What information must each line of a proof provide in order to demonstrate unequivocally that the conclusion is logically entailed by the premises?

What information must each line of a proof provide in order to demonstrate unequivocally that the conclusion is
logically entailed by the premises?

What information must each line of a proof provide in order to demonstrate unequivocally that the conclusion is
logically entailed by the premises? (mark all that apply below) [5 pts]
☐ A statement, which is either an assumption made for the sake of proof, or a validly derived inference.
☐ A line number, which is provides a way to identify each line in the proof.
☐ An assumption set, which lists all the assumptions on which the given statement depends.
☐ An annotation, which justifies the statement as either an assumption or a validly derived inference.
☐ An XKCD comic strip satirizing the principle of explosion.
2. A statement that is tautologous is logically true, whereas a statement that is contingent is… [2 pts]
i) ☐ logically false.
ii) ☐ sometimes true and sometimes false.
3. Two statements that are logically equivalent share the same truth value on each row of a truth table, whereas two
statements that are contradictory… [2 pts]
iii) ☐ have opposite truth values on each row.
iv) ☐ have at least one row on which the truth values are all true.
4. Identify the conclusion of the following argument: [2 pts]
Either the butler did it or the cook did it, unless it was the doctor. If it was done with a knife, the cook did it.
But it was not done with a knife. So, if the butler didn’t do it, the doctor did.
i) ☐ Either the butler did it or the cook did it, unless it was the doctor.
ii) ☐ If it was done with a knife, the cook did it.
iii) ☐ But it was not done with a knife.
iv) ☐ So, if the butler didn’t do it, the doctor did.
5. Is the wff an atomic proposition, conjunction, disjunction, conditional, biconditional, or negation? [20 pts]
i) Q atomic
ii) P &~S conjunction
iii) ~R→T conditional
iv) (P & Q) Ú R disjunction
v) ~P↔~Q biconditional
vi) (P Ú Q) & S conjunction
vii) ~(R→(T Ú ~S)) negation
viii) S→(R→Q) conditional
ix) ~(S→Q) Ú (~S Ú Q) disjunction
x) ~(~(T Ú P) Ú ~(P & Q)) negation
6. Complete the truth table below, then state whether the sequent valid by marking the correct box. [8 pts]
P Q R P→Q Q→(R & P) P ∴R
i) ☐ The sequent is valid.
ii) ☐ The sequent is invalid.
7. Match the primitive rule of inference to its abbreviation. [18 pts]
i) Given a statement that is a disjunction, A Ú B (at line m), and another statement (at line n) that is a denial of
one of its disjuncts, you may conclude the other disjunct.
☐ &I ☐ &E ☐ ÚI ☐ ÚE ☐ →E ☐ →I ☐ ↔I ☐ ↔E ☐ RAA
ii) Given both a statement and its denial (at lines m and n), you may conclude the denial of the assumption from
which the contradiction is derived (at line k).
☐ &I ☐ &E ☐ ÚI ☐ ÚE ☐ →E ☐ →I ☐ ↔I ☐ ↔E ☐ RAA
iii) A Given a statement that is a biconditional, A↔B (at line m), you may conclude either A→B or B→A.
☐ &I ☐ &E ☐ ÚI ☐ ÚE ☐ →E ☐ →I ☐ ↔I ☐ ↔E ☐ RAA
iv) Given a statement that is a conditional, A→B (at line m), and another statement that is its antecedent,
A (at line n), you may conclude the consequent of the conditional, B.
☐ &I ☐ &E ☐ ÚI ☐ ÚE ☐ →E ☐ →I ☐ ↔I ☐ ↔E ☐ RAA
v) Given two statements A and B (at lines m and n), you may conclude their conjunction.
☐ &I ☐ &E ☐ ÚI ☐ ÚE ☐ →E ☐ →I ☐ ↔I ☐ ↔E ☐ RAA
vi) Given a statement, A (at line m), you may conclude any disjunction having A as a disjunct.
☐ &I ☐ &E ☐ ÚI ☐ ÚE ☐ →E ☐ →I ☐ ↔I ☐ ↔E ☐ RAA
vii) Given two conditional statements having the forms A→B and B→A (at lines m and n), you may conclude a
biconditional statement with A and B as its constituents.
☐ &I ☐ &E ☐ ÚI ☐ ÚE ☐ →E ☐ →I ☐ ↔I ☐ ↔E ☐ RAA
viii) Given a statement, B (at line n), you may conclude a conditional having B as the consequent if the antecedent
appears in the proof as an assumption, A (at line m), and B is derivable from A.
☐ &I ☐ &E ☐ ÚI ☐ ÚE ☐ →E ☐ →I ☐ ↔I ☐ ↔E ☐ RAA
ix) Given a statement that is a conjunction, A & B (at line m), you may conclude either conjunct.
☐ &I ☐ &E ☐ ÚI ☐ ÚE ☐ →E ☐ →I ☐ ↔I ☐ ↔E ☐ RAA
8. Using the translation scheme provided below, translate the following wffs. (Please write neatly.) [10 pts]
P: It was done with a revolver. Q: It was done with poison. R: It was done with a knife.
S: The butler did it. T: The cook did it. U: The doctor did it.
i) P→S
ii) ~Q→~U
iii) ~P→(Q Ú R)
iv) (T↔R) & ~R
v) (~S Ú ~U) & (P Ú Q)
i)
ii)
iii)
iv)
v)
9. Using the same translation scheme, symbolize the following natural language argument into a formal sequent:
Either it wasn’t done with a revolver or the cook didn’t do it, but if a revolver was used, then the butler did it.
However, it wasn’t done with a revolver, but with poison. So, the doctor did it. [8 pts]
i)
10. For each sequent below, construct an indirect truth table. If the sequent is invalid, give an invalidating assignment.
(Please write neatly so I can understand what is going on.)
i) ((P Ú Q) & ~R)→S, P & T, ~R Ú ~T ├ S & T [5 pts]
ii) P Ú Q, Q→S, T , R Ú T ├ S & T [5 pts]
iii) P Ú Q, S→Q, Q, P→Q ├ S [5 pts]
11. Give a proof below for the valid sequent from Q.10. (The proof does not require either →I or RAA) [10 pts]
(Again, please write legibly.)
Extra Credit Challenge
12. Consider the following sequent: R, ~R ├ Q The premises of this argument are an inconsistent set.
Now, consider this next sequent: R ├ Q Ú~Q The conclusion of this argument is a tautology.
Explain clearly and in detail why both arguments must be valid. [5 pts

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