Sorry that it's been so long, you can partly blame health and partly my lack of focus.
Last time I blogged, I made an introductory post on the Principle of Sufficient Reason (PSR) and Principle of Causality (PC) and outlined where I intended to go. I said that my approach would be to start by considering the nature of axioms and then to attempt to establish that the PSR and PC should be considered axioms. For this post, I will attempt the first of these tasks (explaining what axioms are); my next post will attempt to make the case that the PSR and PC are axiomatic.
What are Axioms?
The basic function of logic and argument is to get from premises to conclusions. For example, if we start with the premises "All men are mortal" and "Socrates is a man" we can reach the conclusion "Socrates is mortal." Often, in a complex argument, our premises will be the conclusions from an earlier set premises. This process raises the question of where our first and original set of premises comes from.
These first premises are, in logic and maths, commonly called 'axioms'. Some philosophers use the term 'properly basic beliefs'. Axioms can't be proven in the ordinary way because, as I said, they are the original premises which all subsequent arguments depend on.
Common examples of what are often considered axioms, certainly things treated as axioms by Thomists, include the three laws of logic famously formulated by Aristotle ie. The laws of Identity, Non-Contradiction and the Excluded Middle.
Are Axioms Reliable?
Since axioms can't, in the ordinary sense, be proven, it might appear that they are arbitrary and that we can't have good reason for believing them true. I disagree. My position (I claim no originality here) is that there are axioms which we can know are true.
Thomists and other Aristotelians have traditionally argued that certain axioms are self evident. However, since many people have asserted that all sorts of things are self evident, there is an argument that helps establish the reliability of genuine axioms. Put simply a principle can be regarded as axiomatic and true if that principle is such that it needs to be presumed for any knowledge to be possible and/or if the principle must be invoked to argue against it.
An example will illustrate my point. I once got into an argument with someone who argued that the basic laws of logic have no objective basis. His argument went as follows: "If the laws of logic had objective basis the development that we have seen in the sciences would not have happened. The development that we have seen in the sciences has, however, obviously happened, therefore the laws of logic have no objective basis."
Obviously his opening premise is highly debatable, but his argument has a more serious flaw than that. In making his argument, he was himself invoking the Law of Non-Contradiction. This is a basic point commonly found with Aristotle's basic laws, any attempt to argue against them necessarily invokes these laws, thus assuming their validity. This, for Thomists, helps to underscore the self evidence truth of these laws.
Some, of course, will object that while this demonstrates the necessity of such axioms for human thought, it proves nothing about their objective validity. The argument may go that, yes, for human thinking to work, we must assume these axioms are true, but that gives no reason to think they correspond to any objective reality.
On this view, mathematics, logic, and the like, which all depend on these basic laws, are not necessarily connected to anything objectively verifiable but simply the working out of what follows necessarily from these axioms which are, as it happens, necessary for human thought.
I would point out, in response, that if these laws have no objective validity, then we can't even know what follows from them. We might say that "If the Law of Non Contradiction is True then it follows that 2+2 = 4 can't be both true and false." But in saying this we assume the validity of that Law. If these basic axioms are not true, then the whole edifice must collapse, being built on sand.
If these axioms are not reliable, we can't even know that we have no good reason to believe them because that conclusion is it's self based on reasoning based on these axioms; the argument "We only have good reason to believe the things for which evidence can be put forward, but no evidence can be put forward for these axioms" is based on the laws of identity and non-contradiction.
Put simply, these axioms are self evidently true and ought to be believed.
Last time I blogged, I made an introductory post on the Principle of Sufficient Reason (PSR) and Principle of Causality (PC) and outlined where I intended to go. I said that my approach would be to start by considering the nature of axioms and then to attempt to establish that the PSR and PC should be considered axioms. For this post, I will attempt the first of these tasks (explaining what axioms are); my next post will attempt to make the case that the PSR and PC are axiomatic.
What are Axioms?
The basic function of logic and argument is to get from premises to conclusions. For example, if we start with the premises "All men are mortal" and "Socrates is a man" we can reach the conclusion "Socrates is mortal." Often, in a complex argument, our premises will be the conclusions from an earlier set premises. This process raises the question of where our first and original set of premises comes from.
These first premises are, in logic and maths, commonly called 'axioms'. Some philosophers use the term 'properly basic beliefs'. Axioms can't be proven in the ordinary way because, as I said, they are the original premises which all subsequent arguments depend on.
Common examples of what are often considered axioms, certainly things treated as axioms by Thomists, include the three laws of logic famously formulated by Aristotle ie. The laws of Identity, Non-Contradiction and the Excluded Middle.
Are Axioms Reliable?
Since axioms can't, in the ordinary sense, be proven, it might appear that they are arbitrary and that we can't have good reason for believing them true. I disagree. My position (I claim no originality here) is that there are axioms which we can know are true.
Thomists and other Aristotelians have traditionally argued that certain axioms are self evident. However, since many people have asserted that all sorts of things are self evident, there is an argument that helps establish the reliability of genuine axioms. Put simply a principle can be regarded as axiomatic and true if that principle is such that it needs to be presumed for any knowledge to be possible and/or if the principle must be invoked to argue against it.
An example will illustrate my point. I once got into an argument with someone who argued that the basic laws of logic have no objective basis. His argument went as follows: "If the laws of logic had objective basis the development that we have seen in the sciences would not have happened. The development that we have seen in the sciences has, however, obviously happened, therefore the laws of logic have no objective basis."
Obviously his opening premise is highly debatable, but his argument has a more serious flaw than that. In making his argument, he was himself invoking the Law of Non-Contradiction. This is a basic point commonly found with Aristotle's basic laws, any attempt to argue against them necessarily invokes these laws, thus assuming their validity. This, for Thomists, helps to underscore the self evidence truth of these laws.
Some, of course, will object that while this demonstrates the necessity of such axioms for human thought, it proves nothing about their objective validity. The argument may go that, yes, for human thinking to work, we must assume these axioms are true, but that gives no reason to think they correspond to any objective reality.
On this view, mathematics, logic, and the like, which all depend on these basic laws, are not necessarily connected to anything objectively verifiable but simply the working out of what follows necessarily from these axioms which are, as it happens, necessary for human thought.
I would point out, in response, that if these laws have no objective validity, then we can't even know what follows from them. We might say that "If the Law of Non Contradiction is True then it follows that 2+2 = 4 can't be both true and false." But in saying this we assume the validity of that Law. If these basic axioms are not true, then the whole edifice must collapse, being built on sand.
If these axioms are not reliable, we can't even know that we have no good reason to believe them because that conclusion is it's self based on reasoning based on these axioms; the argument "We only have good reason to believe the things for which evidence can be put forward, but no evidence can be put forward for these axioms" is based on the laws of identity and non-contradiction.
Put simply, these axioms are self evidently true and ought to be believed.
