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## Contributors and Attributions

Suppose you are at a party with 19 of your closest friends so including you, there are 20 people there. But with the approach of predicate logic, we can integrate the two levels of analysis, and say: 1. In order to apply lemma L25, s must not appear in Z. That notion of an instance is important to doing proofs in predicate logic.

Predicate Logic! Some statements cannot be expressed in propositional logic, such as:! All men are mortal.! Some trees have needles.! X > 3.! Predicate logic can express these statements and make inferences on them.

- Prove or disprove.
- In this case it might be better to save this page in your favorites to return to it.
- So oranges are sweet citrus fruits.
- Prove your answers.

## Predicate Logic: Formal Deduction

Learning goals By the end of this lecture, you should be able to: Describe the rules of inference for formal deduction for predicate logic. Prove that a conclusion follows from a set of premises using formal deduction inference rules. CS 245 Logic and Computation Fall 2019 3 / 19

## Propositional and Predicate Logic

Propositional and Predicate Logic CS 536: Science of Programming, Fall 2021 1. Fill in the missing rule names in the proof below of ¬(p 㲗 q ... Using propositional and predicate proof rules, find a predicate equivalent to ¬(∀ x . ∃ y . p(x, y)) that has no negation symbols (i.e., ¬), except possibly in front of p(x, y). Write a formal proof that shows each step needed (don't forget the rule names!). Hint: Use …

predicate of identity, “=”. Think of “everyone except John” as “everyone who is not identical to John”.) ∀x (¬ x = John → love (Mary, x)) or equivalently ∀x (x ≠ John → love (Mary, x)) As in the case of some earlier examples, this is a ‘weak’ reading of except, allowing the possibility of Mary loving John.

In order to apply lemma L25, s must not appear in Z. Also, in order to apply lemma L28, s must not appear in Vu P u. The latter restriction is encoded in the VI rule by requiring that Vu P u be the universal generalization of P s.

In a similar way, the restrictions built in the 3E rule play a pivotal role in proving. L36 Soundness for 3E : Assume that s does not appear in Z, in 3u P u , or in X. You will immediately want to know why the restrictions stated in L36 are not the same as the restriction I required of the 3E rule, that s be an isolated name. These three requirements are the ones which appear in the assumption of L Requiring that s be an isolated name is a superficially stronger requirement from which the other three follow.

Since we are proving soundness, if we carry out the proof for a weaker requirement on a rule, we will have proved it for any stronger requirement. You can see this immediately by noting that if we succeed in proving L36, we will have proved any reformulation of L36 in which the assumption which states the requirement is stronger.

Of course, by making the requirement for applying a rule stronger by making the rule harder to apply , we might spoil completeness-we might make it too hard to carry out proofs so that some valid arguments would have no corresponding proofs.

But when we get to completeness, we will check that we do not get into that problem. Let's turn to proving L In the argument above, we need to invoke UI, and when we do, we remove the quantifier, leaving the expression from line 2 looking like this see line 3 : 1. Given line 1, the conclusion follows immediately now: 5. Pp MP 4,1 I hope you can see how this is a merging of the logic of categories with the logic of propositions. Here it is laid out in six trivial steps: 4. Bp SM 4. Sd MP 2,3 5.

Sd CN 2,4 6. Sx EG 5 Proofs in Predicate Logic Try translating these and working a proof for them: 1. Oranges are sweet. Oranges are citrus fruits. So oranges are sweet citrus fruits. Since tomatoes are vegetables, the tomatoes in the garden are vegetables.

Apples and pears grow on trees, so pears grow on trees. Socrates is a man, and all men are mortal, so Socrates is mortal. Leave a Reply Cancel reply Your email address will not be published. Introduction 2. Basic Concepts 2. Exercises with answers 2. Arguments and Non-arguments 3. Induction and Deduction 4. Argument Pattern Recognition Exercises with answers 4.

Exercises on identification and Evaluation 5. Validity Solutions to Counter-example exercises 5. Informal Fallacies in Reasoning 6. Fallacy Identification Exercises 6. Solutions to Fallacy Identification Exercises 7.

Categorical Logic 7. Exercises on Standard Form and Distribution 7. Tommy Flanagan was telling you what he ate yesterday afternoon. Also, if I had cucumber sandwiches, then I had soda. But I didn't drink soda or tea. What did Tommy eat? The deduction rule is valid. Can you chain implications together? Let's find out:. Instead, you should use part a and mathematical induction. Simplify the statements below so negation appears only directly next to predicates.

What do these concepts mean in terms of truth tables? Suppose further that. So subtract 2 from both sides. What is going on here? Is your friend's argument valid? Hint: What implication follows from the given proof?

Suppose you have a collection of 5-cent stamps and 8-cent stamps. But, let's ask some other questions:. Write out the beginning and end of the argument if you were to prove the statement,. You do not need to provide details for the proofs since you do not know what solitary means. However, make sure that you provide the first few and last few lines of the proofs so that we can see that logical structure you would follow. Clearly state the style of proof you are using.

