Understanding Syntax and Translation in Logic

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Explore the world of syntax and translation in logic through topics such as forming well-formed formulas, identifying main connectives, De Morgan's Laws, Venn diagrams, necessary and sufficient conditions, and more. Discover the language of logic, vocabulary, truth-functional connectives, punctuation marks, and how to translate English sentences into logical statements. Dive into the syntax rules, expressions, and well-formed formulas of logic, understanding the recursive definition and principles behind creating valid logical structures.

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Syntax and Translation Syntax and Translation Gregory 2.3-2.4

Contents Contents WFFs: Well-Formed Formulas of S - Identifying WFFs and Main Connectives Translation - Negation-Conjunction-Disjunction: De Morgan s Laws oA little set theory and more Venn Diagrams - Conditionals: Necessary and Sufficient Conditions - Parentheses: punctuating symbolized sentences

The Language The Language S S Vocabulary: Grammatical Categories - Statement Letters A, B, C, . . . A1, B1, C1, . . . , Z1, A2, . . . - Truth-Functional Connectives , , , , - Punctuation Marks (, ), [, ], {, } Translating English sentences - Statement letters translate simple sentences, which have no proper parts that are themselves sentences - Connectives ( sentence-forming operators ) build more complex sentences from simpler sentences

2.3 The Syntax of 2.3 The Syntax of S S Rules for forming WFFs and Syntax Trees

2.3 Expressions of 2.3 Expressions of S S An expression of S is any finite sequence of the symbols of S (not necessarily grammatical!) What s wrong with the expressions on the right?

WFFs (Well WFFs (Well- -Formed Formulas) of Formed Formulas) of S S (1) If is a statement letter, then is a wff of S (2) If and are wffs of S, then (a) is a wff of S (b) ( ) is a wff of S (c) ( ) is a wff of S (d) ( ) is a wff of S (e) ( ) 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)

Recursive (Inductive) Definition Recursive ( Inductive ) Definition Basis Clause specifies initial members of the set of things to which the term in question applies Sentence letters are WFFs of S Recursive (Inductive) Clause specifies how to generate further members of the set of things to which the term applies Rules for forming sentences using the connectives (parentheses are typographically parentheses, brackets or braces and outer parenthesis may be dropped). Extremal Clause states that the term applies to only those things specified by the basis and recursive clauses ( That s all, folks! ) Nothing is a WFF of Sunless it s a result of finite number of applications of above clauses.

Atomic & Molecular Formulas, Main Connectives Atomic & Molecular Formulas, Main Connectives Any WFF that qualifies simply in virtue of clause (1) of the definition of a WFF is called atomic - The statement letters of S are atomic WFFs - Atomic WFFs therefore do not include any connectives Compound WFFs, constructed according to the Clause 2 rules of the definition are molecular. - Each molecular WFF has exactly one main connective (though may have others so... - Each is each is either of the form , , , , or

Syntactic Concepts and Conventions Syntactic Concepts and Conventions The main connective of a molecular WFF is the connective appearing in the clause of the definition of a WFF cited last in showing to be a WFF. The immediate well-formed components of a molecular WFF are the values of and (in the case of clause (2a) simply ) in the last-cited clause of the definition of a WFF--i.e. the connective that went in last. 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 that are atomic WFFs. Scope: The scope of a connective is that portion of the WFF containing its immediate well-formed component(s).

Building WFFs Building WFFs {[( A B) C] (D E)} is a WFF: how did this happen? A , B , C , D , and E are WFFs by Clause 1 A is a WFF by Clause 2 ( A B) is a WFF by Clause 2 [( A B) C] is a WFF by Clause 2 (D E) is a WFF by Clause 2 {[( A B) C] (D E)} by Clause 2

2.3 WFFs and Syntax Trees: p 49 Examples 2.3 WFFs and Syntax Trees: p 49 Examples WFFs of S Not WFFs of S (9) A73 (15) (A73 ) (10) A73 (16) A73 (18) (? E) (12) (D E) (13) ((A B) C) (13) (A B C) (14) ((A B) ( B C)) (14) ((A B) ( B C))

