(Another recent post titled 'Man is the Animal that Speaks' is also related to some of the thoughts here.)
Whenever you hear the complaint, or reassurance, that logic cannot be proved or is unjustifiable, you are hearing a mystic speak. Mystics speak of things that are closed off from normal discourse, and this accounts for the rather indirect and puzzling ways in which mystics speak. Properly understanding the mysticism of logic can go a great length in clarifying if its mystery is necessary or an accident of history. My intention here is not to avoid the mystery, but to better see it for what it is and to suggest a new beginning. I mean to move very rapidly and in a very aggressive mode.
(If it is not immediately obvious why one may maintain that logic cannot be justified, here is the normal account put rather crudely: logic would require logic to justify itself, which is circular, therefore, impossible.)
Let us look at a 'logical law' which apparently cannot be justified. When someone says "there is a square circle" nothing occurs that would indicate a contradiction, yet we recognize it. Ultimately, it is because we cannot imagine such a shape that we encounter this as a contradiction, but we do not even go through the effort to try since these terms conflict formally (symbolically) for us now.
Why are the "contradictory" terms able to appear side by side with no complaint from reality? Put briefly, the words do not contradict absolutely, but only relative to their meanings in our attempt to imagine. If we were to substitute the meaning of 'red' for 'square' then we would no longer have difficulty imagining such an object. The process of being unable to imagine certain combinations of meanings has been generalized and formalized into logic of the sort that we encounter - and even worse, teach - as some sort of fundamental, unjustifiable, mysterious component of the 'rational'.
Now, we may of course ask: why must such justification be in terms of logic? It certainly seems that formality itself should not demand a formal proof, and that the lack of such a proof is not a problem on the part of formality. Maybe the person who demands the proof should be charged with inconsistency? I agree, and the above was meant to show this to some degree. However, it seems that the mystics have control over logic even in this counter complaint, and we should be very puzzled to find ourselves mystics ourselves as we oppose mysticism in logic.
The formalization of logic (possibly starting in Aristotle's Organon) did not intend to create a mystery cult, but only seemed to want to illustrate some of the existing norms for clear communication.
A long standing project of mine is to establish logic anew, this time keeping logic more closely tied to 'discourse' (λόγος). I have a hypothesis that a great deal of disagreements are really misunderstandings - in fact, I would maintain that most disputes in Philosophy are purely misunderstandings. These disputes are held up from being completed by the mysticism of logic. What would it look like to overcome these hurdles in the mysticism of logic? Perhaps people will be able to pursue wisdom with each other rather than have to characterize each other under 'isms' so that they can fit a nice mutual exclusion in formal logic? Maybe people will notice that these isms aren't opposed as much as people may secretly want them to be (why? In order to be correct? We affirm ourselves at the expense of others quite frequently).
As logicians we should be interested in clear discourse - discourse that serves its purpose of communicating, and helping people to get along and mutually benefit each other in cooperation. We should expect no way of showing that a person is correct or incorrect in what they say through any sort of process called logical, we can only interpret what we took it to mean, and how what it meant is something possible for us and the speaker.
If someone says, "there is a cup on the table, and there is no cup on the table", the general impulse is to see a contradiction and to throw the red flag labeled 'false'. However, logic can never show what is correct or incorrect. The mysticism of logic has blocked us off from clarify what it means for this sentence to be problematic. How does logic pretend to do this? We take the statement "the cup is on the table" and assign it the character 'A', and then "the cup is not on the table" is taken to be "~A". Then we interpret the sentence, formally, as "A & ~A", which is a manifest contradiction. We then stop here, self-satisfied in our grasp of the clever method of formalizing these statements.
If we do not wish to take the formal route, we can note that the meaning of the 'contradictory' sentence implies that if I look at the table I will both see and not see a cup there, and this is something that I cannot imagine. Perhaps this person intends to speak of two different tables, or two different cups? Perhaps the passage of time is left out or only implied in the sentence? There are other options for understanding the sentence, for who would say such a thing?
Try to imagine a being for whom a cup could be and not be on a table at the same time. While we certainly could not imagine the way this being experiences, we can at least posit such an being. The discourse available between us and this entity is problematic, and it will be hard to know the proper way to act on the suggestions of such a being. Now, given that we can imagine such a being, we can see that we find the impossibility of the statement above to be grounded on a certain way of understanding the possibilities for the person who says it to us: we assume that, like us, the cup can either be or not be on the table, and not both. This is where we have a chance to discuss the role of logic.
Logic will investigate the sorts of assumptions we have of other speakers and the ways in which beings like us can discuss things. Perhaps this is Aristotle's intention by his categories that illustrate basic kinds of determinations of objects. Logic can also have an empirical part which examines the ways in which people do tend to talk about things, and to this we can attach our formal apparatus (hopefully with some hesitation).
Here I would like to break off and change topics to a brief discussion of the ancient Greeks. The word we translate as truth from the Greek is alethia (ἀλήθεια). This word more literally means 'un-covering'. In Greek the word we translate as false is pseudos (ψεῦδος) which more literally means dissembling (lying). We can recognize right away how different these words are from our casual usage of their translated counter parts.
Plato, who did not have a science of logic, understood that the only thing convincing in discourse as what is true (what un-covers). Clearly he does not mean by truth here what we assign to propositions as a result of our formal process, or even correspondence; rather, Plato refers to how words are only understood on the basis of how things are un-covered for us (loosely, the way in which we can experience the world). Plato affirms that we cannot understand things in a way that is beyond our capacity to experience (except for negatively - by claiming our own finite nature). This is certainly something to meditate on as we try to found logic anew.
