A Socratic Dialogue
By BRENT SILBY
Through this dialogue we see the problem that arises when we take a relativist stance to truth. Many people have taken a liking to relativism; perhaps because it seems so wonderfully democratic. However, the further one goes down the relativist road, the more difficult it becomes to answer fairly straightforward questions. It is almost as if the relativist tries to use logic to argue that logic doesn’t work.
Persons of the dialogue
John: Western Philosophy is really nothing more than opinionated people talking opinion.
Socrates: Doesn’t philosophy involve forming rational argument? This would seem to be quite different to simply voicing opinion. Is a rational argument the same as an opinion?
John: Argument can be reduced to premises, which can further be reduced to more basic premises. As we repeat this reduction, we find ourselves with the most basic axioms. The axioms of most frameworks are fundamentally opinion are they not? In this way, rational argument amounts to not much more than glorified opinion.
Socrates: Can you think of a scientific axiom or mathematical axiom that is opinion-based rather than reason-based?
John: Wolfram Mathworld defines an axiom as “a proposition regarded as self-evidently true without proof”
Socrates: Such as x=x? This is considered self-evidently true. But is it simply opinion? If so, wouldn’t there be some subjectivity or uncertainty about whether or not it is true?
John: This is just accepted uncertainty, because the results of making this assumption are useful. The results of not making this assumption might also be useful.
Socrates: I’m assuming we can agree that opinion carries a level of uncertainty and subjectivity, right? You suggested that self-evident truths have an accepted level of uncertainty, and are therefore opinions? Is that what you’re saying? I don’t want to characterise your position.
x=x is an example of a self-evident proposition, right? Is there a level of uncertainty in the claim that X=X? If so, what is the nature of that uncertainty?
John: I was saying axioms (in mathematics) are equivalent to opinions, because they are subjective, unprovable. Now, x=x is only subjectively self evident. You’re so accustomed to that useful axiom/ opinion that you probably assume its certainty.
Socrates: Certainly I am used to assuming that x=x is self evident. You’re suggesting that I’m assuming its certainty out of habit and that it is, in fact, only subjectively self evident. But I would have thought that something is self evident if its truth is implied by the definition of the terms used within its formulation. In other words, the evidence for its truth is contained within itself. Isn’t it built into the definition of the component terms of x=x that it is self evident, i.e. true by definition?
Your claim that x=x is only subjectively self evident is interesting. I’m wondering how you have come to know that it is only subjectively self evident. Would you be able to show me why x=x might only be subjectively self-evident? If its truth subjective, then it is possible for it to be false, right? Can you show me how it could be considered false? Perhaps you could provide an example in which x=x yields a contradiction. This would help me understand why you think x=x is only subjectively self-evident.
John: No clear example sorry Socrates. I know it’s a core axiom though. I don’t think it’s one that you’d name casually.
Socrates: I agree. I think it is a core axiom and a good example of a self evident truth. Okay, without a clear example of possible falsehood in this case we cannot say definitively whether or not it is subjective. It is therefore possible that this is an example of a core axiom that is not simply a matter of opinion.
Shall we consider another self evident truth: There are no square circles. Would you say that this is a subjective claim?
John: People with a lot of expertise have attended to what an axiom is and what it means quite closely. If the definition does not have a proof but is regarded as a truth doesn’t sound like opinion to you, then we can’t proceed. If there is a possibility it could be, how about we test where that would go so this conversation has some movement.
Socrates: Wouldn’t the term self evident imply the proof is contained within itself? Axioms are foundations upon which reasoning can be built. It seems they are considered axioms because their truth is so obvious (i.e. built into the definition of terms used) that they need no further proof. They could be considered fundamental truths.
You have suggested that axioms (or self evident truths) are simply a matter of opinion. The implication here is that they are subjective. If this is true, I would expect the possibility of their falsehood. Is it enough to just assume they might be false? Upon what is that assumption based?
