Logic midtem
Select ALL the sentences in negation normal form (NNF). A ~~A (~BvC)&(Dv~E) ~Bv~C ~A&~~B
A (~BvC)&(Dv~E) ~Bv~C
Transform this sentence into NNF. (Don't cite any justifications.) ~(~AvB)
A&~B
Label premises and conclusion to make a valid argument.
All criminals have broken the law. Conclusion All felons have broken the law. Premise All criminals are felons. Premise
Select all the sentences that are contingent.
All police dogs are dogs Rufus isn't Rufus. Rufus is Rufus. Correct! All dogs are police dogs. Correct! Rufus is happy. Correct! Some dogs are police dogs. Correct! No dogs are police dogs.
Evaluate whether each argument is valid or invalid. P == Pia is guilty. Argument 1 1. # Thus, 2. P Argument 2 1. P Thus, 2. # Argument 3 1. P Thus, 2. P Argument 4 1. P&~P Thus, 2. # Argument 5 1. # Thus, 2. P&~P Argument 6 1. Pv# Thus, 2. # Argument 7 1. Pv# Thus, 2. P
Argument 1 1. # Thus, 2. P Valid Argument 2 1. P Thus, 2. # Invalid Argument 3 1. P Thus, 2. P Valid Argument 4 1. P&~P Thus, 2. # Valid Argument 5 1. # Thus, 2. P&~P Valid Argument 6 1. Pv# Thus, 2. # Invalid Argument 7 1. Pv# Thus, 2. P Valid
Name that truth function: what has the following truth function? (Assume reference columns are in canonical form.) T F F F
Conjunction
Which equivalence principles are needed to transform the first sentence into the second? Select ALL that are needed. ~(~AvB)vC (AvC)&(~BvC) DeMorgan's Idempotence Distribution Double Negation Associativity Commutativity
DeMorgan's Distribution Double Negation
Name that truth function: what has the following truth function? (Assume reference columns are in canonical form.) T T T F
Disjunction
Use the truth-table method to evaluate this argument: 1. ~P&~Q Thus, 2. PvQ
Invalid, because of row 4
Select ALL the sentences equivalent to P&Q. ~(~Pv~Q) ~(PvQ) ~Qv~P ~(~P&~Q) Q&P P&~~Q
Q&P P&~~Q
A logical system is a ____________ we construct to study reasoning.
model/tool
What is the term for a sentence with all Ts in its truth function?
tautology
Translate: Neither Quinn nor Raquel is guilty.
~(QvR) or ~Q&~R
Select the minimal set of sentences to make a contradictory set. -If Pia is guilty, then Quinn is guilty. -If Quinn is innocent, then Raquel is guilty. If Quinn is guilty, then Raquel is innocent. -Raquel is innocent. -Quinn is innocent. -Pia is innocent.
-If Quinn is innocent, then Raquel is guilty. -Raquel is innocent. -Quinn is innocent.
Complete the formal proof of ((~QvP)vR)v(SvT) from ~~~~~Q. Note: ~Elim only allows you to drop 2 negations at a time. 1. ~~~~~Q Premise 2. _______ _______ 3. _______ _______ 4. _______ _______ 5. _______ _______ 6. _______ _______
1. ~~~~~Q Premise 2. ~~~Q ~Elim;1 3. ~Q. ~Elim;3 4. ~QvP vIntro;3 5. (~QvP)vR vIntro;4 6. ((~QvP)vR)v(SvT) vIntro;5
Consider this sentence: ~((PvQ)&R) Which connective has wide scope? (Type just the connective symbol, not a word.) Which connective has medium scope? Which connective has narrow scope?
