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Technical reference

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First-order languages and structures

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Definition. A first-order language is a collection of distinct typographical symbols classified as follows:

  1. The equality symbol ; the connectives , ; the universal quantifier and the parentheses , .
  2. A countable set of variable symbols .
  3. A set of constant symbols .
  4. A set of function symbols .
  5. A set of relation symbols .

Thus, in order to specify a language, it is often sufficient to specify only the collection of constant symbols, function symbols and relation symbols, since the first set of symbols is standard. The parentheses serve the only purpose of forming groups of symbols, and are not to be formally used when writing down functions and relations in formulas.

These symbols are just that, symbols. They don't stand for anything. They do not mean anything. However, that deviates further into semantics and linguistic issues not useful to the formalization of mathematical language, yet.

Yet, because it will indeed be necessary to get some meaning out of this formalization. The concept of model over a language provides with such a semantics.

Definition. An -structure over the language , is a bundle consisting of a nonempty set , the universe of the structure, together with:

  1. For each constant symbol from , an element .
  2. For each -ary function symbol from , an -ary function .
  3. For each -ary relation symbol from , an -ary relation on , that is, a subset .

Often, the word model is used for that of structure in this context. However, it is important to understand perhaps its motivation, as follows.

Terms, formulas and sentences

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Definition. An -term is a nonempty finite string of symbols from such that either

  • is a variable symbol.
  • is a constant symbol.
  • is a string of the form where is an -ary function symbol and , ..., are terms of .

Definition. An -formula is a nonempty finite string of symbols from such that either

  • is a string of the form where and are terms of .
  • is a string of the form where is an -ary relation symbol and , ..., are terms of .
  • is of the form where is an -formula.
  • is of the form where both and are -formulas.
  • is of the form where is a variable symbol from and is an -formula.

Definition. An -formula that is characterized by either the first or the second clause is called an atomic.

Definition. Let be an -formula. A variable symbol from is said to be free in if either

  • is atomic and occurs in .
  • is of the form and is free in .
  • is of the form and is free in or .
  • is of the form where and are not the same variable symbols and is free in .

Definition. A sentence is a formula with no free variables.

Assignment functions

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Hereafter, will denote a first-order language, will be an -structure with underlying universe set denoted by . Every formula will be understood to be an -formula.

Definition. A variable assignment function (v.a.f.) into is a function from the set of variables of into .

Definition. Let be a v.a.f. into . We define the term assignment function (t.a.f.) , from the set of -terms into , as follows:

  • If is the variable symbol , then .
  • If is the constant symbol , then .
  • If is of the form , then .

Definition. Let be a v.a.f. into and suppose that is a variable and that . We define the v.a.f. , referred to as an -modification of the assignment function , by

Logical satisfaction

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Definition. Let be formula and suppose is a v.a.f. into . We say that satisfies with assignment , and write , if either:

  • is of the form and .
  • is of the form and .
  • is of the form and .
  • is of the form and or .
  • is of the form and for each element , .

Definition. Let be formula and suppose that for every v.a.f. into . Then we say that models , and write .

Definition. Let be a set of formulas and suppose that for every formula then we say that models , and write .

In the case that is a sentence, that is, a formula with no free variables, the existence of a single v.a.f. for which immediately implies that .

Definition. Let be a sentence and suppose that . Then we say that is true in .

Logical implication and truth

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Definition. Let and be sets of formulas. We say that logically implies , and write , if for every structure , implies .

As a shortcut, when dealing with singletons, we often write instead of .

Definition. Let be a formula and suppose that . Then we say that is universally valid, or simply valid, and in this case we simply write .

To say that a formula is valid really means that every -structure models .

Definition. Let be a sentence and suppose that . Then we say that is true.

Variable substitution

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Definition. Let be a term and suppose is a variable and is another term. We define the term , read with replaced by , as follows:

  • If is the variable symbol , then is defined to be the term .
  • If is a variable symbol other than , then is defined to be the term .
  • If is a constant symbol, then is defined to be the term .
  • If is of the form , then is defined to be the term .

Definition. Let be a formula and suppose is a variable and is a term. We define the formula , read with replaced by , as follows:

  • If is of the form , then is defined to be the formula .
  • If is of the form , then is defined to be the formula .
  • If is of the form , then is defined to be the formula .
  • If is of the form , then is defined to be the formula .
  • If is of the form , then
    • if and are the same variable symbol, is defined to be the formula .
    • else, is defined to be the formula .

Substitutability

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Definition. Let be a formula and suppose is a variable and is a term. We say that is substitutable for in , if either:

  • is atomic.
  • is of the form and is substitutable for in .
  • is of the form and is substitutable for in both and .
  • is of the form and either
    • is not a free variable in .
    • does not occur in and is substitutable for in .

The notion of substitutability of terms for variables corresponds to that of the preservation of truth after substitution is carried out in terms or formulas. Strictly speaking, substitution is always allowed, but substitutability will be imperative in order to yield a formula which meaning was not deformed by the substitution.