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Consider the natural deduction proof given below. Keep in mind that predicate logic natural deduction still includes the 18 rules of inference from propositiona log...

Question

Consider the natural deduction proof given below. Keep in mind that predicate logic natural deduction still includes the 18 rules of inference from propositiona logic: In predicate logic; singular statement functions such as Ds (for example, "Spot is dog are treated the same way propositiona letters from propositiona _ logic; Uppercase letters through Z are used to represent predicates; lowercase etters through represent individual constants; and lowercase etters through represent individua

Consider the natural deduction proof given below. Keep in mind that predicate logic natural deduction still includes the 18 rules of inference from propositiona logic: In predicate logic; singular statement functions such as Ds (for example, "Spot is dog are treated the same way propositiona letters from propositiona _ logic; Uppercase letters through Z are used to represent predicates; lowercase etters through represent individual constants; and lowercase etters through represent individual variables_ Using your knowledge of the natural deduction proof method and the options provided in the drop-down menus_ fill in the blanks to identify the missing information that completes the proof: (2)(Cz Mz ) (z)~Mz (z)~Cz ~Mx ~Cx (z)~Cz (Use the following tabs if you need help remembering any of the natural deduction rules You have learned 50 far:) NATURAL DEDUCTION RULES AND PROOF METHODS Modus Panens (MP) Modus Tollens (MT) Pure Hypothetical Syllogism (HS) Disjunctive Syllogism (DS) Constructive Dilemma (CD) Addition (Add) De Morgan's Rule (DM) Commutalivity (Com) Associativity (Assoc) Simplification (Simp) Canjunction (Conj) Distribulion (Dist) Double Negation (DN) Transposilion (Trans) Materal Implicalion (Impl) Material Equivalence (Equiv) Exportation (Exp) Tautology (Taut) Conditional Proof (ACP CP) Indirect Proof (AIP IP) Universal Instantialion (UI) Universal Generalization (UG) Existential Instantiation (EI) Existential Generalization (EG)



Answers

A statement is in prenex normal form (PNF) if and only if it is of the form
$$
Q_{1} x_{1} Q_{2} x_{2} \cdots Q_{k} x_{k} P\left(x_{1}, x_{2}, \ldots, x_{k}\right)
$$
where each $Q_{i}, i=1,2, \ldots, k,$ is either the existential quantifier or the universal quantifier, and $P\left(x_{1}, \ldots, x_{k}\right)$ is a predicate involving no quantifiers. For example, $\exists x \forall y(P(x, y) \wedge Q(y))$ is in prenex normal form, whereas $\exists x P(x) \vee \forall x Q(x)$ is not (because the quantifiers do not all occur first).
Every statement formed from propositional variables, predicates, $\mathbf{T},$ and $\mathbf{F}$ using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 51 asks for a proof of this fact.
Put these statements in prenex normal form. [Hint: Use logical equivalence from Tables 6 and 7 in Section $1.3,$ Table 2 in Section $1.4,$ Example 19 in Section $1.4,$ Exercises 47 and 48 in Section $1.4,$ and Exercises 48 and $49 . ]$
a) $\exists x P(x) \vee \exists x Q(x) \vee A,$ where $A$ is a proposition not in- volving any quantifiers
b) $\neg(\forall x P(x) \vee \forall x Q(x))$
c) $\exists x P(x) \rightarrow \exists x Q(x)$

Hey, it's clear. So when you read here, So we're gonna let an arbitrary statement in the form if P thank you. It can also be not p not cue. Excuse him, then not pee. So the truth tables of these statements, he too, not P, not Q peas and Q not Q. Then not pee True true phones. This would be true this to be true peace. True Falls, falls truth. So the phones, What was this falls true? True falls true and true Zach. Then if P then Q B falls And if negative, if not Q then not people befalls a swell justice delegation symbol. So we see from the truth table that the statement, If you pee, then Q. And if not, Q than not pee are equivalent to one another, so we can transform an arbitrary statement to a statement and pre next, normal for him that is equivalent to the given scene.

You were asked to express the quantification. They're just unique. X p of x introduced infection once before using universal quantification this Sancho pontifications and logical operators. Okay, so we can be right there. Just accept that you bitch! And for all why P of y implies that X is equal to ay right. So if we have to values or if there is some value X, it's that p of X is true and for all why P Y is also true And this implies that X equals y because we wanted to be a unique value. So if it is unique in X equal to y or there is only one values such that it is true.


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