PHIL 114

(M1) B = Bob was elected; J = John was elected; M = Mary was elected Translate: John and Mary were elected, but Bob wasn't.

((J & M) & ¬B)

(M1) B = Bob was elected; J = John was elected; M = Mary was elected Translate: If either John or Mary was elected, then Bob was too.

((J v M) → B)

(M1) B = Bob was elected; J = John was elected; M = Mary was elected Translate: John was elected if Bob wasn't elected and neither was Mary.

((¬B & ¬M) → J)

(M3) Domain: The world and everything in it. J(x) x is a jellybean R(x) x is red G(x) x is green B(x) x is black Either all of the jellybeans are black, or some of them are red.

((∀x)(J(x) → B(x)) ∨ (∃x)(J(x) & R(x)))

(Q11) ((∀x)B(x) → (∀x)C(x)) [IS/IS NOT] in prenex normal form.

((∀x)B(x) → (∀x)C(x)) [IS NOT] in prenex normal form.

(Q11) ((∀x)B(x) ∨ (∀x)C(x)) [IS/IS NOT] in prenex normal form.

((∀x)B(x) ∨ (∀x)C(x)) [IS NOT] in prenex normal form.

(M2) (A v B) [IS/IS NOT] a literal.

(A v B) [IS NOT] a literal.

(Q7) (A ↔ B) [IS/IS NOT] a literal.

(A ↔ B) [IS NOT] a literal.

(Q7) (B & A) [IS/IS NOT] a literal.

(B & A) [IS NOT] a literal.

(M2) E (E → (B & ¬C)) (B & ¬C) ___ is a positive subformula of ____.

(B & ¬C) is a positive subformula of (E → (B & ¬C))

(Q7) (B v A) [IS/IS NOT] a literal.

(B v A) [IS NOT] a literal.

(Q7) (B → A) [IS/IS NOT] a literal.

(B → A) [IS NOT] a literal.

(M3) Domain: The world and everything in it. J(x) x is a jellybean R(x) x is red G(x) x is green B(x) x is black Either some of the jellybeans are black, or some are red.

(E.x)(J(x) & B(x)) v. (J(x) & R(x))

(M2) (J & ¬M) [IS/IS NOT] a logical consequence of ((J & K) & (L v ¬M)).

(J & ¬M) [IS NOT] a logical consequence of ((J & K) & (L v ¬M)).

(M1) (L & ¬M) (L & M) (¬M → (N & (L & ¬M))) ___ is a subformula of ____.

(L & ¬M) is a subformula of (¬M → (N & (L & ¬M)))

(Q2) B Bob is gathering eggs. C Bob has fed the chickens. D Bob has fed the ducks. P The ducks are playing in the pond. The ducks are playing in the pond, and Bob has not fed them.

(P &~D)

(M2) (P → (Q → R)) [IS/IS NOT] logically equivalent to ((P & Q) → R).

(P → (Q → R)) [IS] logically equivalent to ((P & Q) → R).

(M3) j John h Harry m Monday t Tuesday d Wednesday e The Eiffel Tower l The Tower of London p The Parthenon V(x,y,z) x visited y on z Either Harry visited the Eiffel Tower on Monday or John visited the Tower of London on Tuesday.

(V(h,e,m) v.V(j,l,t))

(M3) j John h Harry m Monday t Tuesday d Wednesday e The Eiffel Tower l The Tower of London p The Parthenon V(x,y,z) x visited y on z Harry visited either the Parthenon on Monday or the Tower of London on Wednesday.

(V(h,p,m) v. V(h,l,d))

(Q11) (∀x)(∀y)(B(x,y) & ¬B(x,y)) [IS/IS NOT] in prenex normal form.

(∀x)(∀y)(B(x,y) & ¬B(x,y)) [IS] in prenex normal form.

(Q11) (∀x)(∀y)¬B(x,y) [IS/IS NOT] in prenex normal form.

(∀x)(∀y)¬B(x,y) [IS] in prenex normal form.

(Q11) (∀x)B(x,a) [IS/IS NOT] in prenex normal form.

(∀x)B(x,a) [IS] in prenex normal form.

(Q3) How many rows would you need in the truth-table for a formula containing 5 different atomic formulae?

32 rows ( 2^atomic formulae = 2^5)

(Q7) A [IS/IS NOT] a literal.

A [IS] a literal.

