1. Identifying Counterexamples
Determine truth value of premises and conclusions.
Remember: an argument is sound iff (i) it’s valid and (ii) has all true premises.
Determine whether two arguments are of the same logical form:
Same framework of logical expressions
Same pattern of same non-logical expressions
Where two arguments are of the same logical form which, if any, is a counterexample to the other?
What does that show about validity?
Argument 1 All dogs are cats. Some dogs are reptiles Some cats are reptiles.
Argument 2 All men are mortals. Some mortals are Greeks. Some Greeks are men.
Argument 3 All mammals are vertebrates. Some vertebrates are cold-blooded animals. Some cold-blooded animals are mammals.

2. Going Formal
Meet the Connectives

3. The Language of Propositional Logic
Syntax (grammar, internal structure of the language)
Vocabulary: grammatical categories
Identifying Well-Formed Formulae (“WFFs”)
Semantics (pertaining to meaning and truth value)
Translation
Truth functions
Truth tables for the connectives

4. The Vocabulary of Propositional Logic
Sentence Letters: A, B, … Z
Connectives (“Sentence-Forming Operators”)
~ negation “not,” “it is not the case that”
⋅ conjunction “and”
∨ disjunction “or” (inclusive)
⊃ conditional “if – then,” “implies”
≣ biconditional “if and only if,” “iff”
“Parentheses”: (, ), [, ], {, and }

5. Sentence Letters Translate “atomic” sentences
Atomic sentences have no proper parts that are themselves sentences
Examples:
It is raining R
It is cold C

6. Sentential Connectives
Connect to sentences to make new sentences
Negation attaches to one sentence
It is not raining ∼ R
Conjunction, disjunction, conditional and biconditional attach two sentences together
It is raining and it is cold R ∙ C
If it rains then it pours R ⊃ P

7. Parentheses, brackets & braces
I’ll go to Amsterdam and Brussels or Calais
This is ambiguous and we can’t tolerate ambiguity!
Brussels
AND
Amsterdam OR
Calais
AND
Amsterdam
Brussels OR
Calais

8. Parentheses, brackets & braces
Grouping devices avoid ambiguity (for “unique readability”):
I’ll go to Amsterdam, and then to either Brussels or Calais A ∙ (B ∨ C)
I’ll either go to Amsterdam and Brussels, or else to Calais (A ∙ B) ∨ C
Amsterdam
Calais
Brussels
AND OR
Amsterdam
Brussels
Calais
AND OR

9. Variables: p, q, … p  (q  r)
Sometimes we want to talk about all sentences of a given form, e.g.
A  (B  C)
F  (M  X)
(K  M)  [(N   O)  P]
So we use variables as place-holders
Each of the above sentences is of the form:
p  (q  r)

10. Plugging into variables
Modus Ponens
Substitution Instance of Modus Ponens 
p  q p q
((A  B)  C)
(D  (E   F))
((A  B)  C)
(D  (E   F))
Variables are like expandable boxes
To do proofs in logic you have to see how sentences plug into those boxes.

11. Plugging into variables
Modus Ponens
Substitution Instance of Modus Ponens 
p  q p q
((A  B)  C)
(D  (E   F))
((A  B)  C)
(D  (E   F))
Variables are like expandable boxes
To do proofs in logic you have to see how sentences plug into those boxes.

12. The Grammar of Propositional Logic
Constructing WFFs (Well-Formed Formulae)
Identifying WFFs
Identifying main connectives

13. Rules for WFFs A sentence letter by itself is a WFF A B Z
The result of putting  immediately in front of a WFF is a WFF A  B   B  (A  B)  ( C  D)
The result of putting  ,  ,  , or  between two WFFs and surrounding the whole thing with parentheses is a WFF (A  B) (  C  D) ((  C  D)  (E  (F   G)))
Outside parentheses may be dropped A  B   C  D (  C  D)  (E  (F   G))

14. WFFs
A sentence that can be constructed by applying the rules for constructing WFFs one at a time is a WFF
A sentence which can't be so constructed is not a WFF
No exceptions!!!
woof

