Using FOL to Describe the Properties of Systems

Unlike natural languages, such as English, the language of first-order logic is completely formal, so that it can be mechanically determined whether a given expression is well formed. There are two key types of well-formed expressions: terms, which intuitively represent objects, and formulas, which intuitively express statements that can be true or false. The terms and formulas of first-order logic are strings of symbols, where all the symbols together form the alphabet of the language.

Alphabet

As with all formal languages, the nature of the symbols themselves is outside the scope of formal logic; they are often regarded simply as letters and punctuation symbols.

It is common to divide the symbols of the alphabet into logical symbols, which always have the same meaning, and non-logical symbols, whose meaning varies by interpretation. For example, the logical symbol $\land$ always represents "and"; it is never interpreted as "or", which is represented by the logical symbol $\lor$. However, a non-logical predicate symbol such as $Phil(x)$ could be interpreted to mean "x is a philosopher", "x is a man named Philip", or any other unary predicate depending on the interpretation at hand.

Logical symbols

Logical symbols are a set of characters that vary by author, but usually include the following:

• Quantifier symbols: ∀ for universal quantification, and ∃ for existential quantification
• Logical connectives: ∧ for conjunction, ∨ for disjunction, → for implication, ↔ for biconditional, ¬ for negation. Some authors use Cpq instead of → and Epq instead of ↔, especially in contexts where → is used for other purposes. Moreover, the horseshoe ⊃ may replace →; the triple-bar ≡ may replace ↔; a tilde (~), Np, or Fp may replace ¬; a double bar $\|$, + $+$, or Apq may replace ∨; and an ampersand &, Kpq, or the middle dot ⋅ may replace ∧, especially if these symbols are not available for technical reasons. (The aforementioned symbols Cpq, Epq, Np, Apq, and Kpq are used in Polish notation).
• Parentheses, brackets, and other punctuation symbols. The choice of such symbols varies depending on context.
• An infinite set of variables, often denoted by lowercase letters at the end of the alphabet x, y, z, ... . Subscripts are often used to distinguish variables: $x_0, x_1, x_2, ...$ .
• An equality symbol (sometimes, identity symbol) = (see § Equality and its axioms below).

Not all of these symbols are required in first-order logic. Either one of the quantifiers along with negation, conjunction (or disjunction), variables, brackets, and equality suffices.

Other logical symbols include the following:

• Truth constants: T, V, or ⊤ for "true" and F, O, or ⊥ for "false" (V and O are from Polish notation). Without any such logical operators of valence 0, these two constants can only be expressed using quantifiers.
•     logical connectives such as the Sheffer stroke, Dpq (NAND), and exclusive or, Jpq.

Non-logical symbols

Non-logical symbols represent predicates (relations), functions and constants. It used to be standard practice to use a fixed, infinite set of non-logical symbols for all purposes:

• For every integer n ≥ 0, there is a collection of n-ary, or n-place, predicate symbols. Because they represent relations between n elements, they are also called relation symbols. For each arity n, there is an infinite supply of them:
  
  

  Pⁿ₀, Pⁿ₁, Pⁿ₂, Pⁿ₃, ...

• For every integer n ≥ 0, there are infinitely many n-ary function symbols:
  

  
  

  f ⁿ₀, f ⁿ₁, f ⁿ₂, f ⁿ₃, ...

When the arity of a predicate symbol or function symbol is clear from context, the superscript n is often omitted.

In this traditional approach, there is only one language of first-order logic. This approach is still common, especially in philosophically oriented books.

A more recent practice is to use different non-logical symbols according to the application one has in mind. Therefore, it has become necessary to name the set of all non-logical symbols used in a particular application. This choice is made via a signature.

Typical signatures in mathematics are {1, ×} or just {×} for groups, or {0, 1, +, ×, <} for ordered fields. There are no restrictions on the number of non-logical symbols. The signature can be empty, finite, or infinite, even uncountable. Uncountable signatures occur for example in modern proofs of the Löwenheim–Skolem theorem.

Though signatures might in some cases imply how non-logical symbols are to be interpreted, interpretation of the non-logical symbols in the signature is separate (and not necessarily fixed). Signatures concern syntax rather than semantics.

