-----------------------------------------------------------------------------------------

 ------------------------------------------------

Propositional logic is an improvement over traditional Aristotelian logic, yet it fails to deal with many valid arguments experienced in ordinary situations.  There are arguments whose validity can not be established within the context of the sentential calculus. An example presented by Robert R Stoll in his book, Sets, Logic, and Axiomatic Theories, tells the story:

As Stoll expresses the limitation:

The point is that expressions like:

and

aren't themselves sentences, yet they are central to the predicate calculus.

The first expression is a universal quantifier, referring here to all of the rational numbers, and often written as (x). Another form is the existential quantifier, which refers to the existence of at least one of something in a class, and which is written as ($ x).

The second expression is a predicate.

Constants and variables in the system are identified as terms.

Back to Index

-----------------------------------------------

Symbols represent both the subject and the predicate of a proposition. Universal quantifiers are used to denote whether the proposition is universal or particular. So at bottom we are dealing with propositional functions of n variables (or n-place predicates). For example:

x is the oldest

This sounds like a proposition, but it's a function of one variable and is true or false depending on x. It can be symbolized as:

(x)(x is the oldest)

which is read 'for all x (in a particular domain) x is the oldest'

Another example is:

x is older than y

This is a function of two variables, or a 2-place predicate. Its truth-value depends on x and y. It is symbolized as:

(x)(y)(x is older than y).

Another example is a function of three variables.

x says y is older than z

The process can be continued for any number of variables.

When their variables are assigned specific values (known as constants), propositional functions become simple propositions. Hurray! For example, giving x, y, and z in the last function, the values, Mary, Jim, and Sally, yields the simple proposition:

Mary says Jim is older than Sally.

Back to Index

---------------------------------------------

In Ordinary Logic Makes a Statement or Two I review the categorical sentences of ordinary language. These sentence types play a role in Aristotelian logic, but not in propositional logic. In predicate calculus, though, they're back again and have a prominent part, using universal and existential quantifiers. In the following table I've compared their traditional form with the form they enjoy in the predicate calculus:

We can interpret the predicate forms as follows:

1. If x is one thing, then x is another thing.

(Example: All men are mortal.)

2. If x is one thing, then x is not the other thing.

(Example: All men are not mortal.)

3. Some x is one thing and another thing.

(Example: Some men are mortal.)

4. Some x is one thing and not another thing.

Back to Index

---------------------------------------------

Elsewhere we look at the argument structure of syllogisms for categorical sentences, and we develop both truth-table and deductive procedures for propositional logic. Here we get hold of the argument structure of the predicate calculus. But in either case the goal is the same: to use the argument structure to deduce true statements from presumed true statements -- but the class of statements (information) will not be the same. The approach is intuitive, which means deductions won't yet be drawn from a very tightly structured set of axioms in accordance with precise defining rules of transformation.

A proof is a process of deciding that the statement to be proved follows from another statement or a collection of statements. There are degrees of formality of the deductive process depending on how tightly the system is axiomatized. No matter how careful one gets, however, it's always problematical when using a language to talk about itself. We also have to rely on intuition, which, as powerful as it might be, isn't infallible! Try as hard as we might to be formal, insight leads proof, and its fallibility always leaves room for error.

Deductive proofs in a formal system have to do with statements normally called theorems. In the formal deductive method, theorems are derived from a collection of theorems that have special status in the formal system, and it's these theorems that are identified as the axioms.  (Here, Brown prefers the terms postulates and demonstrations of consequences.) The derivation uses specific transformation rules to change the axioms (along with theorems that have already been proven) in acceptable ways and so generate the additional theorems.

Formal theories accept as much as possible of reasoning that's generally regarded as valid, and reject arguments deemed questionable. So, for example, reasoning by contradiction is often frowned on. So is reasoning that depends on a process involving an infinite number of steps. For the most part, therefore, only direct proofs and constructive proofs are allowed.

Real advantage is derived from deductive systems (vs. truth tables, say) in the sense that the formalization can reveal similarities among apparently diverse structures. The logic of sets and the logic of propositions, for instance, have significant similarities as instances of the more general theory of Boolean Algebra.

Another supporting argument for constructing formal systems is that uninhibited use of intuition leads to paradoxes, or self-referencing statements, like:

This statement is false.

This self-referencing statement is false if true, and true if false. It is inherently contradictory, or so the argument goes.

This argument, though, is weakened by the work of G. Spencer Brown, who developed a calculus of forms in which self-referencing statements can have a truth value different from true or false (namely a third truth value, called imaginary). This proposed solution is the analog to the solution developed in algebra to allow imaginary numbers to deal with self-referencing equations like.

x² + 1 = 0.

This equation has solutions only if imaginary numbers are allowed. The solutions are:

+Ö -1 and -Ö -1

Similarly, paradoxes would have solutions if imaginary statements were allowed.

For examples, see The Elements of Logic, by Stephen F Barker. 

Back to Index

-------------------------------------------

Top of Page