## A MORE STRUCTURED LANGUAGE

Despite the power of propositional and categorical logic, these forms of logic suffer from some severe limitations.

• Some arguments that are clearly valid cannot be shown valid in our system.
• Propositional logic misses the internal structure of sentences.
• Syllogistic logic cannot deal with more than 5 terms, nor can it deal with relations.

We need a new, more powerful, tool:  Predicate Logic.

Central Concepts in Predicate Logic

Propositional logic takes as its basic, atomic units statements, linking them with logical connectives.  While using the same 5 logical operators as propositional logic, predicate logic, in contrast, uses a different set of basic, or atomic, components:  predicates (property constants), individual variables, individual constants, and quantifiers.

PREDICATES: These express properties of things and relations between things. Predicates are either monadic (applying to only one object) or polyadic (relating at least 2 objects each other).

Monadic Predicates: Express properties of objects.  Monadic predicates apply to only one object at a time.  Monadic predicates are represented by a capital letter followed by exactly one variable letter, e.g., ‘Fx’, Gx’, Hx’ , say, respectively, "x (a variable quantity) has property F", "x has property G", and "x has property H".

Polyadic Predicates:   Polyadic predicates  express a relation between 2 or more objects.  Polyadic or relational predicates apply to ordered sets of objects--"Jane is taller than Bill" does not say the same thing as "Bill is taller than Jane".  Relations may be two-termed, or three-termed, or more.  Relational predicates are represented by a capital letter followed by an ordered set of variable letters, the number of elements in the set being determined by the number of things related by the predicate.  "Rxy" says "x bears the relation R to y", "Gxyz" says "x stands in the B relation with regard to y and z (e.g., x is Between y and z)".

The order of the terms is: first the subject, second the object term.

Let ‘a’ designate John, ‘b’ designate Sheila, and ‘Rxy’ mean ‘ x loves y’. Then ‘John loves Sheila’ = ‘Rab’, whereas ‘Sheila loves John’ = ‘Rba’.  Sadly, some relational predicates are not symmetrical, that is, Rab does not logically entail Rba.

To take an example of a three-place predicate, let ‘Rxyz’ = 'x introduced y to z'.  Adding 'c' to designate Helen, 'Racb' says that John introduced Helen to Shiela.  Changing the order of the variables changes the meaning of the sentence.

Another example: let ‘Gxyz’ be defined as ‘x is between y and z’.

Let’ a’ be ‘St. Louis’, ‘b’ be ‘Chicago’ and ‘c’ be ‘Memphis.

"St. Louis is between Chicago and Memphis" is properly symbolized as 'Gabc'.

INDIVIDUAL VARIABLES:  Individual variables are true variables, taht is, they are place holders in formulaic expressions.  Individual variables in predicate logic work just as do variables in algebra, they stand in for a value, but have no fixed value of their own. Individual variables are represented by lower case letters taken from the end of the alphabet ‘x’, ‘y’, ‘z’.

They can be bound by quantifiers.

Sentences containing true individual variables (also called free variables) are called open sentences and have no truth value.  If 'Hx'= 'x is happy', then Hx says only that x, whoever that is, is happy.  Since the value of x is open, we can’t assign a truth value to Hx.  Open sentences are not well-formed, as they have no truth value.

In order to turn an open sentence into a closed sentence that is well-formed and has a truth value, all of the free variables in the open sentence must be bound.  Free individual variables can be bound in two different ways:

• Individual variables may be replaced with individual constants.
• Individual variables may be bound by quantifiers whose scope includes the variables.

Individual Constants:   Individual constants represent specific individuals or objects.  Individual constants are  are names of things, either real things in the universe or (when made clear in advance) in a specified universe of discourse.  Individuals constants are represented by  lower case letters from the beginning of the alphabet: ‘a’, ‘b’, ‘c’.

When an individual constant (or ordered set of constants) follows a predicate, the statement made by the predicate asserts that the individual named has a certain property.   So, if  'Hx' = ‘x is happy’ and 'Txy' = ‘x is taller than y’ and ‘a’ be the individual constant for Alice and ‘b’ the individual constant for Bob,  'Hb' says that 'Bob is happy' and  'Tba' says that 'Bob is taller than Alice'.

