infallible, insofar as this awareness is what makes the state the state that it is, rather than another. The awareness cannot be wrong because there is nothing metaphysically distinct from it for it to be wrong about. In addition, the meta-belief is justified by the awareness because it is a description of the content the awareness is of. By being consciously aware of the content of the first-order state, one seems able to judge whether or not the meta-belief accurately reflects it. Therefore, BonJour says, the first-order constitutive awareness satisfies the requirements for foundationalism (by providing a justification for basic meta-beliefs while itself needing no justification) and avoids the foundationalist dilemma.
Justifying Beliefs About Sensory Experience and the Physical World
BonJour then extends this idea from providing a justification for non-apperceptive meta-beliefs about the content and character of occurrent first-order beliefs or thoughts to providing a justification for analogous meta-beliefs about the content and character of sensory experiences (for example, experiencing the sight of a particular, non-conceptual tree). Some people have claimed that sensory experience is non-propositional because it is non-conceptual (that is, the awareness of the content of sensory experience is not in conceptual terms), therefore it cannot function in a justificatory relation with a meta-belief about the experience (which would, of course, have conceptual and propositional content).
BonJour accepts the view of sensory experience as non-conceptual, but claims that it can still hold justificatory power because it can function in a descriptive relation with the meta-belief - the meta-belief purports to accurately describe conceptually a non-conceptual object. As to whether or not this description is accurate, BonJour parallels his earlier argument for awareness of first-order beliefs. Constitutive of a sensory experience is a non-apperceptive awareness of the content of the experience, and this awareness places one in an ideal position to judge whether or not a given meta-belief about the experience accurately describes it.
Therefore, it is not only meta-beliefs about the content and character of first-order beliefs which are justified by this dilemma-avoiding awareness, but also meta-beliefs about the content and character of sensory experience. It should be noted that in both cases, the justification only applies because the states are conscious states - were they not, the awareness would not exist and therefore the meta-beliefs would not be justified. In this way, BonJour has built an epistemological foundation, but it is still left to get from this justified foundation to justifying our general beliefs about the physical world.
In order to connect the foundation he has laid out to a physical world, BonJour appeals to the strong correlation (indeed, a structural isomorphism) between the content of sensory experiences and the idea of situations in a three-dimensional physical world. He bases this on the character of certain sensory experiences everyone seems to have. For example, one often experiences certain sets of two-dimensional images (say, the set of experiences that come when one does what would commonsensically be thought of as moving around a physical object to view it from various angles) which, when fitted together in a regular way (based on the angle one would commonsensically consider one's self to be at if one were indeed moving around a physical object), construct a three-dimensional object of the sort we commonsensically think of (a box or a cat or another three-dimensional physical object).
This view appears further supported by the way visual images are presented when one does what is commonsensically considered moving around in three-dimensional space (that is, in the same way "moving around an object" seems to provide a coherent and regular way to fit two-dimensional images into a three-dimension object, "moving away from an object", for example, seems to provide a diminishment in the size of the two-dimensional images in a way that fits coherently and regularly with the commonsensical idea of being in a three-dimensional physical world moving away from an object). This idea seems yet further supported by the way two-dimensional experiences seem to replace one another in a way that fits coherently with the idea of a three-dimensional world where one object is blocking the view of another (entering the line of sight at a closer point).
Based upon these aspects of experience that everyone seems to have, BonJour concludes that those experiences (including their isomorphism with the idea of a physical world) must be explained, and are explained by the existence of a three-dimensional physical world which one moves around in. BonJour calls this explanation the quasi-commonsensical hypothesis.
BonJour then proceeds to support the quasi-commonsensical hypothesis by showing that there is no better explanation for the isomorphism. He divides possible explanations up into two categories: analog and digital. Analog explanations explain the features of experience by appeal to features of objects in a physical world that are represented in that experience (the quasi-commonsensical hypothesis falls into this category). Digital explanations explain the features of experience as a representation, caused by some agent or mechanism (such as Berkeley's God or Descartes' demon or a brain-vat), of the experiences one would have if a physical world existed and one were in it (but where, in fact, such a world does not exist and/or one is not in it).