~ & v
Which connective has narrow scope in each sentence? Write the symbol, not a word. ~A&B Cv~D ~(A&B) ~(Cv~D)
~A&B Cv~D ~(A&B) ~(Cv~D)
Which connective has wide scope in each sentence? Write a symbol, not a word. ~A&B Cv~D ~(A&B) ~(Cv~D)
~A&B Cv~D ~(A&B) ~(Cv~D)
Transform this sentence into NNF. (Don't cite any justifications.) ~((AvB)&~(CvD))
~A&~B)v(CvD) (~A&~B)v(CvD) (~A&~B)vCvD ((~A&~B)vC)vD
Which sentences will create a tautology, when put in place of the X? Select ALL that apply. Xv~(~P&(~QvP)) R ~P (PvQ) Q ~Q P
~P ~Q
Consider this pattern: R%~R What connective would you put in place of % in order to make a tautological falsity? (Write the symbol, not a word.) What connective would you put in place of % in order to make a tautology?
& v
Which of these are valid (always)? Select ALL that are true. -An argument's premises guarantee the conclusion is true. -An argument with necessarily true premises -An argument with a necessarily true conclusion -An argument with contradictory premises -A circular argument -An argument with a contradictory conclusion
-An argument's premises guarantee the conclusion is true. -An argument with a necessarily true conclusion -An argument with contradictory premises -A circular argument
Select the minimal set of sentences to make a contradictory set. -Quinn is innocent. -If Pia is innocent, then Quinn is innocent. -Pia is guilty. -If Quinn is innocent, then Raquel is innocent. -Raquel is innocent. -Pia is innocent.
-Pia is guilty. - Pia is innocent.
In the truth function of a sentence with 9 atomics, how many rows would there be? (Write an Arabic number.)
512
Fill in the blanks. _______ is the principles we reason with. _______ is the study of the principles we reason with.
Logic logic/Formal logic/Symbolic logic
Name that truth function: what has the following truth function? (Assume reference columns are in canonical form.) F T
Negation
Select all the sentences that are logically true.
Rufus is a dog. Correct! Rufus is Rufus. No dogs are police dogs. No police dogs are dogs. All dogs are police dogs. Correct! All police dogs are dogs
Assess this argument: 1. Seattle is in Washington. 2. If Seattle is in Washington, then it is in the US. Thus, 3. Seattle is in the US.
Sound
Are these sentences equivalent? Av(B&C) and (AvB)&C
no
Label each sentence a tautological contradiction, a logical contradiction, or neither. (By "logical contradiction" we mean a sentence that is a logical contradiction but not a tautological contradiction.) P == Pia is guilty. Z == Pia is Pia. 1. ~P _______ 2. ~Z _______ 3. Pv# Neither 4. # _______ 5. P&# _______ 6. ~Z&# Tautological falsity 7. P&~Z _______ 8. ~Zv# _______ 9. ~# _______ 10. ~~# Tautological falsity
1. ~P Neither 2. ~Z Logical falsity 3. Pv# Neither 4. # Tautological falsity 5. P&# Tautological falsity 6. ~Z&# Tautological falsity 7. P&~Z Logical falsity 8. ~Zv# Logical falsity 9. ~# Neither 10. ~~# Tautological falsity
Complete the formal proof of SvQ from ~~(P&Q)&R 1. ~~(P&Q)&R Premise 2. _______ _______ 3. _______ _______ 4. _______ _______ 5. _______ _______
1. ~~(P&Q)&R Premise 2. (P&Q)&R &Elim;1 3. P&Q &Elim;2 4. Q &Elim;3 5.SvQ vIntro;4
1. Can augmentation possibly destroy inductive support? 2. Can augmentation possibly destroy validity?
1.) yes 2.) no
There are many ways to do a formal proof of ~Pv~(~Q&(~PvQ)). Your job is to figure out which way we chose and complete the proof. Since there are no premises, we number a subproof assumption as line 1. Hint: always use reductio shortcut.
