∀x bird(x) fly(x).→ • 2. . EXAMPLES 1.4.1 #4 and #5 illustrate the following fundamental fact: Although the statements "Some are…" and "Some aren't…" sound similar, they do not Since there is every man so will use , and it will be . 6. a. 2, then x2! John's father loves Mary's mother 3. | Propositional Logic-Study of declarative sentences, statements about the world which can be given a truth value-Dealt very well with sentence components like: not, and, or, if, …, then-Limitations: Cannot deal with modifiers like there exists, all, among, only. It adds the concept of predicates and quantifiers to better capture the meaning of statements that cannot be adequately expressed by propositional logic. Even though penguins are also birds, they cannot fly. Penguins are birds 3. The predicate in this question is "fly(bird)." Because all birds are able to fly, it will be portrayed as follows. Our convention will be to capitalize at least the rst letter of constant symbols and use lowercase for variables. Convert your first order logic sentences to canonical form. All birds have wings. The more direct translation to Prolog would then be: bird (X) :- fly (X). Rats cannot fly. Tài liệu liên quan. Use predicate logic to state the following sentences. Every man respects his parent. Conclusion: . "Flying things" is a plural noun; we can count flying things. All Germans speak at least two languages • All the triangles are above all the circles. 3. (c) move(x,y,z) (move x from y to z) consist of? Chapter 1b Propositional Logic II (SAT Solving and Application) Discrete Mathematics II BK TPHCM. All birds fly. First-order logic is also known as Predicate logic or First-order predicate logic. ∀x bird (x) →fly (x). All the triangles are blue. "Fly" is a verb, not a plural noun. F(x) = x can fly . Tweety is a penguin. We cannot say it in propositional logic. E.g., "For every x, x > 0" is true if x is a positive integer. First, the higher the frequency, the stronger the logic can be. It has two parts. could be written symbolically as (x(B(x) ( F(x) where. The predicate in this question is " fly (bird) ." Because all birds are able to fly, it will be portrayed as follows. Later we might discover that Fred is an emu. . b. (1) Symbolize the following argument using predicate logic, (2) Establish its validity by a proof in predicate logic, and (3) "Evaluate" the argument as well. Solution: Preconditions (a set of fluents that have to be true for the ope rator to be "Not all integers are . FOL is sufficiently expressive to represent the natural language statements in a concise way. F(x) : x can fly. Ans:- P(x): x is a bird. Unit-1 Predicate Logic 9 All birds can fly. The first type of defaults is readily formalized but the other, as some researchers have noticed, is difficult to deal with. Chapter 2 Predicate Logic Discrete Mathematics II BKTPHCM. James has a friend named Sean, a penguin. Solution: A predicate that can be true or false, depending on the situation/state [2 points] What does the definition of an operator (e.g. Valid 6. Predicate Logic Predicate logic have the following features to express propositions: Variables: x;y;z, etc. For instance, it can join simple sentences or clauses by logical connectives to represent more complex sentences. . 1. The set of premises in each argument are actually consistent. - Some birds can't fly. 1. 2. All of the subject will be distributed in the class defined by the predicate. Changes in knowledge base might have far-reaching effects. x Predicates: 2 : T ;, 3 : T ;, etc. . using predicates penguin (), fly (), and bird () . Consistency — not all deductions may be correct. A/--,4}) and let E be Th({--,E}) (the set of all predicate logic formulas derivable from ---A). An intended logical way to write "All birds cannot fly" could be { x ∣ Birds (x) } ≠ { x ∣ Fly (x) } Similarly to how someone would say "everyday is not your birthday" or "all that glitters is not gold". All noncats are things that cannot run at more than 50 miles an hour. = Only birds are flying things. Almost all species of birds can fly. Do \not all birds can y" and \some bird cannot y" have the same meaning? Not all birds can y. Propositional logic cannot capture the detailed semantics of these sentences. Represent statement into predicate calculus forms : "Not all birds can fly". Not all birds can fly. FMSE lecture 06. What Donald cannot do, can noone do. a. 2. Let us assume the following predicates bird(x): "x is bird" fly(x): "x can fly". 