Going Formal Meet the Connectives The Language of

Going Formal Meet the Connectives

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

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 }

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

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 – If it rains then it pours R∙C R⊃P

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

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) Amsterdam Brussels AND OR Calais – I’ll either go to Amsterdam and Brussels, or else to Calais (A ∙ B) ∨ C Amsterdam OR Calais AND Brussels

Variables: p, q, … • 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)

Plugging into variables Modus Ponens p q Substitution Instance of Modus Ponens ((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.

Plugging into variables Modus Ponens p q Substitution Instance of Modus Ponens ((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.

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

Rules for WFFs 1. A sentence letter by itself is a WFF A 2. Z The result of putting immediately in front of a WFF is a WFF A 3. B 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) 4. ( C D) (( C D) (E (F G))) Outside parentheses may be dropped A B C D ( C D) (E (F G))

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

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)

Hints: When it’s not a WFF • You can't have two WFFs next to one another without a twosided 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)

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)

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)

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)

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)

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.

Translation

Conditionals & Biconditionals If P then Q P, if Q P only if Q P Q Q P 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

Conditionals If P then Q P, if Q P only if Q P Q Q P 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

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!

Not both and & neither/nor Not both P and 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)

Not both and & neither/nor Not both P and 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

Not both and & neither/nor Not both P and 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

De. Morgan’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. ”