Formal Logic/Predicate Logic/Free and Bound Variables

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Free and Bound VariablesEdit

Informal notionsEdit

The two English sentences,

If Socrates is a person, then Socrates is mortal,
if Aristotle is a person, then Aristotle is mortal,

are both true. However, outside any context supplying a reference for 'it',

(1)    If it is a person, then it is mortal,

is neither true nor false. 'It' is not a name, but rather an empty placeholder. 'It' can refer to an object by picking up its reference from the surrounding context. But without such context, there is no reference and no truth or falsity. The same applies to the variable 'x' in

(2)    If x is a person, then x is mortal.

This situation changes with the two sentences:

(3)    For any object, if it is a person, then it is mortal,
(4)    For any object x, if x is a person, then x is mortal.

Neither the occurrences of 'it' nor the occurrences of 'x' in these sentences refer to specific objects as with 'Socrates' or 'Aristotle'. But (3) and (4) are nonetheless true. (3) is true if and only if:

(5)    Replacing both occurrences of 'it' in (3) with a reference to any object whatsoever (the same object both times) yields a true result.

But (5) is true and so is (3). Similarly, (4) is true if and only if:

(6)    Replacing both occurrences of 'x' in (4) with a reference to any object whatsoever (the same object both times) yields a true result.

But (3) is true and so is (4). We can call the occurrences of 'it' in (1) free and the occurrences of 'it' in (3) bound. Indeed, the occurrences of 'it' in (3) are bound by the phrase 'for any'. Similarly, the occurrences 'x' in (2) are free while those in (4) are bound. Indeed, the occurrences of 'x' in (4) are bound by the phrase 'for any'.

Formal definitionsEdit

An occurrence of a variable   is bound in   if that occurrence of   stands within a subformula of   having one of the two forms:

 
 

Consider the formula

 

Both instances of   are bound in (7) because they stand within the subformula

 

Similarly, both instances of   are bound in (7) because they stand within the subformula

 


An occurrence of a variable   is free in   if and only if   is not bound in  . The occurrences of both   and   in

 

are free in (8) because neither is bound in (8).


We say that an occurrence of a variable   is bound in by a particular occurrence of   if that occurrence is also the first (and perhaps only) symbol in the shortest subformula of   having the form

 

Consider the formula

 

The third and fourth occurrences of   in (9) are bound by the second occurrence of   in (9). However, they are not bound by the first occurrence of   in (9). The occurrence of

 

in (9)—as well as the occurrence of (9) itself in (9)—are subformulae of (9) beginning with a quantifier. That is, both are subformula of (9) having the form

 

Both contain the second third and fourth occurrences of   in (9). However, the occurrence of (10) in (9) is the shortest subformula of (9) that meets these conditions. That is, the occurrence of (10) in (9) is the shortest subformula of (9) that both (i) has this form and (ii) contains the third and fourth occurrences of   in (9). Thus it is the second, not the first, occurrence of   in (9) that binds the third and forth occurrences of   in (9). The first occurrence of   in (9) does, however, bind the first two occurrences of   in (9).


We also say that an occurrence of a variable   is bound in by a particular occurrence of   if that occurrence is also the first (and perhaps only) symbol in the shortest subformula of   having the form

 


Finally, we say that a variable   (not a particular occurrence of it) is bound (or free) in a formula if the formula contains a bound (or free) occurrence of  . Thus   is both bound and free in

 

since this formula contains both bound and free occurrences of  . In particular, the first two occurrences of   are bound and the last is free.

Sentences and formulaeEdit

A sentence is a formula with no free variables. Sentential logic had no variables at all, so all formulae of   are also sentences of  . In predicate logic and its language  , however, we have formulae that are not sentences. All of (7), (8), (9), and (10) above are formulae. Of these, only (7), (9), and (10) are sentences. (8) is not a sentence because it contains free variables.

ExamplesEdit

All occurrences of   in

 

are bound in the formula. The lone occurrence of   is free in the formula. Hence, (11) is a formula but not a sentence.

Only the first two occurrences of   in

 

are bound in the formula. The last occurrence of   and the lone occurrence of   in the formula are free in the formula. Hence, (12) is a formula but not a sentence.

All four occurrences of   in

 

are bound. The first two are bound by the universal quantifier, the last two are bound by the existential quantifier. The lone occurrence of   in the formula is free. Hence, (13) is a formula but not a sentence.

All three occurrences of   in

 

are bound by the universal quantifier. Both occurrences of   in the formula are bound by the existential quantifier. Hence, (14) has no free variables and so is a sentence and as well as a formula.