Syntax tree showing how WFF was built Syntax tree showing how WFF was built

2.3.2 Syntactic Concepts and Conventions 2.3.2 Syntactic Concepts and Conventions Atomic Formula, Molecular Formula - Any wff that qualifies simply in virtue of clause (1) of the definition of a wff (that is, any wff that just is some statement letter), is called an atomic formula, wff, or sentence. By analogy, all other wffs are molecular. Outer parentheses (which aren t doing any work) may be dropped, e.g. - (A (B D)) is OK -- and A (B C) is also OK but - A B C is NOT OK (note ambiguity)

Identifying WFFs and Main Connectives Identifying WFFs and Main Connectives Write the main connective of each sentence in the space provided if it is a WFF; if it s not a WFF write X ___ [(A B) C] (D ___ A B E) ___ (A B) C (D X ___ (A B) C E) ___ A (B C) ___ (A B) [C (D X E)] ___ A B C X ___ A ___ ( A) X ___ [A (B C)]

2.4 Symbolization 2.4 Symbolization Translating English Sentences into the Language of S

Negation, Conjunction, Disjunction Negation, Conjunction, Disjunction S doesn t allow compound subjects or predicates E.g. Dick and Jane both went = Dick went and Jane went = D J DeMorgan s Laws: neither nor... ; not both...and... ( ) ( ) Neither nor Not and not ( ) ( ) Not both and Not or not Examples: (Y B) or Y B She was neither young nor beautiful. (H E) or H E You can t both have your cake and eat it.

De Morgans Laws in Logic and Set Theory De Morgan s Laws in Logic and Set Theory the negation of a disjunction is the conjunction of the negations; and the negation of a conjunction is the disjunction of the negations. ( ) ( ) Neither nor Not and not ( ) ( ) Not both and Not and not the complement of the union of two sets is the same as the intersection of their complements; and the complement of the intersection of two sets is the same as the union of their complements. (A B)c = (Ac Bc) (A B)c = (Ac Bc) Complement A union B = complement A intersection complement B Complement A intersection B = complement A union complement B

Venn Diagrams represent sets Venn Diagrams represent sets Things that are neither A nor B Things that are both A and B Things that are B but not A Things that are A but not B A B U

Union, Intersection, Complement Union, Intersection, Complement Fig 1 represents A union B in yellow - Blue is complement of A union B--and intersection of complements of A and B Fig 2 represents A intersection B in yellow - Blue is complement of A intersection B-- and union of complements of A and B

Venn Diagrams represent sets Venn Diagrams represent sets You can t both have have cake and eat it. Nothing here! Havers Eaters U

Venn Diagrams represent sets Venn Diagrams represent sets She was neither young nor beautiful. She isn t anywhere here! Young Beautiful U

Translation: not both Translation: not both and, neither...nor and, neither...nor You can t both have your cake and eat it. (H E) or H E or H E or E H She was neither young nor beautiful. (Y B) or Y B Sweet or cheese but not both. (S C) (S C) Working hard is neither necessary nor sufficient for success [(S W) (W S)]

2.4.2 Translation Exercises 2.4.2 Translation Exercises 1 The Falcons do not score a goal G 2 The Falcons and the Mustangs win their semifinal matches. W V 3 Either the Falcons or the Mustangs win the Cup. F M 4 Santacruz plays for the Falcons but Khumalo does not. S K 5 Khumalo either does or does not play for the Falcons. K K

2.4.2 Exercises 2.4.2 Exercises 6 Neither the Falcons nor the Mustangs win the cup (F M) 7 Both the Falcons and the Mustangs don t win the cup F M 8 Not both the Falcons and the Mustangs win the Cup (F M) 9 Either the Falcons or the Mustangs do not win the Cup F M ( note De Morgan equivalence to (F M) above! ) 10 Santacruz plays for the Falcons and captains the Falcons, but does not play for the Mustangs. (S C) Z