1: |~(~Pv~(~Q&(~PvQ))) Assume 2: | |~P Assume 3: | |~Pv~(~Q&(~PvQ)) vIntro;2 4: | | # #intro;1,3 5: | P ~intro;2-4 6: | | ~(~Q&(~PvQ)) Assume 7: | | ~Pv~(~Q&(~PvQ)) vintro;6 8: | | # #intro;1,7 9: | ~Q&(~PvQ) ~intro;6-8 10: | ~Q &elim;9 11: | ~PvQ &elim;9 12: | | ~P Assume 13: | |# #intro;5,12 14: | | Q Assume 15: | |# #intro;10,14 16: | # velim;11,12-13,14-15 17: ~Pv~(~Q&(~PvQ)) ~Intro;1-16
(Type Arabic numerals.) How many truth values do our systems have? How many truth values can a sentence have at one time?
2 1
A, B and C are three residents of the Island of Knights and Knaves. Exactly one of them is a werewolf. A says, "I am a werewolf." B says, "I am a werewolf." C says, "At most one of us is a knight." What can you infer? A: _______ B: _______ C: _______
A: Knave B: Knave C: Knight Werewolf: C
Next to the Island of of Knights and Knaves is the Island of Randoms, where all the residents (Randoms) randomly speak truly or falsely, in a totally random manner. You meet three people, A, B and C, and you know each of them is either a resident of the Island of Knights and Knaves or the Island of Randoms. You also know the following: 1. Exactly one of A, B and C is werewolf. 2. Exactly one of them is a knight. 3. The knight is the werewolf. This is what you hear: A says, "I am not a werewolf." B says, "That is true." C says, "B is not a Random." A: _______________ B: _______________ C: _______________ Werewolf: _______________
A: Random B: Knight C: Random Werewolf: B
Here is an argument about a group of friends and their pets. 1. Someone owns a dog and a parrot. 2. Someone owns a cat and a pig. 3. Someone owns a horse. 4. If someone owns a dog and a cat, then they also own a snake. 5. If someone owns a parrot or a horse, then they also own a fish. 6. If someone owns a pig, then they also own a dog. Thus: 7. Someone owns a fish. Here is a proof, broken into steps: Step 1: Proof: We know someone owns a cat and pig (premise 2). Step 2: From premise 3 it then follows that they also own a horse. Step 3: Since they own a parrot or horse, it follows they own a fish (premise 5). Step 4: Hence, someone owns a fish. Done. Evaluate where this proof is good or bad; if it's bad, decide where it first goes wrong.
Bad: step 2
Which equivalence principles are needed to transform the first sentence into the second? Select ALL that are needed. (Dropping parentheses around a single sentence does not require a principle.) ~Av~~(B&B) ~AvB Commutativity Associativity Idempotence DeMorgan's Distribution Double Negation
Idempotence Double Negation
Assess this argument: 1. Seattle is in the US. 2. If Seattle is in Washington, then it is in the US. Thus, 3. Seattle is in Washington.
Unsound: invalid
Label each claim true or false, in light BOOL's truth-table semantics. All tautologies are necessarily true. All atomic sentences are contingent. Some taut-falsities are contingent.
true true false
If the premises entail the conclusion, then the argument is ______________ .
valid
1. Imagine this is your premise: ~(P&Q)v(R&S) If you did proof by cases on it, what are your cases (in order)? Remember to drop outer parentheses, so don't write (R&S)! _______________ and _______________ 2. Imagine this is your premise: ~Av~~(BvC) What are your cases (in order)? _______________ and _______________ 3. Imagine this is your premise: (TvU)v(VvW) What are your cases (in order)? Drop outer parentheses! _______________ and _______________ 4. Imagine this is your premise: Dv~(E&F) What are your cases (in order)? _______________ and _______________
1. ~(P&Q) and R&S 2. ~A and ~~(BvC) 3. TvU and VvW 4. D and ~(E&F)
Complete this formal proof of B from ~A and AvB in as few lines as possible. Do cases from left to right! (Left-hand disjunct is the first case.) 1. ~A ______ 2. AvB Premise 3. | _______ _______ 4. | _______ _______ 5. | _______ _______ 6. | _______ _______ 7. _______ _______
1. ~A Premise 2. AvB Premise 3. | A Assume 4. | # #Intro;1,3 5. | B #Elim;4 6. | B. Assume 7. B. vElim;2,3-5,6
Which equivalence principles are needed to transform the first sentence into the second? Select ALL that are needed. (A&B)&C C&(A&B) DeMorgan's Commutativity Distribution Idempotence Double Negation Associativity
Commutativity
If X is some complex sentence that is a tautology of BOOL, then what is ~X? If X is some complex sentence that is a tautology of BOOL, then what is ~~X?
