**SYNTAX**

**Propositional Logic:**

Mention

Use

&

Consider the following pair of sentences:

1. Los Angeles has two words.

2. Los Angeles has nearly four million inhabitants.

In 2, we want to

use

the name of the city to make a claim concerned with

the city of Los Angeles

itself.

In 1, we want to

mention

the name of the city in order to make a claim concerned with

the linguistic expression

that names the city.

In particular, we will often want to explicitly discuss the nature of a linguistic expression regardless of what it may denote or refer to in a language.

In logic, it is important to make a clear distinction between a

linguistic expression

and whatever that expression may

denote.

The difference between the

MENTION

of a term and the

USE

of a term (which is just the difference between the occurrences of

'Los Angeles'

in Sentence 1 and Sentence 2) allows us to make this distinction.

We will use

quotation marks

to make it clear that we intend to mention an expression and not merely use it.

Notice that this is a MENTION, not a USE. The

city of Los Angeles

does not occur in Sentences 1 and 2, but the

linguistic expression

which names it does.

By placing a linguistic expression in quotation marks, we create a name for it. Thus, in this example, 'Los Angeles' is a name for the name of the city of Los Angeles, and can be used to properly express the claim that the name of the city is composed of two words.

3. 'Los Angeles' has two words.

Examples

Mention in English

4. 'Barbara' is not my name.

5. 1 is a number, but '1' is an Arabic numeral.

6. 'Colorless green ideas sleep furiously' is a

grammatical sentence of English.

7. * Buffalo buffalo, but the sentence 'buffalo buffalo

buffalo' is not true.

Curious?

http://en.wikipedia.org/wiki/Buffalo_buffalo_Buffalo_buffalo_buffalo_buffalo_Buffalo_buffalo

*

a note

on

Propositional Logic.

We will now introduce a

formal language

that will allow us to study the structure of a certain class of valid arguments. This is the language of

We will proceed in three stages

when we introduce a formal language:

We will specify the

syntax

or

grammar

of the language, which will tell us which strings of symbols count as sentences of the language.

We will specify a

range of interpretations

for the language, which will tell us how to interpret the formal language and what it is for a sentence of the language to be true under some interpretation of the language.

We will offer a

translation manual

, which will allow us to move from sentences of English to sentences of the formal language.

finer

Think of the language of Propositional Logic as an instrument for the study of arguments under a resolution.

When we use this instrument, we will abstract from various aspects of their formulation in English to focus on the formal skeleton of the argument.

Take, for example, one of the arguments we discussed in the previous tutorial.

Example 1

If the battery is dead, then the laptop will not work. The battery is dead. Therefore the laptop will not work.

Example 2

The laptop will not work. This is because the laptop will not work if the battery is dead, and the battery is dead.

Example 3

The battery is dead. That means the laptop will not work, for the laptop will not work if the battery is dead.

All three are stylistic variants of the same argument, which we can express in the language of propositional logic as follows:

1.

2.

3.

In order to present the

syntax (or grammar) of Propositional Logic,

we will proceed in two

stages:

We will specify the

vocabulary

of Propositional Logic, which is the stock of symbols we can use to form well-formed sentences of the language.

We will specify

strict rules for the formulation of sentences

of Propositional Logic out of symbols in the vocabulary of the language.

We will address the first stage in this tutorial and will take up the second and third stages in tutorials 3 and 4, respectively.

( )

The Vocabulary of Propositional Logic

The stock of symbols of the language of propositional logic includes:

Sentence Letters

(with or without subscripts)

Sentence letters will be used to

translate simple sentences of English

. The language of Propositional Logic will allow us to combine

sentence letters

with

connectives

in order to form

more complex sentences

.

Propositional Connectives

Parentheses

( , )

Propositional Connectives

The intended interpretation of each of these symbols ( ) is given by five connectives used in in English to form more complex sentences out of simpler ones:

Propositional Connectives and English Connectives

Name of the connective

English connective

Symbol

negation

disjunction

conjunction

material conditional

material biconditional

'it is not the case that'

'or'

'and'

'if... then...'

'... if and only if ...'

'it is not the case that'

+

'snow is blue'

=

'it is not the case that snow is blue'

'snow is white'

+

'or'

+

'grass is green'

=

'snow is white or grass is green'

'snow is white'

+

'and'

+

'grass is green'

=

'snow is white and grass is green'

'if ... then ...'

+

'2 is prime'

+

'a square root of 4 is prime'

=

'if 2 is prime, then a square root of 4 is prime'

'... if and only if...'

+

'2 is prime'

+

'a square root of 4 is prime'

=

'2 is prime if and only if a square root of 4 is prime'

We now know that there are three kinds of symbols in the vocabulary of Propositional Logic (PL). The next task is to specify strict rules for the formation of sentences of the language.