If all of the free individual variables in a formula are replaced with individual constants, the open sentence becomes closed and acquires a truth value.

QUANTIFIERS:  Quantifiers tell us of how many objects the predicate is asserted. If we want to assert a predicate of all objects, we use the universal quantifier,"(∀x)". For example, "(∀x)Mx" says that, for all x, x is mortal; or more idiomatically, all things are mortal, everything is mortal. If we want to assert a predicate of some objects (at least one), we use the existential quantifier, "(x)". For example, "(x)Mx" says that, for some (at least one) x, x is mortal; or idiomatically, something is mortal.

The existential quantifier ‘‘ means ‘There exists at least one thing such that...’.

The universal quantifier ‘"‘ means ‘Everything is such that...’.

Quantifiers  bind variables and thus convert expressions containing variables into sentences. For example, ‘Fx’ is not a complete sentence; it is a predicate with a free (unbound) variable.

But ‘(x)Fx’ is a complete sentence. It means ‘There is at least one thing such that it is F’ (i.e. ‘Something is an F’).

Also ‘(x)Fx’ is a complete sentence. It means ‘Everything is such that it is F’ (i.e. ‘Everything is an F’).

Quantifiers refer to a universe of discourse, the set of things we are talking about. By default, the universe of discourse contains everything whatsoever that can be referred to.  We may choose to stipulate that the universe of discourse is some smaller set of things, e.g. the set of all human beings, or the set of rational numbers.  In such cases the quantifiers range over only that set of things. For example,  if the universe of discourse is humans, and 'MX'='x is mortal', then "("x)Mx" would then assert that all human beings are mortal, and "(x)Mx" would assert that at least one human being is mortal.

The existential quantifier has existential import; that is, it asserts the existence of something. "(
x)Mx" asserts the existence of at least one thing having property M. The universal quantifier doe not have existential import. So "(∀x)(Fx Gx)" asserts that all F's —if any— are G's; it does not presuppose that any F's exist.  Remember, this is the Boolean interpretation of the square of opposition.

Quantifiers have scope, namely, the first whole proposition, simple or compound, to their right. In this sense, they have the same scope as the negation sign. "Gx" is inside the scope of the quantifier in "(
"x)(Fx Gx)" but outside in "("x)Fx Gx".  The set of groupers immediately following a quantifier marks its scope.  Variables inside the scope of a quantifier are bound by that quantifier; otherwise they are free. A variable is only bound by a quantifier that ranges over it (i.e., that uses the same variable letter); hence the 'x' in  'Hx' is bound in "("x)Hx" but not in "("y)Hx", even though it is inside the scope of the quantifier in both cases.  Accordingly. '(x)Fx Gx' is not ambiguous. It is an open sentence that means '[(x)Fx] Gx', not a closed A-type proposition meaning '(x)(Fx Gx)'.

A variable may occur more than once in an expression, free in some occurrences and bound in others.  The 'x' in 'Fx' is bound, but it is free in 'Hx'  in '(
x)Fx Gx'. Instead of speaking of free and bound variables, it is better to speak of free and bound occurrences of variables.

The apparatus of quantifiers and variables allows us to express general facts about situations without naming specific objects.

Expressing Categorical Propositions in Predicate Logic

Any statements that we can express in propositional or categorical logic can be expressed in predicate logic.  For example, here is the way to express each of the four categorical propositions in predicate logic:

 Type Categorical Expression Predicate Expression A All F's are G's (∀x)(Fx → Gx) E No F's are G's (∀x)(Fx → ~Gx) I Some F's are G's (∃x)(Fx ● Gx) O Some F's are not F's (∃x)(Fx ● ~Gx)

Two other useful symbolizations to remember are:

'Only F's are G's' is properly symbolized as '(x)(Gx Fx)'

All and only F's are G's is a combination of "All F's are G's" and "Only F's are G's".  From propositional logic you recall that 'P if and only if Q' is symbolized as a biconditional.  'P  Q' is logically equivalent to '([P Q) ● (Q P)].  By parity of reasoning,  'All and only F's are G's' is properly symbolized as (x)(Fx Gx).

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