This completes the proof. The game TENZI comes with 40 six-sided dice each numbered 1 to 6. Suppose you roll all 40 dice. Suppose that each number only came up 6 or fewer times. That's a total of 36 dice, so you must not have rolled all 40 dice. Suppose you roll 10 dice, but that there are NOT four matching rolls.

If we only had three different values, that would be only 9 dice, so there must be 4 different values, giving 4 dice that are all different. By properties of logarithms, this implies. But this is impossible as any power of 7 will be odd while any power of 10 will be even. For each of the statements below, say what method of proof you should use to prove them. Then say how the proof starts and how it ends.

## Propositional Logic, Truth Tables, and Predicate Logic (Rosen, …

Statements in Predicate Logic P(x,y) ! Two parts: ! A predicate P describes a relation or property. ! Variables (x,y) can take arbitrary values from some domain. ! Still have two truth values for statements (T and F) ! When we assign values to x and y, then P has a truth value.

We can use predicate logic (first-order logic) to express all of these. CS Logic and Computation Fall 6 / Would you really use predicate logic? Examples of predicate logic in CS so far: 1. Every well-formed formula has an equal number of left and right brackets. 2. If there does not exist a formal deduction proof from the. Some tautologies of predicate logic are analogs of tautologies for propo-sitional logic (Section ), while others are not (Section ). Proofs in predicate logic can be carried out in a manner similar to proofs in propositional logic (Sections and ). In Section we discuss some of . Predicate Logic is similar: two statements are equivalent if they have the same truth values but must account for Any Predicate definition:P(x) might be x is odd or x is > 0 Any universe/set over quantifiers including a universe of infinite objects can’t use truth tables anymore Need a .

Logical Equivalences Involving Predicates \u0026 Quantifiers (Part 1)

To extend the proof for sentence logic, we need to prove rule soundness for the four new predicate logic rules. Two are easy applications of definitions and lemmas given in section EXERCISES Apply lemma L30 to prove lemma L Apply lemma L3 1 to prove lemma L L35 Soundness for VI : Assume that the name s does not occur in Z or in Vu P u.

Let's consider an arbitrary interpretation, I, in which all the sentences Predicate Logic Proof Exercises Z are true. What will it take for Vu P u to be true also in I? Lemma L28 tells us that given any name, s, not appearing in Vu P uwe need only show that P s is true in all s-variants of I. But this is easy. The assumption that s does not occur in Z allows us to apply lemma L25 as follows: I is a model for Z.

Since s does not occur in Z, L25 tells us that any s-variant of I is also a model of Z. You should carefully note the Predicate Logic Proof Exercises restrictions which play crucial roles in this demonstration.

In order to apply lemma L25, s must not appear in Z. Also, in order to apply lemma L28, s must not appear in Vu P u. The latter restriction Predicate Logic Proof Exercises encoded in the VI rule by requiring that Vu P u be the universal generalization of P s. In a similar way, Predicate Logic Proof Exercises restrictions built in the 3E rule play a pivotal role in proving.

L36 Soundness for Mature Shitting : Assume that s does not appear in Z, in 3u P uor in X. You will immediately want to know why the restrictions stated in L36 are not the same as Predicate Logic Proof Exercises restriction I required of the 3E rule, that s be an Predicate Logic Proof Exercises name.

These three requirements are the ones which appear in the assumption of L Requiring that s be an isolated name is a superficially stronger requirement from which the other three follow.

Since we are proving soundness, if we carry out the proof for a weaker requirement on a rule, we will have proved it for any stronger requirement. You can Ehefrau Swinger this immediately by noting that if we succeed in proving L36, we will have proved any reformulation of L36 in which the assumption which states the requirement is stronger. Of course, by making the requirement for applying a rule stronger by making the rule harder to applywe might spoil completeness-we might make it too hard to carry out proofs so that some valid arguments would have no corresponding proofs.

But when Paderborn Sie Sucht Sex get to completeness, we will check that we do not get into that problem. Let's turn to proving L Assume that I is a model for Z and 3u P u. Since s does not appear in 3u P uthere is an s-variant, I, of I, such that P s is true in I. Since s does not appear in 3u P u Predicate Logic Proof Exercises in Z, and since I and I, differ only as to s, lemma L25 tells us that 3u P u and Zntcrpretationr, Soundness, and Completeness for Predicate Logic Completeness for Predicate Logic Derivations Z are also true in I.

Predicate Logic Proof Exercises, since s is assumed not to appear in X and I and I, differ only as to s, lemma L25 again applies to tell us that X is true in I. The proof is a trivial extension of the proof for sentence logic, Predicate Logic Proof Exercises to fix the ideas you should carry out this extension.

EXERCISE Prove T You only need to extend the inductive step in the proof of T5 to cover the cases of the four quantifier rules. Paul Teller UC Davis. The Primer was published in by Prentice Hall, since acquired by Pearson Education. Pearson Education has allowed the Primer to go out of print and returned the copyright to Professor Teller who is happy to make it available without charge for Dogging Google Maps and educational use.

Contributors and Attributions Paul Teller UC Davis.

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