So, to move forward. I’m wondering, does the claim that an axiom “does not have a proof” (as you stated above) amount to the same thing as an axiom does not require a proof, as in the definition I recently read. Could it be the case that axioms do have proofs, but when used in argument the proof is not explicitly required because it can be safely assumed?
John: In mathematics proofs are built from axioms and proofs already proven. Axioms so not need proof once you’re underway because they are your initial assumptions. This doesn’t inherently make them true or proven. Usually axioms would not be so badly chosen as to be a meaningless start point to build all following deductions from.
Socrates: That makes sense. The axioms are the initial assumptions, which are accepted as true, and upon which reasoning follows. As you say, their acceptance doesn’t inherently make them true or proven. Then again, the basic assumptions are, as you indicate, presumably not chosen at whim. So they are, perhaps, accepted as true with good reason. Perhaps this is where the term self evident comes in.
Okay, so the thought is that because axioms do not require proof they are therefore subjective matters of opinion, right? This, itself, would appear to be an axiom that is placed prior to all other axioms. Perhaps the most fundamental of all axioms is: all axioms are subjective opinions. If I understand what you’ve said throughout the discussion, this is a position you would agree with, right?
Socrates: Thanks John. So we have now defined the most fundamental of all axioms as: axioms are subjective opinions.
But what does this mean? If we apply this axiom to itself, we find that this most basic of axioms is, itself, just a matter of opinion. Does it not follow that it could be false? If so, it could be the case that this basic axiom is not merely a matter of opinion? But wouldn’t that make it a counter example to itself? Wouldn’t this be like saying: The fundamental axiom is that all axioms are matters of opinion. But being a matter of opinion means this axiom could be false, in which case this axiom is not a matter of opinion. This is a contradiction because the axiom states that all axioms are matters of opinion.
I think we might call this a reflexive paradox. Are we getting into trouble here?
I think you made a good point above when you indicated that axioms are chosen with good reason. I agree with that point. If you are correct, wouldn’t we expect the most fundamental of all axioms to be free of contradiction?
John: You’ve not stated all your assumptions. Might i be correct in assuming you’re adopting some assumption like: something must be either true or false, it can not be both or neither.
I think more along the lines: if this is a base assumption, what perspectives and results does it offer and are they new or useful.
This sounds a lot like theory of what is knowable. Did anyone get anywhere definitive on that?
Socrates: You mention the possibility that a base assumption could be both true and false. I wonder, is that true? Or is it false? Does it make sense to say it is both true and false that our base assumptions can be both true and false?
I have no assumptions. I am an explorer. In order to find suitable assumptions, a starting point would be to see where your current assumptions lead us. I am therefore working from your assumption that the most fundamental of all axioms is that axioms are subjective matters of opinion. It follows from that axiom that axioms might not be matters of opinion. In other words, the assumption that axioms are matters of opinion could be false. Can we agree on this?
John: Sure, that axiom is an opinion. Are you assuming a reflexive type axiom where you’ve decided you’re allowed to apply axioms to themselves? I don’t agree. I guess you’re going to suggest logicaly nullifying the axiom because it has lead to contradiction / paradox in a particular case. This is a valid approach, if we’ve agreed to assume that is how to interpret this type of result. But, a paradox / inconsistency does not have to imply a broken axiom.
Socrates: If we agree that all axioms are matters of opinion (I think this is what was stated above), and if this is itself an axiom, then doesn’t it follow that it should apply to itself? Perhaps you could re-word the axiom to read: all axioms except this one are matters of opinion. Is that better?
John: You ask “doesn’t it follow”? No, it does not follow. When you ask this, you are making assumptions. We haven’t yet agreed how axioms interact or what our logic is. I’m assuming there is more than one logic possible
Socrates: Could you explain why it does not follow? Perhaps an example would help. Could you find me an example of a set in which all members have property X, and one member does not have property X? That would help me understand your position.
John: Perhaps you could provide an example of a fact you hold as true which does not sit on some axiom of belief.