Whenever you hear the complaint, or reassurance, that logic cannot be proved or is unjustifiable, you are hearing a mystic speak. Mystics speak of things that are closed off from normal discourse, and this accounts for the rather indirect and puzzling ways in which mystics speak. Properly understanding the mysticism of logic can go a great length in clarifying if its mystery is necessary or an accident of history. My intention here is not to avoid the mystery, but to better see it for what it is and to suggest a new beginning. I mean to move very rapidly and in a very aggressive mode.
(If it is not immediately obvious why one may maintain that logic cannot be justified, here is the normal account put rather crudely: logic would require logic to justify itself, which is circular, therefore, impossible.)
Let us look at a 'logical law' which apparently cannot be justified. When someone says "there is a square circle" nothing occurs that would indicate a contradiction, yet we recognize it. Ultimately, it is because we cannot imagine such a shape that we encounter this as a contradiction, but we do not even go through the effort to try since these terms conflict formally (symbolically) for us now.
Why are the "contradictory" terms able to appear side by side with no complaint from reality? Put briefly, the words do not contradict absolutely, but only relative to their meanings in our attempt to imagine. If we were to substitute the meaning of 'red' for 'square' then we would no longer have difficulty imagining such an object. The process of being unable to imagine certain combinations of meanings has been generalized and formalized into logic of the sort that we encounter - and even worse, teach - as some sort of fundamental, unjustifiable, mysterious component of the 'rational'.
Now, we may of course ask: why must such justification be in terms of logic? It certainly seems that formality itself should not demand a formal proof, and that the lack of such a proof is not a problem on the part of formality. Maybe the person who demands the proof should be charged with inconsistency? I agree, and the above was meant to show this to some degree. However, it seems that the mystics have control over logic even in this counter complaint, and we should be very puzzled to find ourselves mystics ourselves as we oppose mysticism in logic.
The formalization of logic (possibly starting in Aristotle's Organon) did not intend to create a mystery cult, but only seemed to want to illustrate some of the existing norms for clear communication.
A long standing project of mine is to establish logic anew, this time keeping logic more closely tied to 'discourse' (λόγος). I have a hypothesis that a great deal of disagreements are really misunderstandings - in fact, I would maintain that most disputes in Philosophy are purely misunderstandings. These disputes are held up from being completed by the mysticism of logic. What would it look like to overcome these hurdles in the mysticism of logic? Perhaps people will be able to pursue wisdom with each other rather than have to characterize each other under 'isms' so that they can fit a nice mutual exclusion in formal logic? Maybe people will notice that these isms aren't opposed as much as people may secretly want them to be (why? In order to be correct? We affirm ourselves at the expense of others quite frequently).
As logicians we should be interested in clear discourse - discourse that serves its purpose of communicating, and helping people to get along and mutually benefit each other in cooperation. We should expect no way of showing that a person is correct or incorrect in what they say through any sort of process called logical, we can only interpret what we took it to mean, and how what it meant is something possible for us and the speaker.
If someone says, "there is a cup on the table, and there is no cup on the table", the general impulse is to see a contradiction and to throw the red flag labeled 'false'. However, logic can never show what is correct or incorrect. The mysticism of logic has blocked us off from clarify what it means for this sentence to be problematic. How does logic pretend to do this? We take the statement "the cup is on the table" and assign it the character 'A', and then "the cup is not on the table" is taken to be "~A". Then we interpret the sentence, formally, as "A & ~A", which is a manifest contradiction. We then stop here, self-satisfied in our grasp of the clever method of formalizing these statements.
If we do not wish to take the formal route, we can note that the meaning of the 'contradictory' sentence implies that if I look at the table I will both see and not see a cup there, and this is something that I cannot imagine. Perhaps this person intends to speak of two different tables, or two different cups? Perhaps the passage of time is left out or only implied in the sentence? There are other options for understanding the sentence, for who would say such a thing?
Try to imagine a being for whom a cup could be and not be on a table at the same time. While we certainly could not imagine the way this being experiences, we can at least posit such an being. The discourse available between us and this entity is problematic, and it will be hard to know the proper way to act on the suggestions of such a being. Now, given that we can imagine such a being, we can see that we find the impossibility of the statement above to be grounded on a certain way of understanding the possibilities for the person who says it to us: we assume that, like us, the cup can either be or not be on the table, and not both. This is where we have a chance to discuss the role of logic.
Logic will investigate the sorts of assumptions we have of other speakers and the ways in which beings like us can discuss things. Perhaps this is Aristotle's intention by his categories that illustrate basic kinds of determinations of objects. Logic can also have an empirical part which examines the ways in which people do tend to talk about things, and to this we can attach our formal apparatus (hopefully with some hesitation).
Here I would like to break off and change topics to a brief discussion of the ancient Greeks. The word we translate as truth from the Greek is alethia (ἀλήθεια). This word more literally means 'un-covering'. In Greek the word we translate as false is pseudos (ψεῦδος) which more literally means dissembling (lying). We can recognize right away how different these words are from our casual usage of their translated counter parts.
Plato, who did not have a science of logic, understood that the only thing convincing in discourse as what is true (what un-covers). Clearly he does not mean by truth here what we assign to propositions as a result of our formal process, or even correspondence; rather, Plato refers to how words are only understood on the basis of how things are un-covered for us (loosely, the way in which we can experience the world). Plato affirms that we cannot understand things in a way that is beyond our capacity to experience (except for negatively - by claiming our own finite nature). This is certainly something to meditate on as we try to found logic anew.
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