For example, in English, there are rules which tell us that

'snow is white and and and and'

is

not

a sentence, but that the examples considered earlier

are

sentences. The

syntactic

rules for Propositional Logic will be much more straightforward than those for English.

( )

Sentences of Propositional Logic (PL)

All sentence letters are sentences of PL

If and are sentences of PL, then each of the following are sentences of PL:

Nothing else is a sentence of PL.

a.

b.

c.

d.

e.

These rules allow us to associate a

construction tree

with each sentence. These tree for a given sentence describes how the sentence has been formed out of simpler sentences.

Example 1

Notice that if a formula is not merely a sentence letter, then it has a

main connective

, which is the last connective added in the construction. The main connective appears only at the very 'root' of the construction tree for the formula.

Sentence letters are the most basic constituents of any sentence of PL, so every branch of the construction tree for a sentence will end in a sentence letter.

Example 2

Construction Tree

Construction Tree

The main connective in this formula is a

disjunction.

Example 3

Construction Tree

The main connective in this formula is the

conditional

.

Example 4

Construction Tree

The main connective in this formula is a

negation

.

A string of symbols is

not

a sentence of PL if it cannot be constructed out of sentence letters, connectives, and parentheses according to the strict rules of formation detailed so far. We can identify problem

atic strings of

symbols when we set out to provide

a construction tree

for the formula in question.

Example 1

Formation Failures

The question mark indicates that none of our rules of sentence formation can be used to justify the construction of the relevant string of symbols. In this case, the problem is on the first level of the tree: none our rules allow us to construct a sentence by merely placing a sentence letter between two parentheses.

Not a sentence of PL.

(Remember, each node of the construction tree for a sentence must itself be a well-formed sentence.)

Example 3

Formation Failures

Although Rule 2 (Clause b) would allow us to construct a formula from and by inserting an arrow and placing the result between two parentheses, none of the formation rules allow us to construct a formula by inserting an arrow and adding

only a right parenthesis

to the resulting string of symbols.

?

?

Example 3

Formation Failures

?

?

?

?

There are two potential construction trees for this formula, each problematic.

In both cases, the problem arises because the formation rules do not allow us to form a sentence by

merely

placing an arrow between two sentences. Clauses (b)-(e) require that we add left and right parentheses whenever we insert a logical connective between sentences, in order to avoid ambiguity.

**( )**

Two Notational Conventions

We often leave off parentheses when we write arithmetical equations involving binary operations like

+

and

x

. For example, the arithmetical expression '2 + 3 x 5' may seem ambiguous between (2+(3 x 5) and ((2+3) x 5) depending on which operation is applied first. In practice, however, we are never confused because we take

'2 + 3 x 5' to abbreviate: '2+(3 x 5)'

and

'2+(3 x 5)' to abbreviate: '(2+(3 x 5))'

This is because we take two

notational conventions

for granted in the language of arithmetic.

One may omit the outer parentheses when forming an arithmetical term out of + or x and simpler arithmetical terms.

One may omit the inner parentheses when there is a unique way to restore them on the assumption that multiplication is applied before addition.

'

(

2 + (3 x 5)

)

'

'2 +

(

3 x 5

)

'

Notational Conventions

Arithmetic

With these in mind, we will adopt two parallel conventions for the language of propositional logic.

that the second convention does not allow us to omit the inner parentheses in

Notice

'2+ (2+2)'

Otherwise, the expression ' 2 + 2 + 2 ' would be ambiguous between:

((2+2) + 2)

and

(2+ (2+2))

That is, there would not be a unique way to restore the parentheses.

One may omit the outer parentheses of a complex formula and write (for example) as an abbreviation for the well-formed formula .

One may omit inner parentheses when there is a unique way to restore them on the assumption that conjunction and disjunction take precedence over the conditional and the biconditional.

Notational Conventions

Propositional Logic

So, for example, we will take

to abbreviate

and

to abbreviate

that the second notational convention does not allow us to omit the inner parentheses in expressions like

Notice

As in the arithmetical cases, there is no unique way to restore the parentheses given only the assumption that conjunction and disjunction take precedence over the conditional and biconditional.

(How would you go about finding the construction tree for these formulas? Without the parentheses, we are unable to identify the main connective or distinguish the simpler sentences which compose the more complex one.)

It is important to remember that these

conventions

do not belong to the

official syntax

of the language. It is still true that is

not

a well-formed formula of the language of propositional logic; it is just that we have

agreed to let it abbreviate

a certain well-formed formula of the language.

We have now

specified a formal language

and have offered

strict rules of sentence formation

, which should allow us to determine whether or not a given string of symbols of the language counts as a sentence. In the next tutorial, we will explain how to

interpret

this formalism and how to define important logical concepts such as

consistency

and

validity

.

Formation rules for