## 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.

### 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 ∧)

## 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

### Exercise 1 (of 2):

### Exercise 2 (of 2):

### Help Video:

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