2.4.2 Exercises 2.4.2 Exercises 11 Either Nakata and Khumalo both play for the Falcons, or Khumalo plays for the Mustangs. (N K) O 12 Nakata plays for the Falcons, and Khumalo plays for either the Falcons or the Mustangs N (K O) 13 Either Khumalo, Nakata, or Santacruz plays for the Falcons K (N S) or (K N) S ( is associative!) 14 Either both teams score a goal, or neither team wins the Cup. (G H) (F M) ( consider De Morgan equivalence for (F M ) 15 Santacruz plays for and captains the Falcons, and either they score a goal or they don t win the Cup (S C) (G F)

Translating Conditionals and Biconditionals Translating Conditionals and Biconditionals

Translating Conditionals and Biconditionals Translating Conditionals and Biconditionals Antecedent sufficient for consequent; consequent necessary for antecedent. Example: If x is a dog then x is a mammal oBeing a dog is a sufficient condition for being a mammal. oBeing a mammal is a necessary condition for being a dog. Necessary and Sufficient Conditions If P then Q, P only if Q, Q, if P P is sufficient for Q, Q is necessary for P P if and only if Q P is necessary and sufficient for Q (and vice versa)

Necessary Necessary & & Sufficient Sufficient Conditions Conditions Necessary and sufficient mean exactly what you think they mean! Necessary means required - Being at least 21 is a necessary condition for drinking legally in California. Sufficient means enough - A blood alcohol level of exactly 0.08 is a sufficient condition on being legally drunk in California.

Necessary and Sufficient Conditions Necessary and Sufficient Conditions

Necessary Necessary & & Sufficient Sufficient Conditions Conditions The state of affairs described in the antecedent is asserted to be a sufficient condition on the state of affairs described in the consequent. The state of affairs described in the consequent is asserted to be a necessary condition on the state of affairs described in the antecedent. is sufficient for If it rains then it pours. is necessary for

Antecedent Antecedent Sufficient Sufficient/Consequent /Consequent Necessary Necessary (1) If someone is a mother then they re female - If you know that someone is a mother, that s enough to tell you that person is female, so being a mother is a sufficient condition on being female. - Being a mother is not a necessary condition on being female since you can be female without being a mother. - Being female is necessary for being a mother: if someone is not female they can't possibly be a mother. Thus (1) says that being a mother is a sufficient condition on being female and being female isa necessary condition on being a mother.

Necessary, Sufficient, Both, or Neither? Necessary, Sufficient, Both, or Neither? Being a dog is __________ for being a mammal. sufficient Being a mammal is __________ for being a dog. necessary Paying your tuition is __________ graduation from USD. necessary Getting a 95% is __________ for getting an A in this course. sufficient neither Coming to every class meeting is __________ for passing this course. x being odd is __________ for xy being odd. necessary x being even is __________ for xy being even. sufficient necessary sufficient x and y being both being odd is __________ for xy being odd.

Contrapositive Contrapositive The contrapositive of a conditional is the result of flipping and negating its antecedent and consequent. - The contrapositive of If P then Q is If NOT-Q then NOT-P. A conditional is logically equivalent to its contrapositive - P and Q are logical equivalent iff they necessarily have the same truth value. This explains why we understand the consequent of a conditional as a necessary condition for the antecedent. - If Amalasuntha is a dog then Amalasuntha is a mammal. - If Amalasuntha is not a mammal then Amalasuntha is not a dog.

WFF!

2.4.4 Exercises 2.4.4 Exercises 1 If Nakata plays for the Falcons, then the Falcons win the Cup. N F 2 The Falcons score a goal, if Khumalo plays for them. K G 3 The Falcons score a goal only if Khumalo plays for them. G K 4 The Falcons score a goal if and only if Khumalo plays for them. G K 5 Santacruz plays for the Falcons, assuming she doesn t play for the Mustangs. Z S

2.4.4 Exercises 2.4.4 Exercises 6 The Falcons winning their semifinal match is necessary for their winning the Cup. F W 7 The Falcons winning the final match is sufficient for their winning the Cup. L F 8 The Falcons winning the final is match is necessary and sufficient for winning the Cup. L F 9 The Mustangs win the Cup unless the Falcons score a goal. M G or M G etc. 10 Unless Santacruz doesn t play for the Falcons, the Mustangs will not win the Cup. S M or S M or M S or S M

