Tutorial 16: Symbolization using the quantifiers

Topic
Logical System

Tutorial 16 Symbolization using the quantifiers.

2013

Skill to be acquired in this tutorial:

To learn how to use the Universal and Existential Quantifiers in symbolizing propositions.

Bergmann[2008] The Logic Book Sections 7.4

The Tutorial

In Predicate Logic there are two new logical connectives, the Universal Quantifier (∀x) and the Existential Quantifier (∃x). These are used for symbolizing certain English constructions (they also have their own rules of inference and their own semantics, which we will learn about later).

The Universal Quantifier (∀x) is read in English 'For all x,' or 'Whatever x you choose,'. The x here is a variable; that is, it is a term and so is like a name but unlike a constant term, a for example, it does not name a specific object in particular. When a Universal Quantifier appears it is followed by a formula known as its scope. Let us put the scope in brackets for the time being. The following is an example of a Universally Quantified formula

(∀x)(Fx)

and it is read in 'Whatever x you chose, x is F'. So, for example, if you wished to symbolize

Everything thinks.

you would first re-cast this as

Whatever x you chose, x thinks.

and then, using the convention T = thinks, symbolize this to

(∀x)(Tx)

The Existential Quantifier (∃x) is read in English 'There is an x such that...'. The x here is a variable and when an Existential Quantifier appears it is followed by a formula known as its scope. The following is an example of an Existentially Quantified formula

(∃y)(Fy)

and it is read in 'There is a y such that y is F'. So, for example, if you wished to symbolize

Something thinks.

you would first re-cast this as

There is a y such that y thinks.

and then, using the convention T = thinks, symbolize this to

(∃y)(Ty)

The symbols a,b,c...l are the constant terms (used for depicting Arthurs and Beryls and suchlike) and the lower case letters m,n,o...z are the variable terms. Usually it will not matter which variable you use-- 'There is a y such that y thinks' and There is a x such that x thinks' mean one and the same thing.

It is good practice or style to use parantheses to show the scope, as examples (∀x)(Tx), (∀x)(Tx∨Gx), and (∀x)(Tx⊃Gx). But if the parantheses are omitted we need to remember that the quantifiers have, to use a technical term, 'high precedence' and so bind tightly to whatever is immediately following them (only negation has higher precedence). So, as examples,

(∀x)Tx means (∀x)(Tx)
(∀x)∼Tx means (∀x)(∼Tx)
(∀x)Tx∨Gx means (∀x)(Tx)∨Gx (ie the scope of the quantifier does not run the full remaining length of this formula)
(∀x)Tx⊃Gx means (∀x)(Tx)⊃Gx (ie the scope of the quantifier does not run the full remaining length of this formula)

and similarly for the existential quantifier eg

(∃x)(Tx&Gx) of course means (∃x)(Tx&Gx) but
(∃x)Tx&Gx means (∃x)(Tx)&Gx (the scope is not the rest of the formula, and the main connective of the whole formula is the &)

The Logic Book and individual constants and variables

Thus far we have met individual constants such as 'a', which we have sometimes used to symbolize 'Arthur', and variables, for example 'x' which do not name specific objects in particular. In The Logic Book , they use {a,b,c,...,v} as constants and {w,x,y,z} as variables. This is perfectly acceptable convention; and it has advantages-- for example, letters later in the alphabet can still have mnemonic value such as 'v' for 'Violet'. But as these systems develop, they can be used for mathematics in particular various advanced systems will use what is known as mathematical induction. Mathematical induction is pretty well always laid out in terms of a number, n. And with induction, n is going to be a variable. In sum, as we advance we will want n to be a variable. So here, the symbols {a,b,c...l} are the constant terms (used for depicting Arthurs and Beryls and suchlike) and the lower case letters {m,n,o...z} are the variable terms. This matters. Some of the rules that we will meet permit steps using constants and do not permit the same steps using variables.

Exercise to accompany Predicate Tutorial 6

Remember, the program uses the following conventions

a = ARTHUR
b = BERYL
c = CHARLES
Sx = STUDIES
Tx = THINKS
Ax = ANGRY
Bx = BOLD
Cx = CHEERFUL
Nx = NINCOMPOOP
Px = PHILOSOPHER

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