In this approach, every non-logical symbol is of one of the following types:

• A predicate symbol (or relation symbol) with some valence (or arity, number of arguments) greater than or equal to 0. These are often denoted by uppercase letters such as P, Q and R. Examples:
  • In P(x), P is a predicate symbol of valence 1. One possible interpretation is "x is a man".
  • In Q(x,y), Q is a predicate symbol of valence 2. Possible interpretations include "x is greater than y" and "x is the father of y".
  • Relations of valence 0 can be identified with propositional variables, which can stand for any statement. One possible interpretation of R is "Socrates is a man".
• A function symbol, with some valence greater than or equal to 0. These are often denoted by lowercase roman letters such as f, g and h. Examples:
  • f(x) may be interpreted as "the father of x". In arithmetic, it may stand for "-x". In set theory, it may stand for "the power set of x".
  • In arithmetic, g(x,y) may stand for "x+y". In set theory, it may stand for "the union of x and y".
  • Function symbols of valence 0 are called constant symbols, and are often denoted by lowercase letters at the beginning of the alphabet such as a, b and c. The symbol a may stand for Socrates. In arithmetic, it may stand for 0. In set theory, it may stand for the empty set.

The traditional approach can be recovered in the modern approach, by simply specifying the "custom" signature to consist of the traditional sequences of non-logical symbols.

Formation rules

The formation rules define the terms and formulas of first-order logic. When terms and formulas are represented as strings of symbols, these rules can be used to write a formal grammar for terms and formulas. These rules are generally context-free (each production has a single symbol on the left side), except that the set of symbols may be allowed to be infinite and there may be many start symbols, for example the variables in the case of terms.

Terms

Only expressions which can be obtained by finitely many applications of rules 1 and 2 are terms. For example, no expression involving a predicate symbol is a term.

Formulas

1. Predicate symbols. If P is an n-ary predicate symbol and t₁, ..., t_n are terms then P(t₁,...,t_n) is a formula.
2. Equality. If the equality symbol is considered part of logic, and t₁ and t₂ are terms, then t₁ = t₂ is a formula.
3. Negation. If $\varphi$ is a formula, then $\lnot \varphi$ is a formula.
4. Binary connectives. If $\varphi$ and $\psi$ are formulas, then $\varphi \rightarrow \psi$) is a formula. Similar rules apply to other binary logical connectives.
5. Quantifiers. If $\varphi$is a formula and x is a variable, then $\forall x\varphi$ (for all $\varphi$holds) and $\exists x\varphi$ (there exists x such that $\varphi$) are formulas.

Only expressions which can be obtained by finitely many applications of rules 1–5 are formulas. The formulas obtained from the first two rules are said to be atomic formulas.

For example:

$\forall x\forall y(P(f(x))\rightarrow \neg (P(x)\rightarrow Q(f(y),x,z)))$

is a formula, if f is a unary function symbol, P a unary predicate symbol, and Q a ternary predicate symbol. However, $\forall x\,x\rightarrow$ is not a formula, although it is a string of symbols from the alphabet.

The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way - by following the inductive definition (i.e., there is a unique parse tree for each formula). This property is known as unique readability of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a proof of unique readability.

This definition of a formula does not support defining an if-then-else function ite(c, a, b), where "c" is a condition expressed as a formula, that would return "a" if c is true, and "b" if it is false. This is because both predicates and functions can only accept terms as parameters, but the first parameter is a formula. Some languages built on first-order logic, such as SMT-LIB 2.0, add this.

Notational conventions

For convenience, conventions have been developed about the precedence of the logical operators, to avoid the need to write parentheses in some cases. These rules are similar to the order of operations in arithmetic. A common convention is:

• $\lnot$ is evaluated first
• $\land$ and $\lor$ are evaluated next
• Quantifiers are evaluated next
• $\to$ is evaluated last.

Moreover, extra punctuation not required by the definition may be inserted - to make formulas easier to read. Thus the formula:

$\lnot \forall xP(x)\to \exists x\lnot P(x)$

might be written as:

$(\lnot [\forall xP(x)])\to \exists x[\lnot P(x)]$.

