What is truth TLDR: Truth should be viewed as a defined concept, rather than one that inherently exists in logic I enrolled in non-classical logic because I was interested in the philosophy behind logic, so I was rather disappointed when I discovered that the course was nearly entirely math. Despite this, I have come to a much better understanding of what it means for something to be. The mantra of postmodern theorists is that truth is a construct and I was quite surprised to discover that this is true in a sense, but not in the sense they meant. When trying to understand truth, the actual structure of the world is actually irrelevant for the most part (Ill hold off addressing this unless someone asks me in the comments), so we can work within a simple model. We define a four dimensional world - with three spatial dimensions and a fourth dimension for time. Our universe is fully described by providing a collection(possibly infinite) of co-ordinates in the four dimensional world where there are particles and labelled each point in the set with the kind of primitive particle at that point. We can then define the meaning of various terms such as cat or dog by defining certain a collection (typically infinite) of states, where each state is a group of particles that corresponds to the desired category. Here we could (theoretically) answer the question Is collection A of particles a cat? by seeing if the collection corresponding to the category cat contains a set of particles in the same configuration as A(1). In this model, truth is well defined - to say that a proposition about a collection of particles is true is to simple say that the particles are in the appropriate set. We call these simple set membership tests primitives Next we add the operations and, or and not. This allows us to construct formulas such as: a (set of particles) is an apple (in the apple set) and b (set of particles) is a banana (in the banana set). Before truth was simply defined by primitives - membership in a set. Now truth is derived by combining the truth value of primitives together. We have extended our original notion of truth. Next we add more operations such as for all and there exists, this allows us to make comments about the state of the world as a whole. This is called first order logic. Not all of these questions are answerable, but for all of them the truth value is well defined. Next we attempt to extend logic by allowing statements to refer to themselves. This is where we run into trouble. This statement is false is the classical example - whether we set it to true or false we run into a contradiction. At first the idea that there are statements that are not true and not false is mindblowing, but it seems much less so if we consider the statement, I am a blaugh, where the word blaugh is undefined. But if we want to truly understand we this statement is false cant be defined, well have to dive in a bit further. Consider the statement, A, Statement B is false. We look at statement B, determine its truth value and then use it to determine the truth value of A itself. Statement B may refer to more statements itself, so we may have to follow the path through statements C, D and E before we have a solution. As long as this path concludes at some point, it makes sense to extent the definition of truth to handle these kinds of statement. But if the path never ends, then talking about truth makes no sense. So we can only use this algorithmic definition to extend truth to non-self-referential sets. But what about those self-referential statements that can be assigned a truth value - This statement is either true or false. Here we could extent our definition of truth to cover these statements by allowing us to substitute both true and false in and if exactly one of these is consistent, to take that as the truth value. Once we adopt the framework of truth being a matter of definition, rather than truth being something that already exists and which we discover, these paradoxes become much less mysterious. If we get a contradiction, it is because we extended truth in flawed manner. Perhaps there is a definition of truth that covers every case, perhaps not - I havent done enough logic to be able to answer this question. But the reason why the view of truth as a concept existing in its own right is so popular is because after each extension we end up with a general definition that covers the previous case. And this view of truth really shouldnt be an automatic assumption. (1) When we are answering whether a collection of particles is a cat, it is implicitly assumed that all particles have the same time co-ordinate. When we are answering whether a collection is an event, such as a volcanic eruption, we actually need to consider particles with different time co-ordinates. Also, here it is convenient to define an equivalence class of states. This just means that we say some states - those we can obtain from each other by moving the object or changing the time - are equivalent. Then we can define the collection of particles for a category, such as cat, by providing simply the relative position of particles. Otherwise, wed have to say that collection X of particles on Thursday is a cat and collection X of particles on Wednesday is a cat and collection x is a cat when it is centered on coordinates (0,0,0) and also a collection when it is centered on (1,1,1)
Posted on: Tue, 24 Jun 2014 06:04:13 +0000
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