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The first-order logic with identity.

Now we go one step further than in the previous post introduced

SL.

We now consider the internal structure of atoms
that is not decomposable into smaller statements statements. This requires but one deeper understanding of logical languages.
The
blocks
a logical language are the things from which you can merge the sentences (statements). The
syntax of a language is the study of how the elements must be arranged so that they arise from sets. A calculus is a system that defines the syntax of a logical language. As
predicate refers to a sequence of words of a
natural language which contains at least zero space, and which is a declarative sentence, if we substitute in each of the spaces a proper name. Spaces are numbered. The number of vacancies, which contains a predicate is the arity of the predicate. For example is "Bruno is a bear" a set because you can assign it a truth value, at least if we know what Bruno is meant. This rate is also a zero-place predicate, as the proper name has already been used. "_
one is a bear" is executed against a one-place predicate. Here you can use any name, and depending on which one starts with, there are different truth values to the statement.
"_ one eats more berries per day as _
2 "is a two place predicate. You can use any name (Bear) and check the truth value. With predicates we therefore relations between" treat individuals. "But what if we want to say that all Bears are mammals. Then we had to give each bear a name and so "one _
is a mammal" evaluate for each bear. But it's also more elegant, namely
quantifiers as or . The former means "to all" and the latter "there exists one." The idea is that you can paste into empty spaces and thus a predicate quantifiers replaced by one less step. For example, I now x for any bears. Then we can say, " x: x is mammal," which is to say as much as "Every bear is a mammal." A shorter notation is obtained if we show "one _ is a mammal" just write "S_ a . Then the whole " xSx." Is It read: "For all x, Sx the set" This set Sx is nothing but the title "S_ a " in which we have x. If we say only that there is a bear is a mammal, then that means simply " xSx. " together with the connectives is this structure predicates and quantifiers, the so-called


predicate logic of the first stage. It would clarify that is not what the word" shall mean with identity. "The identity is an equal sign between two names put, you write a = b, meaning that designate the names a and b are the same object. You can import identity with two rules, the identity elimination and identity introduction. The identity elimination says that if one has shown that a = b, any one in each set can often replace a by b, without changing the truth value of the sentence. Mind you would have this but really can make
each set. The introduction of identity , simply says that the truth value of a sentence that has to do with a, does not change when a by itself, so a replaced.
The introduction of identity seems to be redundant, but it is not. With no further unquestionable statements, called axioms .
statements, of which it is believed
simply that they are true and which would otherwise be challenged. Typically, the formulation of axioms would be purely arbitrary. However, there are many unquestionable things, that assumption is not arbitrary, but actually usable. So now we know roughly what is the logical Grundlge
for the formulation of Zermelo-Fraenkel set theory
.
Full details the first-order logic with identity be found for example in
Lecture Notes Introduction to Logic by Dr. Klaus Dethloff (University of Vienna ).