In some fields, it is common to use infix notation for binary relations and functions, instead of the prefix notation defined above. For example, in arithmetic, one typically writes "2 + 2 = 4" instead of "=(+(2,2),4)". It is common to regard formulas in infix notation as abbreviations for the corresponding formulas in prefix notation, cf. also term structure vs. representation.

The definitions above use infix notation for binary connectives such as$\to$. A less common convention is Polish notation, in which one writes $\rightarrow$, $\wedge$ and so on in front of their arguments rather than between them. This convention is advantageous in that it allows all punctuation symbols to be discarded. As such, Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read. In Polish notation, the formula:

$\forall x\forall y(P(f(x))\rightarrow \neg (P(x)\rightarrow Q(f(y),x,z)))$

becomes "$∀x∀y→Pfx¬→ PxQfyxz$".

Free and bound variables

In a formula, a variable may occur free or bound (or both). One formalization of this notion is due to Quine, first the concept of a variable occurrence is defined, then whether a variable occurrence is free or bound, then whether a variable symbol overall is free or bound. In order to distinguish different occurrences of the identical symbol x, each occurrence of a variable symbol x in a formula φ is identified with the initial substring of φ up to the point at which said instance of the symbol x appears. Then, an occurrence of x is said to be bound if that occurrence of x lies within the scope of at least one of either $\exists x$ or $\forall x$. Finally, x is bound in φ if all occurrences of x in φ are bound.

Intuitively, a variable symbol is free in a formula if at no point is it quantified: in $∀y P(x, y)$, the sole occurrence of variable x is free while that of y is bound. The free and bound variable occurrences in a formula are defined inductively as follows.

Atomic formulas

If φ is an atomic formula, then x occurs free in φ if and only if x occurs in φ. Moreover, there are no bound variables in any atomic formula.

Negation

x occurs free in ¬φ if and only if x occurs free in φ. x occurs bound in ¬φ if and only if x occurs bound in φ

Binary connectives

x occurs free in (φ → ψ) if and only if x occurs free in either φ or ψ. x occurs bound in (φ → ψ) if and only if x occurs bound in either φ or ψ. The same rule applies to any other binary connective in place of →.

Quantifiers

x occurs free in ∀y φ, if and only if x occurs free in φ and x is a different symbol from y. Also, x occurs bound in ∀y φ, if and only if x is y or x occurs bound in φ. The same rule holds with ∃ in place of ∀.

For example, in $∀x ∀y (P(x) → Q(x,f(x),z))$, x and y occur only bound, z occurs only free, and w is neither because it does not occur in the formula.
Free and bound variables of a formula need not be disjoint sets: in the formula $P(x) → ∀x Q(x)$, the first occurrence of x, as argument of P, is free while the second one, as argument of Q, is bound.

A formula in first-order logic with no free variable occurrences is called a first-order sentence. These are the formulas that will have well-defined truth values under an interpretation. For example, whether a formula such as $Phil(x)$ is true must depend on what x represents. But the sentence $∃x Phil(x)$ will be either true or false in a given interpretation.

Example: ordered abelian groups

In mathematics, the language of ordered abelian groups has one constant symbol 0, one unary function symbol −, one binary function symbol +, and one binary relation symbol ≤. Then:

• The expressions $+(x, y)$ and $+(x, +(y, −(z)))$ are terms. These are usually written as $x + y$ and $x + y − z$.
• The expressions $+(x, y) = 0$ and $≤(+(x, +(y, −(z))), +(x, y))$ are atomic formulas. These are usually written as $x + y = 0$ and $x + y − z  ≤  x + y$.
• The expression $(\forall x\forall y\,[\mathop {\leq } (\mathop {+} (x,y),z)\to \forall x\,\forall y\,\mathop {+} (x,y)=0)]$ is a formula, which is usually written as $\forall x\forall y(x+y\leq z)\to \forall x\forall y(x+y=0)$. This formula has one free variable, z.

The axioms for ordered abelian groups can be expressed as a set of sentences in the language. For example, the axiom stating that the group is commutative is usually written $(\forall x)(\forall y)[x+y=y+x]$.