1. (some birds can fly) Negation: birds b, b cannot fly. 1. . • First-order logic is also known as Predicate logic or First-order predicate logic. • Assuming that birds usually fly, and tweety is a bird, when can we conclude that tweety flies? Saying as: 'It is not the case that all things which are birds can fly.' we could code this alternatively as: ∃x (B(x) ∧ ¬F(x)) Saying as: 'There is some x which is a bird and cannot fly.' To get a feel for what kind of reasoning must predicate logic be able to support, let us consider the following argument: "No books are . (all birds can't fly) Definition: Universal Conditional Quantifier: A universal conditional statement is in the form: x if P(x) then Q(x) Example: x R if x! Although we have not yet de ned the semantics of rst-order logic lets consider some example formulas along with their intuitive natural language interpretations. Predicate Logic Outline • Predicate logic • Predicate logic as formal language • Quantifiers • Parse Trees • Replacing free variables • Scope of quantifiers • Mixing quantifiers • Order of quantifiers Propositional logic • It deals with sentence components like not, and, or and if… then. Use mathematical induction to prove that, for n≥1, 12 + 22 + 32 + ….. + n2 = n (n+1) (n+2)/6 4. In this question the predicate is "fly(bird)." And since there are all birds who fly so it will be represented as follows. Domain for x is all birds. In common sense reasoning two typical types of defaults are encountered. Tweety is a penguin 2. B(x) = x is a bird. Consider the statement, " is greater than 3″. They love to eat fish. Some Examples of FOL using quantifier: All birds fly. • 1. Changes in knowledge base might have far-reaching effects. Consistency — not all deductions may be correct. c. Mary and Sue have the same paternal grandfather. 5 Predicates x > 3 value of propositional function P at x P(x) denotes predicate If an object is not to the right of all the squares, then it is not blue. Examples: Is ò T P1 ó True or False Is T is a great tennis player ó True or False? Not only is there at least one bird, but there is at least one penguin that cannot fly. This branch of logic specifies the logical relationships among claims that can be expressed in the forms "All Xs are Ys," "No Xs are Ys," "Some Xs are Ys," and "Some Xs are not Ys." Developed by Aristotle inthe fourth century B. C. E., categorical logic is also known as Aristotelian or traditional logic. - We don't have the same bug as "some birds can fly" with the implication because we're doing a universal quantification and not an existential one. Semantically equivalent formulas. ∃x∀y is not similar to ∀y∃x. Predicate Logic More powerful Express a wide range of statements in mathematics and computer science. Be sure to define all predicates, constants, and variables. Birds except penguins can fly 2. Modularity sacrificed. Let us assume the following predicates bird(x): "x is bird" fly(x): "x can fly". Represent statement into predicate calculus forms : "Not all birds can fly". Predicate logic is an extension of Propositional logic. Because there is every man so will use ∀, and it will be portrayed as follows: A sentence like "birds can fly" reads "for all x, if x is a bird, then x can fly." Equivalently this reads, "either x isn't a bird, or x can fly." "Birds cannot fly" reads "there doesn't exist some x such that x is a bird and x can fly." Predicate Logic Anvesh Komuravelli 1 Why Predicate Logic? Subject Predicate Sentence 3.8: Only birds fly. ∧Ak → B, that is, all the statements are in the Horn form. 1. All birds have wings. Valid 8. 1. The predicate in this question is " respect (x, y)," where x=man, and y= parent. What is a predicate? 1.4 Predicate Logic. Example: No birds have gills Type I and O proposition. There is no predicate-logic formula with u and v as its only free variables and R its only predicate such that holds in directed graphs iff there is a path from u to v. ∀x bird(x) →fly(x). Chapter 2 Predicate Logic Discrete Mathematics II BKTPHCM. In general, a statement involving n variables can be denoted by . Regarding the second question: USING PREDICATE LOGIC Representation of Simple Facts in Logic Predicate Logic CS 3234: Logic and Formal Systems Martin Henz and Aquinas Hobor September 2, 2010 Generated on Tuesday 14 September, 2010, 11:29 1 Syntax of Predicate Logic 1.1 Need for Richer Language Propositional logic can easily handle simple declarative statements such as: Student Peter Lim enrolled in CS3234. It overcame some of the problems in representing logical issues using propositional logic. Predicate Logic Question 3 (10 points) Write out the following statements in first order logic: All birds can fly. Some automobiles are not Fords. All birds can fly (1) Penguin is a bird (2) Then you may conclude Penguin can fly. Solution for Express the following sentence in Predicate Logic(Define Ontology first and use it.) | Propositional Logic-Study of declarative sentences, statements about the world which can be given a truth value-Dealt very well with sentence components like: not, and, or, if, …, then-Limitations: Cannot deal with modifiers like there exists, all, among, only. "Not all cars are expensive" is equivalent to "Some cars are not expensive", . 