Ravages‘ post about god and religion (trackbacked to my previous post) has attracted a lot of debate. The one most interesting to me was about logic. It is amusing how engineering, scientists swear by logic like it was the holy grail of truth and knowledge. I used to be one of them, till I read the Robert Persig book ‘Zen and the art of motorcycle maintainence’. In that book Phaedrus questions and challenges rationality itself. That got me thinking about logic and its validity. How logical is logic itself?
Fortunately for me, I am a computer science student, and I have the technical background to take a crack at this question. I know, a student of philsophy would be better equipped than I am, but when you are dealing with computational theory the lines get blurred, exteremly blurred. One of the assistant professors in computer science did his bachelors in philosophy and while pursuing his masters, specialized logic and stumbled upon computer science, going on to do a PhD in it. One of the best teachers I have taken a course under.
What is logic? Unfortunately we enounter a stumbling block at the very begining here. There no universally accepted definition for logic. The most general one I could come across was in MSN Encarta. It says “Logic is a science dealing with the principles of valid reasoning and argument. The study of logic is the effort to determine the conditions under which one is justified in passing from given statements, called premises, to a conclusion that is claimed to follow from them.” This is a fairly generic definition that includes most, but not all, of what logic is. This also gives me enough ammo to attack it with.
Traditionally, logic is bivariate. That is to say that there can be only two values to a statement, viz., ‘true’ and ‘false’. Logic also says all that is not false is true. It works well when you are dealing with the limited universe as defined by your premises. But the real question is, how valid can this be in reality? Not very. Consider the statement ‘It will rain tomorrow’, is it true? It is possible, but we cannot say it is true. This is not allowed in traditional logic. A statment can either be true or false, nothing else. How are we to try and understand the world, the natural, the supernatural or the divine with something as limited as this?
Granted, that we can do better than this. We do have multi-valued logic systems. But they fail too. They cant take you very far without stumbling on the rules that they created themselves. To illustrate one such example with bivariate logic, consider the statement ‘This statement is false’. Its simply a statement talking about itself. Can we assign a truth value to it? Just try and you will realize that it is not possible. You will run into contradictions either way. How useful is a tool that isnt powerful enough to talk about itself?
One might argue that is indeed the nature of all tools and intelligence. We havent been able to figure out how our own brain works, and logic is an invention of man, how can u expect logic to be able to explain itself? Going by that argument, our quest for Truth is an excercise in futility because the Truth is much bigger and more profound that ourselves, and if we arent powerful enough to understand how we work, then how can we be expected to understand Truth?
Going back to the definition, logic as a system can be employed only under the existance of premises, i.e., statements that are assumed to be true. In mathematics, they are refered to as axioms. What if we dont have any assumptions? What we seek is the ultimate truth, the truth above and beyond all truth. We wish to gain the knowledge that will explain the universe we live in, starting from nothing, void, no assumptions. Call it what you wish, enlightenment, nirvana, Truth, God. You choice of the name is your choice of the path that you wish to follow to get to the destination. How do you employ logic here? It isnt even valid anymore.
The natural question to ask is, how does logic work so well for science, if it fails miserably trying to explain nature? After all, isnt science a quest for understanding nature? Logic does remarkably well for science and engineering simply because the language and universe of science and engineering is mathematics. It is a severly limited universe, and one invented by man. My math professor said it best when he explained “What are numbers? Numbers are things you do arthmetic with. What is arthmetic? Arthmetic is what you do with numbers.” That’s how math starts off. Those are its assumptions – existance of numbers and arthmetic.
This makes me question science as a tool to understanding the unverse. It is such an adhoc, weak and approximate means. It isnt even consistent with itself. In fact Godel, one of the most influential logicians of the twentieth century, proved that any self-consistent recursive axiomatic system powerful enough to describe integer arithmetic will allow for “true” propositions about integers that can not be proven from the axioms. So science cannot be right about things simply because the logic system that it relies on is not consistent, is flawed. Science was always a quest for explainations, never the truth anyway.