The Language The Language SS

: Æ Ç ! $

Simple and Compound StatementsSimple and Compound Statements

• By simple statement we mean a statement which does not contain further statements as parts

• By compound (or complex) statement we mean a statement which has one or more further statements as a part or parts

Simple and Compound StatementsSimple and Compound Statements1. Manet changed painting

2. Manet subverted traditional artistic norms

3. Manet cleared the way for Monet

4. Manet did not change painting

5. Manet changed painting or he subverted traditional artistic norms

6. If Manet did not change painting, then Manet did not clear the way for Monet

7. Monet believed that Manet changed painting

8. Picasso doubted that Monet believed that Manet subverted traditional artistic norms

9. Manet cleared the way for Monet because he subverted traditional artistic norms

10. Manet subverted traditional artistic norms after he cleared the way for Monet

Truth-Functional CompoundsTruth-Functional Compounds

Truth-Functional Compound:A statement is a truth-functional compound iff the truth value of the compound statement is completely and uniquely determined by (is a function of) the truth values of the simple component statements.

Truth-Functional LogicTruth-Functional Logic

Truth-Functional Logic:Truth-Functional Logic is the logic of truth-functional combinations of simple statements. We will study the properties which arguments and statements have in virtue of their truth-functional structure.

We will use capital letters, A-Z, to represent simple statements, and truth-functional connective symbols

: Æ Ç ! $to combine simple statements into more complex statements

Negation—The Hook Negation—The Hook ::

It is not the case that …Not …

• The hook is appended directly to the left of the statement to be negated

• The negation has the truth value opposite that of the negated statement

Conjunction—The Wedge Conjunction—The Wedge ÆÆ

Both … and - - -

• The two components of the conjunction are called the (left and right) conjuncts

• A conjunction is T iff both conjuncts are T

conjuncts

Disjunction—The Vee Disjunction—The Vee ÇÇ

Either … or - - -

• The two components of the disjunction are called the (left and right) disjuncts

• A disjunction is F iff both disjuncts are F

• This is an inclusive ‘or’; not an exclusive ‘or’; i.e. it is T when both disjuncts are T

disjuncts

Material Conditional—Material Conditional—The Arrow The Arrow !!

If …, then - - -… only if - - -

• The left side of the conditional is the antecedent; the right side is the consequent

• The material conditional is F iff the antecedent is T and the consequent is F

• This is not implication, and should not be read as “implies”

antecedent consequent

Material Biconditional—Material Biconditional—The Double Arrow The Double Arrow $$

… if and only if - - -

… iff - - -

• The material biconditional is T when the two components have the same truth value, and F when they do not

• This is not equivalence and should not be read as “equals”

Object Language and MetalanguageObject Language and Metalanguage

Object Language:When one is talking about a language, the object language is the language being talked about

Metalanguage:When talking about a language, the metalanguage is the language in which one is talking about the object language

Use and MentionUse and Mention

Most of the time we use words, phrases, statements to communicate. Sometimes we want to talk about words themselves, so we mention them by enclosing them in quotation marks or by displaying them set off from the main text:

1. Socrates is a Greek philosopher2. ‘Socrates’ is a Greek name3. ‘Socrates’ is a Greek philosopher4. Socrates is a Greek name5. Blah blah blah blah. The name

Socratesis Greek. Blah blah blah blah…

MetavariablesMetavariables

Metavariables:Metavariables are variables of the metalanguage which range over (take as possible values) expressions of the object language.

A, B, C,…, Z, A1, B1,…

Metavariables allow us to speak generally about (mention) the form of expressions in the object language.

Form and InstanceForm and Instance

Syntax & SemanticsSyntax & Semantics

Syntax:Syntax is the study of the signs of a language with regard only to their formal properties—e.g., which shapes are signs the language, what their permissible combinations and transformations are

Semantics:Semantics is the study of language with regard to meaningful interpretations or valuations of the components—e.g., what the signs mean and how the meanings of simpler signs contribute to the meanings of combinations of signs

The Symbols of The Symbols of SS

Statement Letters:

A, B, C,…, Z, A1, B1, C1,…, A2,…

Truth-Functional Connectives:: Æ Ç ! $

Punctuation Marks:( )

Definition 2.3.1 (Expression of S). An expression of S is any finite sequence of symbols of S

Expression:

A1))BZ))!Æ:

)($$$B32

:(A Æ :B)

