In the Beginning (there were symbols)

For no good reason


*note this is really for my own personal reference—I figured it would be a good way to cram certain concepts into my noggin, and being the layperson that I am at some of this stuff, having to explain it on this page should help. So… here goes.

Predicate Calculus. Symbols. Symbolization. Commutativity. Gödel.

Okay, so we’ve got some symbols, namely:

‘~’ (tilde) stands for what we would commonly think of as ‘not’ or ‘negation’, i.e., “TheZenFly is ‘not’ a penguin” may be symbolized as “~P”, where “P” stands in place of the predicate.

‘→’ (conditional) usually sits in the middle of a conditional statement (If P then Q), such as: “If TheZenFly is a penguin, then Superman is a monkey”, Thus symbolized as “P→Q”, where ‘P’ stands for ‘penguin’ and ‘Q’ stands for ‘monkey’.

‘&’ or ‘^’ (“and”) is used to connect two statements (yeah, English grammar), such as “P ^ Q”… or “TheZenFly is a penguin ‘and’ Superman is a monkey”.

‘V’ stands for the common usage of the word ‘or’, again connecting two separate statements, i.e., “Either TheZenFly is a penguin ‘or’ Superman is a monkey” (P V Q).

One should note that ‘or’ is used in the sense of ‘one or the other, or both’—in this sense, it is an ‘inclusive or’, meaning that statements on either side of the connector (V) could both occur to make the whole true. The symbol ‘V’ with a line under it would indicate a function of ‘one or the other, but not both’—in this case the connector (V) would be considered ‘exclusive’.

I should note that this symbol isn’t really a V, but nonetheless resembles one, and this is all I currently have at my disposal for proper symbolization…

‘↔’ stands for ‘if and only if’, and again, sits between two propositions, i.e., “TheZenFly is a penguin ‘if and only if’ Superman is a monkey” (P ↔ Q)

digging deeper…

Variables, typically symbolized by lowercase letters (x, y, z, a, b, c,…), are meant specify individuals, or ‘subjects’, in the grammatical sense. In the above examples, ‘TheZenFly’ and ‘Superman’ could be symbolized as such: ‘x (TheZenFly) is a P’, and ‘y (Superman) is a Q’ respectively.

Okay, a couple more—noting that I can’t symbolize existential quantifiers because my word processor insists on typing in Greek after entering one (I hope to resolve this minor annoyance at some point). So we’ll just say that:

“For All” is typically symbolized with an upside-down ‘A’, and is meant to specify, as the name implies, all members, or subjects (propositional variables) of a particular set of objects, hence: “For All x, Px” (for all ZenFlys, ZenFly is a penguin, or ‘Ax (Px)’)

“There Exists” is typically symbolized with a backwards ‘E’, and is meant to specify a particular member of a particular set of objects, e.g., “There Exists an x, such that Px” (There exists a ZenFly such that TheZenFly is a penguin, or ‘Ex (Px)’

I’ll have to come up with some better examples…

Also, it occurs to me that I’ve read entire books to prepare myself for this, and it is hard to explain without writing a book myself. In other words, there is a lot of jargon and other hoo-ha that I won’t be able to spend time explaining, though I’ll do my best to stay on planet earth for whoever might be reading. The above mentioned symbols/concepts are essential, so please, bear with me.

Okay, so… certain Mathematical operations can be symbolized using the above mentioned terms. For example we know that addition and multiplication is commutative (1 + 2 = 2 +1). When speaking in terms of propositions instead of numbers, we can say the same of propositional variables, e.g., AxAy (For All x and for All y) (x^y = y^x).

Similarly, we could say that multiplication distributes over addition with the symbolization: AxAyAz, x(y^z) = (x .y) ^(x .z)

So what’s the point? Well, such a system of symbolization can be used to exemplify the logical relations between propositions, kind of like packing propositions into little boxes for the purpose of ridding a set of a propositions of all the gobbly-gook, so that once one is aware of the concepts involved with the symbols, they can easily check the logical validity of such relations. Gödel, the main focus of this project, came up with a couple proofs (involving his incompleteness theorems), showing that any kind of axiomatic system with which one would claim to base any kind of Mathematical truth is necessarily incomplete, or lacking, as it were. In other words, there is no such thing as a logically ‘airtight’ system—there will always be a hole in one’s theories. This is apparently pretty huge. Why? Because it calls everything into question. Ultimately, all of the Mathematical ‘truths’ that we claim to be ‘absolute’ or ‘irrefutable’ are necessarily refutable in some way. I’m guessing this has something to do with the issue of self-reference and the assumptions that one must start with to get their proverbial feet off the ground (this, of course, is just an intuitive guess).

Alas, that’s about as far as I’ve gotten, and Gödel’s proofs are extremely complex (really, I’ve hardly begun to understand), so I shan’t continue to oversimplify. More to come soon…