This question asks us to examine introspection, the process (literally) of looking inwards. Further to this, in mentioning that we should start from our own case it implies that this process will be used for further purposes. This assumes foundationalism or at least a foundationalist approach.
Introspection is a process that we engage in frequently; if someone asks our opinion we must examine our mental states to ascertain the answer. In philosophy introspection has been used as a method to determine facts about people and the world around us. The special philosophical qualities attributed to introspection include that knowledge obtained in this way is by some special method or has particular reliability or certainty.
Essentially the logic of introspection can be seen in the following way:
Uncertain epistemic knowledge > introspection provides certain knowledge > stable base for all knowledge
This illustration also highlights one of the key drivers for the theory, that of scepticism. Introspection was seen as infallible and omniscient, thus providing certainty compared to other epistemic knowledge.
This in turn highlights that introspection was seen as different from normal observation. Essentially the observation of mental states was seen as direct, without any intervening mechanisms. This metaphysical point made an individuals inner world distinct from his observation of the outer world. This can be partially attributed to a dualist perspective.
This leads us to our final observation on the philosophical attributes of introspection: that mental states are private and 'belong' to the individual. This again has its roots in dualism, in the description of the ontology of mind, as material monist theories do not allow for privacy of mental states
Taken together these empirical, metaphysical and ontological considerations show how introspection was seen to be of such import.
The search for infallible knowledge was driven by foundationalism, which held that all knowledge could be built upon a fact that was indubitable and incorrigible. Introspection held the promise of being the tool that could provide that knowledge.
Finally due consideration of religion as an influence should be given as God was invoked (e.g. by Descartes) to make the logical leap between inner and outer knowledge. Without God this leap might not have been seen as so probable.
Introspection as a method has a considerable advantage as a tool because all of us possess it. It can be used simply and easily. It also requires deep thought and a quiet place in which to think - this has a certain romantic appeal to philosopher and public alike.
The role of introspection as more reliable is also intuitively appealing. The fact that you are an authority on your own inner states is born out in practice on a daily basis. This is even enshrined in language - to challenge someone over a state that they ascribe to themselves (e.g. "no you don't feel that") is simply not considered meaningful.
The logic applied to the use of introspection to glean reliable knowledge - as Descartes postulated with the cogito (I think, therefore I am) also seems entirely straightforward and reasonable. In associating reason and logic with the mind, surely when we look within the mind we must find truth and light?
Needless to say introspection as a method and as a starting point is not perfect. In order to understand the objections we must first understand the logical implications of starting from introspection.
If the starting point of the examination of the world is internal, then the inherent assumption is that my access to my internal states is privileged in that they are mine, only I have access to them and I know them all without fault. However this forms the limits of my introspection. No one else can know my states and so by implication I can never know their states.
This leads to solipsism: in being certain of my own mind I can never leave its confines. The entire world remains doubtful to me. I can never know if the people around me also share this predicament because I cannot know if they have minds or are merely automatons. This forms the fundamental problem of philosophy: the problem of other minds.
Many philosophers have attempted to solve the problem of other minds. The responses and counter responses are essentially based upon attempting to prove that knowledge of the inner can be applied to the outer in an attempt to solve the problem of other minds.
This is the one fundamental objection to the use of introspection that transcends all others. The use of introspection as a starting point creates the problem of other minds, thus to solve the problem of other minds no amount of reasoning on how to apply knowledge of the inner to the outer will solve the issue. The only way to tackle the problem of other minds is to attack the use of introspection as a starting point itself.
In considering the basis for why one might use introspection we have touched upon ontological, metaphysical and epistemic reasoning. However, taken individually or together each of these viewpoints lead inexorably to solipsism and the problem of other minds because each creates a division between the inner and outer worlds.
The epistemic approach assumes that mental states are objects awaiting perception. Mental states cannot be objects of knowledge as they are illusory with no real existence. In addition, if it makes no sense to doubt mental states then it is also impossible for them to be certain. Neither certainty nor doubtfulness can apply to something that does not materially exist!
This leads us on to the ontological confusion of possession. This assumes that people own mental states. However, we don't 'have' a thought in the same way that we 'have' a car or house or bank account. Mental states do not have existence in their own right, you cannot pluck one from your head and place it on the table, and in the same way it is absurd to talk of owning them.
Finally the metaphysical approach assumes that there is a difference between inner and outer states. Now if we attempt to start from our own case and move outwards then this will appear to be the case, However we could equally attempt to start from the viewpoint that there is one case and we have one perspective upon it.
These key arguments are from Wittgenstein who proposed the private language argument to solve the problem of other minds. Wittgenstein saw 'epicycle upon epicycle' of modification upon the same original idea: that of starting with our own case. To solve the problem of other minds that was thus created he attacked the very idea of introspection as a philosophical tool.
Wittgenstein highlighted the impossibility of a private language. The fact that we communicate successfully using language shows that we must experience the world in the same way as if not how could we attach the same words to the same concepts? Meaning of words must be public in order for communication to take place.
Wittgenstein continued that psychological concepts are not acquired by abstraction from one's own case, neither are they inherent at birth. Instead these concepts are social, communicated between and learned from others and have a specific linguistic and cultural context.
This reasoning has two implications: firstly that the problem of other minds need not exist if we regard the single source of our knowledge of ourselves and of the world, namely language. This is the source of knowledge for both first person and third person perspectives - there is no metaphysical divide because the source of both is the same thing!
Secondly, the fact that the ontological, metaphysical and epistemic assumptions led to introspection and thus solipsism and thus the problem of other minds was the result of a confusion of language. We might say that we 'have' an idea, but this is not true in the sense that we might 'have' an object. Although the same word is used, the meaning in one case should be different. In the same way referring to the observation of mental states objectifies them and thus implies properties. In fact the states do not have existence in their own right. Lastly the assumption that language and concepts were somehow separate led to the assumption that there was a fundamental division between first and third party perspectives.
Thus, in assessing of the view that we start from our own case we can conclude that starting with introspection leads to solipsism and the problem of other minds and that the only way to solve this problem is to reject introspection as a starting point. A public, shared meaning must be the starting point for any useful knowledge about the world.
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