(Q7) ((P ∨ ¬Q) & (R ∨ Q)) A. Conjunctive normal form B. Disjunctive normal form C. Both conjunctive and disjunctive normal form D. Neither

A. Conjunctive normal form

(Q7) (R & (P v ¬R)) A. Conjunctive normal form B. Disjunctive normal form C. Both conjunctive and disjunctive normal form D. Neither

A. Conjunctive normal form

(Q1) The gerbil is hiding from the cat in the teapot. A. Declarative B. Interrogative C. Imperative

A. Declarative

(M1) Which of the following words and phrases are premise indicators? A. because B. since C. for (the reason that) D. thus E. as a result

A. because B. since C. for (the reason that)

(Q6) Which of the binary connectives are communtative? A. conjunction B. disjunction C. the conditional

A. conjunction B. disjunction

(Q1) Which of the following words and phrases are premise indicators? A. for (the reason that) B. because C. thus D. since E. as a result

A. for (the reason that) B. because D. since

(Q1) Which of the following words and phrases are conclusionindicators? A. hence B. therefore C. inasmuch as D. accordingly E. for (the reason that)

A. hence B. therefore D. accordingly

(Q2) (G → ((¬K ∨ J) & ((H & E) & F))) Select all subformula: A. (G → (¬K ∨ J)) B. (¬K ∨ J) C. (K ∨ J) D. (G ∨ K) E. (E & F) F. (H & E) G. there are additional subformulae not listed

B. (¬K ∨ J) F. (H & E) G. there are additional subformulae not listed

(Q2) B Bob is gathering eggs. C Bob has fed the chickens. D Bob has fed the ducks. F Bob is chasing the fox away. P The ducks are playing in the pond. Q The ducks are quacking loudly. S The chickens are playing in the pond. ¬(D & C) A. Bob hasn't fed both the ducks and the chickens. B. Bob hasn't fed both the chickens and the ducks. C. Bob hasn't fed the ducks and he hasn't fed the chickens. D. Bob hasn't fed the chickens and he hasn't fed the ducks.

B. Bob hasn't fed both the chickens and the ducks.

(M2) (¬P v (Q & R)) A. Conjunctive normal form B. Disjunctive normal form C. Both conjunctive and disjunctive normal form D. Neither

B. Disjunctive normal form

(Q7) ((P & Q) ∨ R) A. Conjunctive normal form B. Disjunctive normal form C. Both conjunctive and disjunctive normal form D. Neither

B. Disjunctive normal form

(M1) Was it the gerbil or the cat that got into the teapot? A. Declarative B. Interrogative C. Imperative

B. Interrogative

(Q1) Why are there paw print-shaped tea stains everywhere? A. Declarative B. Interrogative C. Imperative

B. Interrogative

(M3) What is the top-level structure of the well-formed formula (R(a,b) & R(b,a))? A. It is an atomic formula. B. It has a connective as its main operator. C. It has a quantifier as its main operator. D. It has a top-level structure not listed above.

B. It has a connective as its main operator.

(M3) What is the top-level structure of the well-formed formula ¬R(a,w)? A. It is an atomic formula. B. It has a connective as its main operator. C. It has a quantifier as its main operator. D. It has a top-level structure not listed above.

B. It has a connective as its main operator.

(M2) What are the contradictions that will be candidates for any application of an indirect rule of the following derivation? (P → T) (¬T → ¬R) (¬P → R) ... Goal: T A. P and ¬P B. R and ¬R C. T and ¬T

B. R and ¬R

(Q3) Which of the truth-tree rules are branching rules? A. conjunction B. disjunction C. conditional D. negated conjunction E. negated disjunction F. negated conditional G. double negation

B. disjunction C. conditional D. negated conjunction

(Q7) ((P & ¬R) & Q) A. Conjunctive normal form B. Disjunctive normal form C. Both conjunctive and disjunctive normal form D. Neither

C. Both conjunctive and disjunctive normal form

(Q7) (P & Q) A. Conjunctive normal form B. Disjunctive normal form C. Both conjunctive and disjunctive normal form D. Neither

C. Both conjunctive and disjunctive normal form

(Q1) Don't let the cat and the gerbil knock over the teapot again! A. Declarative B. Interrogative C. Imperative

C. Imperative

(Q1) Don't let the cat chase the gerbil into the teapot! A. Declarative B. Interrogative C. Imperative

C. Imperative

(M3) When constructing a truth-tree for a valid argument, which of the following statements is true? A. The tree cannot be completed. B. The tree can be completed, and might or might not be closed when complete. C. The tree can be completed, and will be closed when complete.