Socrates: It is a little premature to move on to this question. We have not yet finished exploring the question as to whether or not axioms are a simply matters of belief (or opinion). So, shall we put your question to one side until we settle on what an axiom is?
I note that you have not yet answered my earlier question. Recall, I suggested that if all axioms are matters of opinion, and if this is itself an axiom, then doesn’t it follow that it should apply to itself? Your response was “no”. I then asked why it does not follow. This has not been answered yet. I’m also interested in seeing an example of one of these “other logics” you mention.
John: (no response)
Socrates: Let’s pause here and summarise our discussion so far. I think a recap is important to ensure we do not skip past any important questions or issues. Here is a summary of what we’ve covered.
You began with the claim that Western Philosophy amounts to nothing more than “opinionated people talking opinion”. I suggested that Philosophers form rational argument, which is quite the opposite to voicing opinion. I asked whether you think rational argument is the same as opinion. Your response was that axioms (i.e. the foundation upon which arguments follow) are fundamentally opinion. You then stated that an axiom is a proposition regarded as self-evidently true without proof.
I asked whether an axiom such as x=x is just opinion, i.e. that there is a level of subjectivity or uncertainty as to its truth. You responded by suggesting that x=x is only subjectively self evident. In reply I proposed that its self evidence is built into the definition of its component terms. I asked you to show me a way in which x=x could be considered false. The existence of such an example would show that its truth is subjective (i.e. a matter of opinion). You could not provide an example. I then asked whether the self evident truth there are no square circles is a subjective claim. You did not answer that question.
As we moved on, you indicated agreement with the notion that the most fundamental of all axioms is: all axioms are subjective opinions. I then pointed out a problem with this position, because as an axiom, it would apply to itself, which would mean it could be false, i.e. it would serve as a counter example to itself.
You didn’t agree that this was a problem and suggested that it would only be a problem if we assume that axioms could apply to themselves. However, I responded that if the axiom states that all axioms are matters of opinion, then it must follow that it applies to itself. But you said it doesn’t follow. You went on to suggest the existence of other logics.
I asked why it doesn’t follow. I asked for an example to help make the point. The type of example I asked for was one in which a set in which all members have property X, contains one member that does not have property X. You did not give me an example.
Here are my questions that remain unanswered.
1. show me a possible scenario in which x=x is false
2. is there are no square circles a subjective claim?
3. why should the axiom “All axioms are matters of opinion” not apply to itself?
4. what is an example of a set in which all members have property X, and one member of that set does not have property X?
5. What is an example of “other logics”?
You said earlier that this looks like the beginning of a theory of knowledge. If so, would it not be wise to have answers to these questions before proceeding? If we leave these questions unanswered, we might find ourselves in hot water further down the line when they inevitably come back to haunt us.
John: Socrates, at this time I cannot answer those questions.
Socrates: Do you think the difficulty in answering such questions may stem from your assumption that axioms are only subjectively true, i.e. matters of opinion?
John: I may not be able to answer the questions above, but it does not follow that axioms are anything more than subjective opinions.
Socrates: Perhaps. Nevertheless, you cannot answer my questions. Now, would you agree that these questions would pose no problem if we accept that there are some basic axioms that are more than mere matters of opinion? In other words, these questions become irrelevant if we accept the existence of fundamental axioms that have an objective truth value?
John: I would have to agree that these questions would pose no problem if the basic axioms were not subjective matters of opinion.
Socrates: Given the difficulty in answering the questions above, would you agree that it is possible that the basic axioms upon which we build reasoning are objective truths, and not merely matters of opinion?
John: The way you formulate it, I have to agree that it is possible.
Socrates: If forming a rational argument involves logic, and if logic is based upon fundamental axioms, and if axioms might be objective truths, would you accept that there is a possible difference between a rational argument and an opinion?
John: Again, the way you have constructed this, I have to accept the possibility.
Socrates: So it is possible that Western Philosophy is not simply “opinionated people talking opinion”?
John: It is possible.
Socrates: Shall we accept this conclusion for now and adjourn until the above questions can be answered?