Taut-falsity Tautology
"If Uma didn't have a motive, then it's likely that she didn't do it." Which word indicates this is an inductive inference? _______________ "Stan's fingerprints were on the weapon. That guarantees he is guilty." Which word indicates this is a deductive inference? _______________
likely guarantees
What is meaning? What is grammar? What is truth? What is scope? What is use?
meaning = Semantic grammar = Syntactic truth = Semantic scope = Syntactic use = Pragmatic
1. A proof is a step-by-step ____________ . (Write one word; two possible correct answers.) 2. Informal proofs in this class are written in which language? 3. Formal proofs that we will soon learn are written in which language? _____________ 4. Informal proofs should start with the word ______________ and end with the word _________________ . 5. In a good proof, every step is ______________and_________________ .
1. explanation or demonstration 2. English or natural language 3. BOOL 4. Proof // done or QED 5. Obvious // valid
"Given that Pia and Raquel are innocent, we know Quinn is guilty." What is the signal word used? What kind of signal word is it? (Write premise or conclusion)
1. given given that "given" "given that" 2. premise
Complete this formal proof of P from PvP. The vertical bar | marks a line in a subproof. The justification "Assume" marks a subproof assumption. Note: Subproofs can have 1 line; there is no need to reiterate the assumption. When citing vElim with one-line subproofs, there will be no dash. 1. PvP Premise 2. | _______Assume 3. | _______Assume 4. P _______
1. PvP Premise 2. | P Assume 3. | P. Assume 4. P vElim;1,2,3
Which are good definitions of entailment? Select all that apply. -When the conclusion is true, the premises must be true. -When some premises make it likely that a conclusion is true. -When the premises are true, the conclusion cannot be false. - It is impossible for the premises to be true and the conclusion false.
-When the premises are true, the conclusion cannot be false. - It is impossible for the premises to be true and the conclusion false.
Complete this formal proof of ~(~A&~B) from A in as few lines as possible. Justify subproof assumptions with Assume. 1. A Premise 2. | _______ _______ 3. | _______ _______ 4. | _______ _______ 5. _______ _______
1. A Premise 2. | ~A&~B Assume 3. | ~A &Elim;2 4. | # #Intro;1,3 5. ~(~A&~B) ~Intro;2-4
Complete this formal proof of BvA from AvB in as few lines as possible. The vertical bar | marks a line in a subproof. Write the justification "Assume" to mark the start of a subproof assumption. 1. AvB Premise 2. | _______ _______ 3. | _______ _______ 4. | _______ _______ 5. | _______ _______ 6. _______ _______
1. AvB Premise 2. | A assume 3. | BvA vIntro;2 4. | B assume 5. | BvA vIntro;4 6. BvA velim;1,2-3,4-5
Idempotence has two forms, one for conjunction and one for disjunction: P&P ⇔⇔ P PvP ⇔⇔ P Complete these formal proofs of the same idea. First, we prove conjunction from left to right (the left-hand side is the premise, and the right-hand side is the conclusion). 1. P&P Premise 2. ______ _______ Second, prove the conjunction form from right to left: 1. P Premise 2. _______ _______ 3. _______ _______ Third, prove the disjunction form from right to left: 1. P Premise 2. _______ _______ (You'll learn how to prove the disjunction form from left to right in next chapter.)