C. The tree can be completed, and will be closed when complete.

(Q6) According to DeMorgan's laws, a disjunction of negations is equivalent to what? A. a conjunction of negations B. a negated disjunction C. a negated conjunction

C. a negated conjunction

(Q7) For which of the following formulae is ((M & N) & ¬L) the canonically constructed disjunctive normal form equivalent? A. ((M & N) → L) B. ((M & N) → ¬L) C. ¬((M & N) → L) D. (¬(M & N) → L)

C. ¬((M & N) → L)

(M1) Match each connective name with the symbol we have chosen to use for that connective: Conjuction [ ] Disjunction [ ] The Conditional [ ] Negation [ ] - → - & - v - ¬

Conjuction [&] Disjunction [v] The Conditional [→] Negation [¬]

(M1) Match each connective name with the corresponding English word or phrase: Conjuction [ ] Disjunction [ ] The Conditional [ ] Negation [ ] - [either...or] - [not] - [and] - [if...then]

Conjuction [and] Disjunction [either...or] The Conditional [if...then] Negation [not]

(M3) Is (∀x)(∀y)(G(x,y) ∨ ¬G(y,x)) a tautology, contingent, or contradictory?

Contingent

(M3) Is (∀x)R(x,x) a tautology, contingent, or contradictory?

Contingent

(Q9) Is (∀x)R(x,x) a tautology, contingent, or contradictory?

Contingent

(M1) B = Bob was elected; J = John was elected; M = Mary was elected Translate: Not all of Mary, Bob, and John were elected.

Either are correct: ¬((M & B) & J) ¬(M & (B & J))

(M3) a Aristotle s Socrates n Xanthippe (Socrates' wife) G(x) x is Greek P(x) x is a philosopher M(x,y) x and y are married L(x) x has a system of logic named after him/her T/F: (¬L(a) ∨ ¬G(a))

False

(M1) G (H → F) ((F → H) & G) ___ is a subformula of ____.

G is a subformula of ((F → H) & G)

(M3) Is ((∀x)B(x) ∨ C(y)) well-formed?

Is ((∀x)B(x) ∨ C(y)) well-formed?

(M3) Is the following formula true or false on the given interpretation? (∀x)(P(x) → G(o,x))

It's false

(M3) Is the following formula true or false on the given interpretation? (∃x)(∀y)G(x,y)

It's false

(M1) B = Bob was elected; J = John was elected; M = Mary was elected Translate: John was elected, but Mary wasn't.

J & ¬M

(M2) (A → (¬S → M)) M ¬S ___ is a positive subformula of ____.

M is a positive subformula of (A → (¬S → M))

(Q2) What is the order of precedence for the connectives (from highest to lowest)? Conjuction [&] Disjunction [v] The Conditional [→] Negation [¬]

Negation [¬] Conjuction [&] Disjunction [v] The Conditional [→] N C D TC

(M3) Is (∀z)(E x) (¬S(z,x) & ¬S(x,z)) well-formed?

No, it is not well-formed.

(Q5) Can negation introduction (¬I) be used to derive any type of formula other than a negation?

No.

(M3) Domain: The world and everything in it. J(x) x is a jellybean R(x) x is red G(x) x is green B(x) x is black If all the jellybeans are black, then none are red.

There are two possible correct answers: ((∀x)(J(x) → B(x)) → ¬(∃x)(J(x) & R(x))) and ((∀x)(J(x) → B(x)) → (∀x)(J(x) → ¬R(x)))

(M3) I(a) = 2 I(b) = 3 I(c) = 4 I(R) = {<2,2,4>,<2,3,6>,<3,2,6>,<3,3,9>} T/F: (R(a,b,c) → R(c,b,a))

True

(M3) I(a) = 3 I(b) = 2 I(c) = 4 I(R) = {<2,2,4>,<2,3,6>,<3,2,6>,<3,3,9>} T/F: ¬R(a,c,b)

True

(Q5) Can negation elimination (¬E) be used to derive a negation?

Yes.

(Q1) Logic is the study of ___.

arguments

(M3) Both 0-place predicate letters and formulae of the form φ(τ1,...,τn) (i.e., an n-place predicate letter predicate letter followed by n constants) are referred to as ___ formulae.

atomic

(M2) The connective introduced in chapter 7 is called the ___.

biconditional

(M3) An occurrence of a variable υ in a formula is ___ just in case that occurrence is in the scope of a quantifier that has υ as its variable of quantification. An occurrence of a variable is ___ just in case it is not bound.

bound, free

(M2) ___ is distributive with respect to ___.

conjunction, disjunction - or - disjunction, conjunction

(Q6) ___ is distributive with respect to ___.

conjunction, disjunction disjunction, conjunction

(M1) Is (A → (B & ¬B)) a tautology, contingent, or contradictory?

contingent

(Q9) Is (∀x)(P(x,x) → (∃z)P(x,z)) a tautology, contingent, or contradictory?

contingent

(Q9) Is (∀y)R(a,y) a tautology, contingent, or contradictory?

contingent

(Q9) Is (∃z)(∀y)(F(y) → (G(z,y) & ¬F(z))) a tautology, contingent, or contradictory?

