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If you aren't really interested in logic but just want an overview of its connection to math, here is one general picture:
You start with a formal language L. You create L "from scratch". Everything that you do will be in connection with L. L will have a set of
symbols. For example, in English, the symbols would be the alphabet, numbers, punctuation symbols, etc. The symbols of L may fall into different categories, just as in English. You may have
propositional symbols, ex. a, b, c, ...
variable symbols, ex. x, y, z...
connective symbols, ex. ~, &, v, ->
predicate or relation symbols, ex. =, < , >, \in
function or operation symbols, ex. +, -, *, /
a special type of function symbols, constants, ex. 0, 1, 2, 3, ...
quantifier symbols, ex. \forall , \exists
punctuation symbols, ex. (, ), {, }, .
and so on, depending on how you intend to use your language. BTW, the symbols of L don't need to have a written form, but in order to talk about L, you represent the symbols of L with written symbols.
You then string the symbols of L together to get
expressions of L. For example, a->~b, =a5+2h64>, 1=5, and 1+1=2 would be expressions. But just as in English, many of the expressions are nonsense. For instance, the English expression "fhdea phradji qu4032rafdjkasjf" doesn't "mean" anything in English. So we single out some expressions as meaningful: the words, sentences, etc. The same is done in L.
You define the
formulas of L. The formulas are the "meaningful" expressions. Your symbol categories will play a large role in defining your formulas, since certain symbols are intended to be used in a specific way. For instance, in English, the period symbol, ., is intended to come at the end of a sentence. The definition of formulas of L might say:
1) a propositional symbol is a formula.
2) If P is a formula, then ~P is a formula.
3) If P and Q are formulas, then ->PQ is a formula.
This is a definition of formulas for a language for propositional logic. It is more complicated for more complicated languages and logics. The set of formulas is what we're really interested in; The formulas are all of the meaningful things that can be said in L.
Once you have the formulas of L, a division happens: You will define a) the basic semantics and b) the calculus (a general term- not the field of math).
You're probably most familiar with the calculus. This is the (possibly empty or infinite) set of
axioms and the set of
inference rules. The axioms, if you have any, are formulas of L. You just choose some formulas! Well, you probably choose them carefully. :) An inference rule has two parts: a set of hypotheses and a conclusion. The hypotheses and conclusion are formulas. A rule states that some conclusion can be inferred from the hypotheses. This is how you get theorems from axioms, i.e., how you prove things. Initially, the axioms act as hypotheses. You apply a rule to an axiom, or set of axioms, and you get another formula- a theorem. You can then take these new theorems, apply rules to them, and get more theorems. Pretty simple, ay? A proof is just a sequence of formulas that obey your inference rules. Of course, figuring out which rules to apply to which theorems (the axioms are also considered theorems) in order to get a specific conclusion is where the intelligence and creativity of the mathematician come into play.
The semanitcs basically tells you what the formulas mean, which are true, and which are false. Remember, the formulas are just meaningless symbols so far. For instance, you may have an equality symbol, =. You need to define the circumstances that make a formula with the equality symbol true. It's kind of difficult to give an example because, well, I haven't gone into the necessary details. I gave the definition of formulas for a language for propositional logic above, so I'll use it. A
valuation V on L (part of the semantics) is a function from the set of formulas of L to the set {T, F} of truth-values. For instance, if P and Q are formulas and V is a valuation,
1) P
V denotes the truth-value assigned to P by V (i.e. the value of V at P).
2) (~P)
V = {F iff P
V = T, T otherwise}.
3) (->PQ)
V = {F iff P
V = T and Q
V = F, T otherwise}.
Pretty simple, right? So you will want to choose your inference rules so that truth is preserved. That is, rules such that if the hypotheses are all true, then the conclusion is also true. Make sense? That way, if you choose axioms that are true, and have truth-preserving inference rules, all of your theorems will be true! That meaningless, mechanical manipulation of symbols that happens with the calculus results in a collection of true statements.
Well, again, it's more complicated for more complicated systems. The valuations are not so simple, and things that worked for propositional logic, like truth tables, don't work anymore.
I can go into more detail about something if you're interested. You can see a calculus for propositional logic (this one has no axioms)
here, to get the idea.
Are you starting to see how it's all built up?
Another part of the semantics for more complicated languages is a structure (or interpretation).
A mathematical structure S consists of:
1) A non-empty set U, called the universe or domain of S; The members of U are called the individuals of S;
2) A set of basic operations on U;
3) A non-empty set of basic relations on U.
An example:
1) the individuals are natural numbers, U = {0, 1, 2, 3, ...};
2) the four basic operations are the designated individual 0, the unary operation s, which assigns to each number n its immediate successor, and two binary operations + and *, which assign to each pair of numbers their sum and product, respectively;
3) the only basic relation is the identity relation {(n, n) : n is in U}.
The connection between the language L and the operations and relations can be seen in the categories I listed above. You have predicate and function symbols in your language. In defining a structure on L, i.e., what the language "means", the predicate symbols become relations and the function symbols become operations and constants.