1. P&P Premise 2. P &Elim;1 1. P Premise 2. P Reit;1 3. P&P &Intro;1,2 1. P Premise 2. PvP vIntro;1
Complete this formal proof of ~(~Pv~Q) from P&Q, following all conventions from the textbook. There are many ways to do it. Your job is to figure out which way we have in mind using these directions: - Pay attention to the subproofs! - When performing similar operations, go from left to right. 1. P&Q Premise 2. _______ _______ 3. _______ _______ 4. | _______ _______ 5. | | _______ _______ 6. | | _______ _______ 7. | | _______ _______ 8. | | _______ _______ 9. | _______ _______ 10. _______ _______
1. P&Q Premise 2. P &elim;1 3. Q &elim;1 4. | ~Pv~Q assume 5. | | ~P assume 6. | | # #intro;2,5 7. | | ~Q assume 8. | | # #intro;3,7 9. | # velim;4,5-6,7-8 10. ~(~Pv~Q) ~intro;4-9
Following the conventions in the book, complete this formal proof by filling in the justifications: 1. P&Q Premise 2. R&(SvT)Premise 3. Q _______ 4. SvT _______ 5. (SvT)&Q _______
1. P&Q Premise 2. R&(SvT)Premise 3. Q &Elim;1 4. SvT &Elim;2 5. (SvT)&Q &Intro;3,4
"Pia is innocent, because she never works with Quinn and Quinn is guilty." What is the premise? _____________________________ What is the conclusion? _____________________________
1. She never works with Quinn and Quinn is guilty. 2. Pia is innocent.
There are many ways to do a formal proof of C&D from ~(~Cv~D). Your job is to figure out which way we chose and complete the proof. Hint: we used reductio shortcut. For example, if you are proving P by reductio, you assume ~P. But when you do the ~Intro step you don't write ~~P, you write P directly. See Section 13.4 in the textbook.
1: ~(~Cv~D) Premise 2: | ~C Assume 3: | ~Cv~D vIntro;2 4: |# #intro;1,3 5: C ~intro;2-4 6: | ~D Assume 7: | ~Cv~D vIntro;6 8: | # #intro;1,7 9: D ~intro;6-8 10: C&D &intro;5,9
Consider this sentence: ~~((~A&~B)v(C&~D)) How many unary-connective instances are there? (Type an Arabic numeral.) How many binary-connective instances are there?
5 3
Select all that are true. -An argument can have no premises. -An argument can have no conclusion.
An argument can have no premises.
Name that truth function: what has the following truth function? (Assume reference columns are in canonical form.) T F
Atomic
Consider this argument: 1. Pia is looking at Quinn. 2. Quinn is looking at Neela. 3. Pia is married. 4. Neela is unmarried. Thus, 5. A married person is looking at an unmarried person. Here is a proof: Proof: Quinn must be married or unmarried. First, consider if he is married. Then, since he is looking at Neela and Neela is unmarried, a married person is looking at an unmarried person. Second, consider if he is unmarried. Then, since Pia is looking at him and Pia is married, we again know that a married person is looking at an unmarried person. In either case, the conclusion follows. Hence we can conclude that a married person is looking at an unmarried person. Done. Correctly label the parts of the proof: "Done" "Pia is looking at him" "we again know that a married person is looking at an unmarried person" "Proof" "Hence we can conclude that a married person is looking at an unmarried person"
Closing Restating a premise Stating an intermediary conclusion Opening Stating the final conclusion
Why is it helpful to identify gaps in a person's argument?
Correct! Because you can assess their reasoning, even if they are making unstated assumptions. Because validity requires true premises. Because their reasoning is flawed if they are making unstated assumptions. Because soundness requires true premises.
Mark all the sentences (in the sense that concerns us).
Correct! Pia is innocent. Prove Pia is guilty! Correct! If Pia is innocent, then Quinn is innocent. Pia. Is pia guilty?