contingent

(M1) A formula of sentential logic is called ____ just in case it is false on every truth-value assignment.

contradictory

(Q3) A formula of sentential logic is called ___ just in case it is false on every truth-value assignment.

contradictory

(Q9) Is ((∀x)P(x) & (∃x)¬P(x)) a tautology, contingent, or contradictory?

contradictory

(Q9) Is ((∀x)R(x,x) & ¬(∀x)R(x,x)) a tautology, contingent, or contradictory?

contradictory

(M1) If an argument is valid, then it has no _____.

counterexamples

(Q3) If an argument is valid, then it has no ___.

counterexamples

(Q5) The double negation rules, DNI and DNE, are called ___ rules because any application of either one of them can be replaced with a derivation that produces the same formula.

derived

(Q7) A formula of sentential logic is in ___ normal form if and only if it is one of the following: A literal; A conjunction of literals; A disjunction of conjunctions of literals.

disjunctive

(Q8) The set of n-tuples of members of the domain of discourse we assign to a given n-place predicate letter in an interpretation is called the ___ of the predicate.

extension

(M2) In what order should the strategies be applied? refutation, inversion, division, extraction, conversion

extraction > conversion > inversion > division > refutation E C I D R Eric's Cock Is Damn Ripe

(M2) The symbol called the ___ is used in derivations to explicitly indicate that a ___ has been derived.

falsum, contradiction

(Q1) A ___ argument is one that has all true premises, and whose premises support the conclusion.

good

(M1) A ___ argument is one that has all true premises, and whose premises support the conclusion.

good or sound

(Q9) The two types of terms introduced in chapter 9 are ___ and ___.

individual constants, individual variables.

(M1) Consider the following argument: ¬(B v D) ¬H ------------ ∴ B Is the argument valid or invalid?

invalid

(M2) Can conjunction introduction (&I) be used to derive any type of formula other than a conjunction?

no

(M2) How many lines should be cited for an application of vIL?

one

(M2) How many lines should be cited for an application of ¬I?

one

(Q5) How many lines should be cited for an application of ¬I?

one

(Q2) A formula ψ is a subformula of a formula φ if and only if ψ appears (as a node) in the ___ of φ.

parse tree

(M3) The two types of basic expressions introduced in chapter 8 are ___ and ___.

predicates, individual constants

(Q8) The two types of basic expressions introduced in chapter 8 are ___ and ___.

predicates, individual constants

(Q9) ∀ and ∃ are called ___.

quantifier symbols

(Q1) Declarative sentences express ___, imperative sentences express ___, and interrogative sentences express ___. {questions, statements, commands}

statements, commands, questions

(Q3) A formula of sentential logic is called ___ just in case it is true on every truth-value assignment.

tautology

(Q9) Is (∀x)(P(x) ∨ ¬P(x)) a tautology, contingent, or contradictory?

tautology

(Q9) Is (∀x)(P(x,x) → (∃z)P(x,z)) a tautology, contingent, or contradictory?

tautology

(Q9) Is ¬(∃x)(F(x,x) & ¬(∃z)F(x,z)) a tautology, contingent, or contradictory?

tautology

(Q9) Is ¬(∃x)(P(x,x) & ¬(∃z)P(x,z)) a tautology, contingent, or contradictory?

tautology

(M2) Two formulae of sentential logic are logically equivalent if and only if they are assigned ___ truth-value(s) on every truth-value assignment.

the same

(Q6) The property of the conditional that the rule HS captures is called ___.

transitivity

(Q5) How many lines should be cited for an application of ⊥I?

two

(Q2) B Bob is gathering eggs. C Bob has fed the chickens. D Bob has fed the ducks. P The ducks are playing in the pond. Bob has fed neither the chickens nor the ducks.

~(C v. D)

(Q2) B Bob is gathering eggs. C Bob has fed the chickens. D Bob has fed the ducks. P The ducks are playing in the pond. It's not the case that if Bob has fed the chickens, then the ducks are playing in the pond.

~(C->P)

(Q2) ¬(A & B) (¬(A & B) & (¬(A ∨ ¬B) ∨ (A & B))) ¬(A ∨ B) ___ is a subformula of ___.

¬(A & B) is a subformula of (¬(A & B) & (¬(A ∨ ¬B) ∨ (A & B)))

(M2) ¬B [IS/IS NOT] a literal.

¬B [IS] a literal.

(Q2) (Q ∨ P) (Q → (R → (¬Q & P))) ¬Q ___ is a subformula of ___.

¬Q is a subformula of (Q → (R → (¬Q & P)))

(Q7) If (φ → ψ) is a tautology, then ___.

ψ is a logical consequence of φ