BonJour proposes that analog explanations are preferable to digital explanations because digital explanations rely on two claims - that the corresponding analog explanation that is being emulated by the agent or mechanism could account for experiences, and that the agent or mechanism can successfully perform this emulation - whereas analog explanations rely only on the former claim. Since digital explanations add that extra claim, and because they are basically parasitic on analog explanations, they seem to be less likely to be true, and thus inferior. In addition, there appears to be no reason to choose one of the innumerable digital explanations to another, whereas the analog explanation they are all parasitic on is singular - it provides a much more simple and elegant explanation.
BonJour further claims that the only possible analog explanation is the quasi-commonsensical hypothesis; therefore it is the best possible explanation and we are thus justified in accepting it. With this final piece of the puzzle, BonJour has completed the construction of a basic internalist foundationalist system by which we can justify our beliefs about sensory experience (seemingly avoiding the dilemma raised to previous internalist foundationalist systems) and also justify our leap from sensory experience to the existence of a commonsensical three-dimensional physical world.
An Alternate Analog Explanation
Unfortunately, there are some holes in BonJour's proposal which on further reflection deem it less promising than it first appears. The most problematic area is BonJour's insistence that the quasi-commonsensical hypothesis is the only analog explanation that accounts for experience. His justification of the hypothesis itself appears to beg the question of the existence of the three-dimensional physical world in that it indirectly appeals to that world (through use of three-dimensional physical object terminology) to justify the putting together of two-dimensional images in a way that is isomorphic to the three-dimensional world in question.
As BonJour himself concedes, the justificatory force of experience in relation to physical object claims depends on the experience being describable independent of three-dimensional physical object terms. He uses them in his account only as a sort of shorthand, claiming that an account can be given without relying on such physical object terms. That does not appear to be the case, however.
Consider an example, where a person is experiencing what common sense would consider looking at a box while circling it. Independent of physical terms, the experience is only a two-dimensional 'picture' within which certain colors are perhaps set off distinctly from certain other colors in a roughly square shape. This experience alone obviously does not evidence the existence of any three-dimensional physical world.
Consider that later the person experiences a slightly altered 'picture', in which the square shape set apart by its difference in color from the rest of the 'picture' becomes a non-square parallelogram. Later, the shape set apart in the picture is altered further, and as more time passes, yet further. Soon, the shape begins to coalesce back into its original square shape. Let us say this process repeats three more times. Again, the fact that the set-apart shape's shape changed in time and even repeated does not seem to reflect a changing-point-of-view movement around a three-dimensional object any more than it does a static-point-of-view view of a two-dimensional 'picture' that changes with time.
Someone in BonJour's place might add that the part of the two-dimensional picture outside of the set-apart-by-color shape changes as well, and that it changes in such a way that, after a time, certain 'pictures' (that is, certain patterns of color/shape) repeat. This might be construed as evidence that the person experiencing these 'pictures' has changed their point of view in a three-dimensional world (moved so that the background changes more than the foreground), and indeed the experiences do fit coherently with such a view. However, and this is the key part, the experiences fit just as easily with a two-dimensional static-point-of-view 'picture' which changes as time passes.
A three-dimensionalist might ask what causes those changes, especially what causes the repetitions or even just similarities in the 'picture', but this question is immaterial. One might say, for example, that the picture is simply a solopsistic mind's dreamscape and that the mind enjoys certain patterns (or for that matter, sets of patterns, to account for repetition in change between 'pictures' as well as single 'pictures') and so chooses to repeat them. Any number of other accounts might be given to explain the patterns in the experiences a person has as viewing a changing two-dimensional 'picture' from a static point of view, and none of them can be ruled out based on any criteria BonJour presents. This two-dimensional explanation in fact appears to be an alternate analog explanation to his three-dimensional one, and there is no reason to prefer the three-dimensional one short of begging the question and assuming three-dimensionality in the first place.
By switching accounts of experience from the physical object terminology BonJour uses for shorthand to a more phenomenological terminology, it becomes clear that there is nothing inherent in a person's experience to suggest the three-dimensional explanation over the two-dimensional one (or perhaps other more drastically different explanations that might come along). Therefore BonJour's explanation, while possible (in that it fits and explains sensory experiences), is not as justified as he claims. He has apparently managed to provide a justificatory basis for sensory experience, which is impressive, but that foundation seems insufficient to justify the intuitive, commonsense view of a three-dimensional physical world in which one moves around. BonJour's account simply does not seem able to go beyond sensory experience and justify our common sense perspective of what that sensory experience is or represents.