2,569. Prof.) Ans:- P(x): x is an integer. Predicate Logic The propositional logic is not powerful enough to represent all types of assertions that are used in computer science and mathematics, or to express certain types of relationship between propositions such as equivalence. Determine if the following logical and arithmetic statement is true or false and justify [3 marks] This problem has been solved! . 4 Negation of Universal Conditional . 2. 3. Consider the premises: P1: Nothing intelligible puzzles me. Some dogs are not collies. 73. 2. For example , Ex.1: All birds fly. • Organize facts about birds as listing of facts (robins fly) (gannets fly) (western grebes fly) (crows fly) (penguins don't fly) (ostriches don't fly) (common loons fly) (fulmars fly) (arctic loons fly) • Approximately 8,600 species of birds in world -Big list -Small in comparison to world population of ~100 billion birds! C. Therefore, all birds can fly. Once a value has been assigned to the variable , the statement becomes a proposition and has a truth or false(tf) value. The logical operations and identities in the previous sections apply to both propositions and predicates. Nor can we show the following logical equivalences: "Not all birds fly" is equivalent to "Some birds don't fly". 2. Given that a P is usually a Q, and given P(a) is true, it is reasonable to conclude that Q(a) is true unless there is good reason not to • Finding that "good reason" is the whole purpose of the all the default reasoning different methods Modularity sacrificed. Aristotle contemplating a bust of Homer by Rembrandt van Rijn. • But logical aspects of natural and artificial languages are much . Here is also referred to as n-place predicate or a n-ary predicate. category. The sentence in predicate logic allows the case that there are no birds, whereas the English sentence probably implies that there is at least one bird. Not all students like both Mathematics and b. ∀x bird(x . Propositional logic is not enough to express the meaning of all statements in mathematics and natural language. Specify what variables you are using for each ATOMIC predicate, and then translate the following statements into predicate logic expressions [3 marks] a. e.g If we know that Fred is a bird we might deduce that Fred can fly. Later we might discover that Fred is an emu. 8xF(x) 9x:F(x) There exists a bird who cannot fly. CS 561, Session 12-13 17 Semantics • Referring to individuals • Jackie • son-of(Jackie), Sam Every man respects his parent. The predicate is a sentence containing a specific number of variables, and becomes a statement when specific values are substituted in place of the predicate variables. Predicate Logic x Variables: T, U, V, etc. P2: Logic puzzles me. "Not all integers are even" is equivalent to "Some integers are not even". Example: All birds have wings Type E proposition. Bow-Yaw Wang (Academia Sinica) Predicate Logic October 13, 202116/156. John likes everyone, that is older than $22$ years old and that doesn't like those who are younger than $22$. Predicate logic and Prolog In 1879 the German philosopher Gottlob Frege gave a more powerful logical reasoning system that lead to the development of predicate logic. The method for writing a First-Order Logic / Predicate Logic • First - order logic or predicate logic is a generalization of propositional logic that allows us to express and infer arguments in infinite modes like - All men are mortal - Some birds cannot fly - At least one planet has life on it 71. Translating an English sentence into predicate logic can be tricky. cEvery bird can fly. FMSE lecture 06. Semantics of Predicate Logic • A term is a reference to an object - constants - variables - functional expressions • Sentences make claims about objects - Well-formed formulas, (wffs) Semantics, part 2 The logic of propositions (also called propositional logic) is an alternative form of knowledge representation, which overcomes some of the weakness of production systems. And since there are all birds who fly so it will be represented as follows. In this question, the predicate is "respect(x, y)," where x=man, and y= parent. Some boys play cricket. First-order logic is another way of knowledge representation in artificial intelligence. . Recall that inferences with modus