Expressions of Expressions of SS

Not an Expression:

A1$)BZ)@!Æ:

)($ga$$B32

:(A Æ :B)

Well-Formed Formulas of Well-Formed Formulas of SSDefinition 2.3.2 (Well-Formed Formula of S). Where

P and Q range over expressions of S,

(1) If P is a statement letter, then P is a wff of S

(2) If P and Q are wffs of S, then

(a) :P is a wff of S

(b) (P Æ Q) is a wff of S

(c) (P Ç Q) is a wff of S

(d) (P ! Q) is a wff of S

(e) (P $ Q) is a wff of S

(3) Nothing is a wff of S unless it can be shown so by a finite number of applications of clauses (1) and (2)

Labeled Syntax TreeLabeled Syntax TreeThe labels on the branches indicate which clause of Definition 2.3.2 they represent

Unlabeled Syntax TreeUnlabeled Syntax Tree

Syntactic ConceptsSyntactic Concepts

Atomic Formula:Any wff which qualifies simply in virtue of clause (1) of Def. 2.3.2 (that is, any wff which just is some statement letter), is called an atomic formula, wff, or statement. By analogy, all other wffs are molecular.

Syntactic ConceptsSyntactic Concepts Main Connective, Well-Formed Components:

• Atomic wffs have no main connective.

• The main connective of a molecular wff R is the connective appearing in the clause of Def. 2.3.2 cited last in showing R to be a wff.

• The immediate well-formed components of a molecular wff are the values of P and Q (in the case of clause (2a) simply P) in the last-cited clause of Def. 2.3.2.

• The well-formed components of a wff are the wff itself, its immediate well-formed components, and the well-formed components of its immediate well-formed components.

• The atomic components of a wff are the well-formed components which are atomic wffs.

Main Connective and ComponentsMain Connective and Components

Main Connective

The wff itself

Immediate well-formed components

Well-form

ed com

pon

ents

Atomic components

Scope:The scope of a connective is that portion of the wff containing its immediate sentential component(s).

:((D ! A) Æ (:C $ A))

Syntactic ConceptsSyntactic Concepts

‘‘Not’, ‘And’, ‘Or’ Not’, ‘And’, ‘Or’

Not Both P and Q:(P Æ Q)

Both Not-P and Not-Q:P Æ :Q

m m

Either Not-P or Not-Q:P Ç :Q

Neither P nor Q:(P Ç Q)

‘‘If’, ‘Only If’, ‘If and Only If’If’, ‘Only If’, ‘If and Only If’

‘if’ alone:

If P, then QP ! Q

Q, if P

‘only if’ alone:

Only if P, QQ ! P

Q, only if P

‘if’ and ‘only if’:

P if and only if Q

P $ QP if Q, and only if Q

If P then Q, and if Q then P

P iff Q

P is necessary and sufficient for QP iff Q

P $ QQ $ P

P is necessary for QFor Q, it is necessary that P

Q only if P

Q ! P

P is sufficient for QFor Q, it is sufficient that P

If P, then Q

P ! Q

Necessary and SufficientNecessary and Sufficient

‘‘Unless’Unless’

P unless QUnless Q, P

:Q ! P:P ! Q P Ç Q

A Translation Key (Interpretation)A Translation Key (Interpretation)

E: Elmo is a monsterG: Grover is a monsterF: Elmo is furryU: Grover is furryR: Elmo is redB: Grover is blueL: Elmo has googly eyesO: Grover has googly eyesS: Elmo can singD: Elmo can danceI: Grover can sing

A: Grover can dance

Another Translation KeyAnother Translation Key

A: Ann goes to the fair

B: Bob goes to the fair

C: Carol goes to the fair

D: Ann drives

E: Bob drives

F: Carol drives

L: Ann eats lots of popcorn

M: Bob eats lots of popcorn

N: Carol eats lots of popcorn

O: Ann pays

P: Bob pays

Q: Carol pays

R: Ann rides the roller coaster

S: Bob rides the roller coaster

T: Carol rides the roller coaster

X: Ann throws up

Y: Bob throws up

Z: Carol throws up

A Third Translation KeyA Third Translation Key

S: Figure 1 is a square

Q: Figure 1 is a quadrilateral

U: Figure 1 has opposite sides equal

A: Figure 1 has equal angles

R: Figure 1 is a rectangle

E: Figure 1 has equal sides