Recall that we can express unique existence as
(1) ∃x(P(x) ∧ ∀y(P(y) → x = y))
In many unique existence proofs, instead of proving (1), we prove the following:
(2) ∃xP(x) (3) ∀x∀y(P(x) ∧ P(y) → x = y) Your task here is to prove (1) from (2) and (3) using the rules of inference for propositional and predicate logic.
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prove that the arguments are valid using rules of inference and laws of predicate logic, (state the laws/rules used) Væ(P(x) + (Q(x) ^ S(x))) 3x(P(x) R(x)) - - .. Ex(R(x) ^ S(x)) - - - (0)H-TE. - – – – – – (24-TE ((x)S_(w))XA ((x)S ^ ()04)XA (2) 1 (x)d)XA
Which lines in the following are not valid? #1 please Put an 'x' at the end of the proof line which has the error and write the number of the error made listed on the Restrictions on Quantifier Inference Rules Which lines in the following are not valid? Explain why in each case (1) 1. (9[(Нх Кх) — Мх] 2. (ЭхХ Нх- Кх) 3. Нх- Кх 2 EI 4. Mx 1,3 MP 5. (Эх)Мх 4 EG 1. (х(Мx D Gx)...
Suppose that we add a new quantifier called exists unique to first- (d) order logic, using the symbol 3! to represent it. It means that there is exactly one element of the universe that satisfies the subsequent formula. In this question, variables will range over the universe of numbers. [1 mark] (0) Is 3!x. x + x 2 valid? Why? Give an example of a valid formula that uses the quantifier, and an example of an unsatisfiable one. Both must...
it is about the classical logic in the subject of formal method: the question is shown as the picture Question 1: Classical Logic [25 marks) a) Answer the following questions briefly but precisely. i. State what it means for an argument to be valid in Predicate Logic. [3 marks ii. Suppose you use resolution to prove that KB = a. Does this mean that a is valid? And why? [3 marks b) Consider the following three English sentences: Sl: If...
1. Formalize the following argument by using the given predicates and then rewriting the argument as a numbered sequence of statements. Identify each statement as either a premise, or a conclusion that follows according to a rule of inference from previous statements. In that case, state the rule of inference and refer by number to the previous statements that the rule of inference used.Lions hunt antelopes. Ramses is a lion. Ramses does not hunt Sylvester. Therefore, Sylvester is not an...
5. Symbolize the following argument and prove it is a valid argument. Let B ( x ) = x is a bear; D ( x ) = x is dangerous, and H ( x ) = x is hungry. Every bear that is hungry is dangerous. There is a hungry animal that is not dangerous. Therefore there is an animal that is not a bear. 6. In order to prove an quantificational argument invalid it is only necessary to find a...
The only 9 rules of inference allowed are: 1. Modus Ponens (MP) 2. Modus Tollens (MT) 3. Hypothetical Syllogism (HS) 4. Disjunctive Syllogism (DS) 5. Constructive Dilemma (CD) 6. Simplification (Simp) 7. Conjunction (Conj) 8. Addition (Add) 9 absorption SECTION ONE: Formal proofs of validity using natural deductions Prove the following argument valid using the nine rules of inference. Copy-and-paste key of symbols: • v - = i Argument Two (1) A5B (2) ( A B ) > C (3)...
Problem 1 [8pt] Prove that the following two Hoare triples are valid. (Hint: in predicate logic Pi equivalent to -P V P2) is a) (4pt) y:= x * 2; y:= y + 3; lu > o) b) (4pt) if (y>2) x := y-1; elsex5-y: fr > 1)
Simplify the following sentences in predicate logic so that all the negation symbols are directly in front of a predicate. (For example, Vx ((-0(x)) + (-E(x))) is simplified, because the negation symbols are direct in front of the predicates O and E. However, Væ -(P(2) V E(x)) is not simplified.) (i) -(3x (P(x) 1 (E(x) + S(x)))) (ii) -(Vx (E(x) V (P(x) +-(Sy G(x, y))))) Write a sentence in predicate logic (using the same predicates as above) which is true...
You are given three boxes filled with chocolates. One box contains only dark chocolates, one box contains only milk chocolates, and one contains both dark and milk chocolates. However, the boxes are labeled incorrectly. You know that you have exactly one box of each, and you also know that each box is definitely labeled wrong. Given the incorrect labeling of the chocolate boxes and the three observations, use Propositional Logic to derive the correct labeling of the second box. You...