2.4.5 Complex Symbolization 2.4.5 Complex Symbolization (1) Assuming Nakata and Khumalo play for the Falcons, the Falcons will win the Cup. (N K) F (2) The Falcons win the Cup only if Santacruz plays for and captains the Falcons. F (S C) (3) Santacruz must play for and captain the Falcons, if they are to win the Cup. F (S C) (4) The Falcons win the Cup if and only if they win both their semifinal and final matches. F (W L) (5) If the Falcons and the Mustangs win their semifinal matches, then either the Falcons or the Mustangs win the Cup, but not both. (W V) [(F M) (F M) or (W V) ( F M)

Parentheses, Braces, & Brackets: Punctuation Parentheses, Braces, & Brackets: Punctuation 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

Parentheses, Brackets & Braces: Punctuation Parentheses, Brackets & Braces: Punctuation Grouping devices avoid ambiguity (for unique readability ): - I ll go to Amsterdam, and then to either Brussels or Calais: A (B C) Brussels AND Amsterdam OR Calais - I ll either go to Amsterdam and Brussels, or else to Calais: (A B) C AND Amsterdam Brussels OR Calais

Complex Symbolization: Punctuation Complex Symbolization: Punctuation (6) If Santacruz and Khumalo don t both play for the Mustangs, then the Mustangs will win the Cup only if Santacruz or Nakata doesn t play for the Falcons. (Z ) [M ( S N)] or ( Z O) [M (S N)] (7) If the Falcons and the Mustangs win their semifinal matches, then either the Falcons or the Mustangs win the Cup, but not both (W V) [(F M) (F M) or (W V) ( F (8) Either the Falcons win the Cup or Nakata doesn t play for them; moreover Nakata doesn t play for them. (F N) N (9) If Santacruz plays for the Falcons, then if she captains the Falcons then the Falcons will score a goal and win the cup. F [C (G F)] M)

2.4.6 Exercises 2.4.6 Exercises 1. Santacruz plays for the Falcons or the Mustangs, but she does not captain the Falcons. (S Z) C 2. Either Nakata and Khumalo play for the Falcons, or Santacruz plays for the Mustangs . (N K) Z 3. Either the Falcons or the Mustangs win the Cup, but not both. (F M) (F M) 4. The Falcons and the Mustangs both win their semifinal match, and the Mustangs or the Falcons don t win the Cup. (W V) ( M F)

2.4.6 Exercises 2.4.6 Exercises 5. Either Santacruz plays for the Falcons and the Falcons score a goal, or Khumalo plays for the Mustangs and the Mustangs score a goal. (S G) (O F) 6. The Falcons win the Cup if and only if either Nakata or Santacruz play for them. (N S) F 7. If the Mustangs won their semifinal match but did not win the Cup, then they did not win the final match. (V M) T 8. The Falcons will not win the Cup unless Nakata plays for them. N F or F N or N F

2.4.6 Exercises 2.4.6 Exercises 9. Unless Santacruz and Khumalo both play for the Falcons, the Mustangs will win the Cup. (S K) M or M (S K) or (S K) M 10. Only if Nakata plays for the Falcons and the Mustangs don t score a goal, will the Falcons win the final match and the Cup. (L F) (N H) 11. For the Falcons to score a goal it is necessary that Santacruz and Khumalo play for them. G (S K) 12. For the Falcons to win the final match and the Cup it is sufficient that Santacruz plays for them and captains them. (S C) (L F)

2.4.6 Exercises 2.4.6 Exercises 13. If Khumalo, Nakata, and Santacruz all play for the Falcons, then the Mustangs will not score a goal nor will they win the Cup. [(K N) S] ( H M) or [(K N) S] (H M) 14. The Falcons score a goal and win the Cup if Santacruz and Khumalo play for them, and only if Santacruz and Khumalo play for them. (S K) (G F) 15. If Khumalo and Santacruz play for the Mustangs, then if Nakata doesn t play for the Falcons, then the Falcons will win the Cup only if the Mustangs don t score a goal. (O Z) [ N (F H)]