1. The Language S
: Æ Ç ! $

2. Simple 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

3. Simple and Compound Statements
1. 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

4. Truth-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.

5. Truth-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

6. 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

7. Conjunction—The Wedge Æ
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

8. Disjunction—The Vee Ç 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

9. Material Conditional— 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

10. Material Biconditional— 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”

11. Object 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

12. Use 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:
Socrates is a Greek philosopher
‘Socrates’ is a Greek name
‘Socrates’ is a Greek philosopher
Socrates is a Greek name
Blah blah blah blah. The name
Socrates
is Greek. Blah blah blah blah…

13. Metavariables
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.

14. Form and Instance

15. Syntax & 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

16. The Symbols of S Statement Letters: A, B, C,…, Z, A1, B1, C1,…, A2,…
Truth-Functional Connectives:
: Æ Ç ! $
Punctuation Marks:
( )

17. Expressions of S
Definition (Expression of S). An expression of S is any finite sequence of symbols of S
Expression:
A1))BZ))!Æ:
)($$$B32
:(A Æ :B)
Not an Expression:
)($ga$$B32
:(A Æ :B)

18. Well-Formed Formulas of S
Definition (Well-Formed Formula of S). Where P and Q range over expressions of S,
If P is a statement letter, then P is a wff of S
If P and Q are wffs of S, then
:P is a wff of S
(P Æ Q) is a wff of S
(P Ç Q) is a wff of S
(P ! Q) is a wff of S
(P $ Q) is a wff of S
Nothing is a wff of S unless it can be shown so by a finite number of applications of clauses (1) and (2)

19. Labeled Syntax Tree
The labels on the branches indicate which clause of Definition they represent

20. Unlabeled Syntax Tree

21. Syntactic Concepts
Atomic Formula: Any wff which qualifies simply in virtue of clause (1) of Def (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.

22. Syntactic Concepts Atomic wffs have no main connective.
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 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
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.

23. Main Connective and Components
Atomic components
Well-formed components
Immediate well-formed components
Main Connective
The wff itself

24. :((D ! A) Æ (:C $ A)) Syntactic Concepts
Scope: The scope of a connective is that portion of the wff containing its immediate sentential component(s).
:((D ! A) Æ (:C $ A))

25. ‘Not’, ‘And’, ‘Or’ Not Both P and Q :(P Æ Q) Both Not-P and Not-Qm
Either Not-P or Not-Q
:P Ç :Q
Neither P nor Q
:(P Ç Q)

26. ‘If’, ‘Only If’, ‘If and Only If’
‘if’ alone:
If P, then Q
P ! Q
Q, if P
‘only if’ alone:
Only if P, Q
Q ! P
Q, only if P
‘if’ and ‘only if’:
P if and only if Q
P $ Q
P if Q, and only if Q
If P then Q, and if Q then P
P iff Q

27. Necessary and Sufficient
P is sufficient for Q For Q, it is sufficient that P
If P, then Q
P ! Q
P is necessary for Q For Q, it is necessary that P
Q only if P
Q ! P
P is necessary and sufficient for Q
P iff Q
P $ Q Q $ P

28. ‘Unless’
P unless Q
Unless Q, P
:Q ! P :P ! Q P Ç Q

29. A Translation Key (Interpretation)
E: Elmo is a monster
G: Grover is a monster
F: Elmo is furry
U: Grover is furry
R: Elmo is red
B: Grover is blue
L: Elmo has googly eyes
O: Grover has googly eyes
S: Elmo can sing
D: Elmo can dance
I: Grover can sing
A: Grover can dance

30. Another 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

31. A 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