15. Main Connective
In constructing a WFF, the connective that goes in last, which has the whole rest of the sentence in its scope, is the main connective.
This is the connective which is the “furthest out.”
Examples
(  C  D)  (E  (F   G))
 ( C  D)

16. Hints: When it’s not a WFF
You can't have two WFFs next to one another without a two- sided connective between them. BAD! AB C  D (E  F)G
Two-sided connectives have to have WFFs attached to both sides. BAD!  A (B  C)  ( D  E) G   H
You can't have more than one two-sided connective at the same level BAD! A  B  C (  C  D)  (E  F   G)

17. Identifying WFFs & Main Connectives∨
1 (S   T)  ( U  W)
2  (K  L)  ( G  H)
3 (E  F)  (W  X)
4 (B   T)   ( C  U)
5 (F   Q)  (A  E  T) X X ≡ X

18. Identifying WFFs & Main Connectives
 1 (S   T)  ( U  W)
X 2  (K  L)  ( G  H)
X 3 (E  F)  (W  X)
 4 (B   T)   ( C  U)
X 5 (F   Q)  (A  E  T)

19. Identifying WFFs & Main Connectives
6  D   [ ( P  Q)  (T  R) ]
7 [ (D   Q)  (P  E) ]  [A  (  H) ]
8 M (N  Q)  ( C  D)
9  (F   G)  [ (A  E)   H]
10 (R  S  T)   ( W   X) ∨ X X ⊃ X

20. Identifying WFFs & Main Connectives
 6  D   [ ( P  Q)  (T  R) ]
X 7 [ (D   Q)  (P  E) ]  [A  (  H) ]
X 8 M (N  Q)  ( C  D)
 9  (F   G)  [ (A  E)   H]
X 10 (R  S  T)   ( W   X)

21. Why should we care about this?
Because in formal logic we determine whether arguments are valid or not by reference to their form.
And that assumes we can identify the form of sentences, i.e. that we can identify main connectives.
In doing formal derivations in particular, we have be able to immediately see what the forms of sentences are in order to formulate strategies.

22. Translation

23. Conditionals & Biconditionals
If P then Q
P  Q
P, if Q
Q  P
P only if Q
P  Q
P if and only if Q
P  Q
Note: A biconditional is a “conditional going both ways”: so P  Q is the conjunction of P  Q and Q  P

24. Conditionals If P then Q P  Q P, if Q Q  P P only if Q P  Q
5 If Chanel has a rosewood fragrance then so does Lanvin. C  L
6 Chanel has a rosewood fragrance if Lanvin does. L  C
8 Reece Witherspoon wins best actress only if Martin Scorsese wins best director. W  S

25. Biconditionals P if and only if Q P  Q
7 Maureen Dowd writes incisive editorials if and only if Paul Krugman does. D  K
A biconditional is a “conditional going both ways”: so P  Q is the conjunction of P  Q and Q  P. “Only if” is only half of “if and only if.” Be careful!

26. Not both and & neither/nor
Not both P and Q
~ (P  Q)
Neither P nor Q
 (P  Q)
You can’t both have your cake and eat it.
~ (H  E)
She was neither young nor beautiful.
 (Y  B)

27. Not both and & neither/nor
Not both P and Q
~ (P  Q)
Neither P nor Q
 (P  Q)
15 Not both Jaguar and Porsche make motorcycles. ~ (J  P)
16 Both Jaguar and Porsche do not make motorcycles.  J  ~ P

28. Not both and & neither/nor
Not both P and Q
~ (P  Q)
Neither P nor Q
 (P  Q)
18 Not either Ferrari or Maserati makes economy cars. 19 Neither Ferrari nor Maserati makes economy cars.  (F  M)
20 Either Ferrari or Maserati does not make motorcycles.  F  ~ M

29. DeMorgan’s Laws ~ (P  Q) is equivalent to  P   Q
“She was neither young nor beautiful” is equivalent to “She was old and ugly” - NOT “She was old or ugly.”
“You can’t both have your cake and eat it” is equivalent to “You either don’t have your cake or you don’t eat your cake” - NOT “You don’t have your cake and you don’t eat your cake.”