In order to put the previous sentence into NNF, which rules are needed? Select ALL and only those that are needed. Double Negation Idempotence Associativity Commutativity DeMorgan's Distribution
Double Negation DeMorgan's
Complete these two formal proofs of the Double Negation equivalence in as few lines as possible. As always, justify subproof assumptions with Assume. One direction: 1. ~~A Premise 2. _______ _______ The other direction: 1. A Premise 2. | _______ _______ 3. | _______ _______ 4. _______ _______
One direction: 1. ~~A Premise 2. A ~Elim;1 The other direction: 1. A Premise 2. | ~A assume 3. | # #intro;1,2 4. ~~A ~Intro;2-3
Symbolize these in BOOL. Follow all conventions from the textbook, and use the same atomic sentences throughout. Pia and Raquel are guilty: Quinn is guilty or Pia and Raquel are guilty: Either Pia is not guilty, or Quinn is guilty and Raquel is innocent:
P&R Qv(P&R) ~Pv(Q&~R)
Translate into BOOL: 1. "Pia is guilty, but Raquel is innocent." 2. "Although Raquel is innocent, Pia is guilty." 3. "Despite the fact that Pia is guilty, Quinn is not."
P&~R ~R&P P&~Q
Select all the sentences that are logically false.
Pia is innocent. No dogs are police dogs. Rufus is Rufus. Pia is guilty. Rufus is a dog. Correct! Rufus isn't Rufus.
Consider the sentence ~Pv(Q&R). Remember to drop outer parentheses if you are only writing a part of a sentence. What's the first conjunct in the sentence? What's the second conjunct? What's the first disjunct? What's the second disjunct?
Q R ~P (Q&R)
What is the truth function for this sentence? ~P&(~Q&P) Write T or F (upper or lower case is acceptable.) Row 1: Row 2: Row 3: Row 4:
Row 1: F Row 2: F Row 3: F Row 4: F
What is the truth function for this sentence? (P&Q)v~R Write T or F (upper or lower case is acceptable.) Row 1: Row 2: Row 3: Row 4: Row 5: Row 6: Row 7: Row 8:
Row 1: T Row 2: T Row 3: F Row 4: T Row 5: F Row 6: T Row 7: F Row 8: T
What is the truth function for this sentence? Pv~(Q&P) Write T or F (upper or lower case is acceptable.) Row 1: Row 2: Row 3: Row 4:
Row 1: T Row 2: T Row 3: T Row 4: T
What is the word for the number of inputs a connective has? What sort of connective is conjunction? (Write a word, not a number.) What sort of connective is negation? What sort of connective is disjunction?
arity binary unary binary
If a conclusion is certain, then it is maximally likely. That means ______________ logic is a special case of ________________logic.
deductive inductive
Learning theorists call a problem that is solvable with effort a _____________ difficulty.
desirable
A tempting but flawed form of reasoning is a ...
fallacy
Logical ability is mostly fixed and not responsive to training. (T/F)
false
Here are two sentences: (1) Pv~R (2) Pv~(QvR) Does (1) entail (2)? (Write Yes or No.) If not, write the row number for one counterexample (just write one Arabic numeral, like 2); if yes, just write None: Does (2) entail (1)? (Write Yes or No.) If not, write the row number for one counterexample (just write one Arabic numeral, like 2); if yes, just write None:
no 6 yes none
At least one of Stan, Tamar, and Uma is guilty. If Uma is innocent, then Tamar is innocent. Stan never works alone. Who can you know is guilty?
uma
A sound argument is _______ and has true _______ .
valid premise
Use the truth-table method to evaluate this argument: 1. PvQ 2. ~QvR Thus, 3. PvR
valis
Select all the sentences equivalent to ~P. ~P&~P ~Pv~P ~Q P ~~~P ~~~~~~P
~P&~P ~Pv~P ~~~P
Select ALL the sentences that are literals. ~Q P&~Q ~~~P ~P ~~P P
~Q ~P P
