8.1: Introduction

The propositional logic we've looked at so far can't handle arguments like 

       All Greeks are mortal

       Socrates is a Greek

       So, Socrates is mortal

The problem is that the important structure in arguments of that sort of argument lies below the level of the proposition. We need to look at the structure of the propositions.

In the example, the validity depends on relations expressed by the terms Greek, Socrates, mortal, and All.

The terms Greek and mortal are general terms - also known as predicates. 

The term Socrates is a singular term. 

The word All is a quantity term. 

We need a logic that can deal with these three types of terms. Such a logic is a Predicate Logic - also known as Quantification Theory (QT), Lower Predicate Calculus (LPC), General Predicate Logic, and First Order Logic.

We will consider a restricted QT called Monadic QT (MQT).

The formal language of MQT is MQL.

Singular reference occurs when reference is made to a specific individual thing. For example:

       Alan and I live in the capital city.

makes singular reference to Alan, to me, and to the capital city

Singular reference may be expressed by

       1.       proper names

       2.       definite descriptions

       3.       singular pronouns.

Some singular references are non-specific, for example:

       A woman lives here

       Someone is tall

We do not treat a man or someone as singular terms. We treat them as quantity terms.

In propositions things are sometimes said to have properties, for example:

       Someone is tall

attributes the property of being tall to someone. We say that that property is predicated of them.

We now introduce the symbols of MQT.

We need symbols for singular reference (typically we'll be thinking of names when we do this.)

We pick small letters from the front of alphabet for our individual constants.

We may wish to set up a translation dictionary for these constants. For example:

       a = Alan

       b = Bill

       c = Clare

(By convention, do not use letters p, q, r, x, y, z. You'll see why.)

We need symbols for properties. 

We pick any big letters for our predicate letters.

We may wish to set up a translation dictionary for these constants. For example:

       B = is a baker

       T = is tall

       L = left town on the noon stage

The attribution of property P to individual i is written by joining the letter and the constant in that order. So:

       Alan is a baker = Ba

       Bill is tall = Tb

       Clare left town on the noon stage = Lc

Those combinations result in propositions. We can apply the logical connectives of PL to then in the natural way. For example:

       Alan is a not a baker = ~Ba

       If Bill is tall then Clare left town on the noon stage = Tb É Lc

Some propositions do not have singular or specific reference. For example:

       Everyone likes Chris

       Most people like Chris

       At least one person likes Chris

We have here quantified reference to all persons, most persons, at least one person. 

The term that expresses quantified reference is a quantifier. Here Everyone, most, and at least one, are quantifiers.

The previous examples were singly quantified because there is reference to just one quantity. There is no necessity to be so restricted.

       At least one person likes everyone

       Anyone anywhere can enjoy anything anyway

Let's stick to single quantification for now. And we'll also restrict ourselves to just two quantities: all and some.

When we have reference to each and every thing we have a universal quantification. 

       Everyone likes Chris

is universally quantified.

When we have reference to at least one thing we have a particular or existential quantification. 

       Someone likes Chris

is existentially quantified.

UNIVERSAL QUANTIFICATION

Terms such as all, every, each, ... are universal quantifiers.

       Each and every thing is not round  -   is a universally quantified negation.

       Each and every thing is round and green  -  is a universally quantified conjunction.

       Each and every thing is round or green  -  is a universally quantified disjunction.

       Each and every thing is round only if it is green  -  is a universally quantified conditional.

And so on.

Note the difference between UQ conjunctions and UQ conditionals. In the example the UQ conjunction says that there are in the whole universe only round green things. The UQ conditional says that if there are any round things in the universe then they are green things too.

Most UQ propositions are UQ conditionals.

Be careful. Some terms can mean either universal or existential quantification. If in doubt ask whether you can replace the term by all or some (and making proper grammatical changes of course) without changing the meaning. If you can then it is universal or existential respectively.

EXISTENTIAL QUANTIFICATION

Terms such as some, at least one, a few, ... are existential quantifiers.

       At least one thing is not round  -   is an existentially quantified negation.

       At least one thing is round and green  -  is an existentially quantified conjunction.

       At least one thing is round or green  -  is an existentially quantified disjunction.

       At least one thing is round only if it is green  -  is an existentially quantified conditional.

and so on.

Note the difference between EQ conjunctions and EQ conditionals. In the example the EQ conjunction says that there is a round green thing. The EQ conditional says that there is something that is such that if it is round then it is green too. One almost never sees EQ conditionals in real life.

Most EQ propositions are EQ conjunctions.

Logicians speak logicese. When they see the sentences

       Everything is round

       Someone is tall

They naturally read these as

       Every x is such that x is round

       There is an x such that x is tall

The x in those paraphrases is the variable of quantification. We generally use the letters x, y, z for this purpose.

We can partially translate some sentences into MQL thus:

       Everything is round

=    Every x is such that x is round

=    Every x is such that Rx 

       Something is round

=    There is an x such that x is round

=    There is an x such that Rx 

And we now introduce two quantifier symbols:

" - which is the universal quantifier symbol to translate all, every,  etc.

$ - which is the existential quantifier symbol to translate some, a,  etc.

and we use them thus:

       Every x is such that Rx 

=    ("x) Rx

       There is an x such that Rx 

=    ($x) Rx