ponens for KB in the Horn normal form are both sound and cont'd Title: Sentences - either TRUE or false but not both are called propositions. This paper establishes a general scheme for . Only two students took both French and Greek in spring 2010 4. Every child is younger than its mother. :o I want to formulate the following statements into formulas of predicate logic. "Not all birds fly" is equivalent to "Some birds don't fly". Bhavin B. Joshi (Asst. Each of those propositions is treated independently of the others in propositional logic. (BI), F(x)) (iii) There is no student in this class who speaks both Greek and Italian. Not all birds can fly x ( B(x) F(x) ) x ( (B(x) F(x) ) B(x) : x is a bird. Birds except penguins can fly 2. Instead, they walk. In this section we look at two operations that generalize the and and or operations to predicates. Some natural problem is not monotonic û non-monotonic logic. Type E - Universal Negative proposition None of the subject will be distributed in the class defined by the predicate. "All birds can fly" is trickier: we want to say something about just birds, but ∀ is going to give us a statement about all objects. . John's father loves… Birds can fly Formalized in PL1, the knowledge base KB results: penguin (tweety) penguin (x) ⇒ bird (x) bird (x) ⇒ fly (x) From there (for example with resolution) fly (tweety) can be derived (Fig. e.g. 2. \Not all birds can y.":(8xBird(x) )Fly(x)) ; which is the same as When you add Penguin cannot fly, then that theorem cannot be proved anymore. Syntax of Predicate Logic • Terms: a reference to an object • variables, •constants, • functional expressions (can be arguments to predicates) . Use predicate logic to state the following sentences. This means that a statement of the form "All A are B" is true even in the odd case where category A has no members. The predicate can be considered as a function. 3. 4.2).4 Evidently the formalization of the flight attributes of penguins is insufficient. - All dogs are mammals. Even adding only the induction axiom for the natural numbers makes the logic incomplete. Valid 9. 4. So, if there is a single pair of odd numbers whose sum is not even, the implication would be false, which is what we want. A predicate with variables (called an atomic formula) can be made a proposition by applying one of the following two operations to each of its variables: assign a value to the variable quantify the variable using a quantifier Let us use predicate GreatThan(x, 1) to represent x >1. universal quantifier for every object x in the universe, x > The predicate in this question is "respect(x, y)," where x=man, and y= parent. The predicate is "fly(bird)." And since there are all birds . F and G, as always, are predicate letters. All penguins are birds. WUCT121 Logic 61 Definition: Truth Set If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true.The truth set is denoted )}{x ∈D : P(x and is read "the set of all x in D such that P(x)." Examples: • Let P(x) be the predicate "x2 >x" with x∈ i.e. E is not grounded in the sense above: If we take E as a belief set (relevant for the . NB: Evaluating an argument often calls for subjecting a critical. Ans : - P ( x ) : x is a bird . It is an extension to . Cumbersome control information. First-order logic is a powerful language that develops information about the objects in a more easy way and can also express the relationship between those objects. Consider the following statements. . The statement " If a predicate p ( n) holds for n, then p ( n + 1) also holds ", or. Provide a resolution proof that tweety can fly. 4. (Jan-2012-win-old)[3] A crow is a bird. At least one bird can fly and swim. 0. NB: Evaluating an argument often calls for subjecting a critical Exs: Some Examples of FOL using quantifier: 1. • 3 birds can't fly. Propositional logic and Predicate logic are fundamental to all logic. Propositional Logic (PL) : A proposition is a statement, which in English would be a declarative sentence. It says that, X is a bird if X can fly (or, if X can fly, then X must be a bird ). Question: (1) Symbolize the following argument using predicate logic, (2) Establish its validity by a proof in predicate logic, and (3) "Evaluate" the argument as well. "Not all birds fly" is equivalent to "Some birds don't fly". (If the argument takes the form of denying that something has a property because the frequency in the population is so low, then the reverse holds and the lower the frequency, the stronger the . (the subject of a sentence), can be substituted with an element from a . All birds can fly . . Example: birds b such that b can fly. Not in general valid *7. (i) Some old dogs can learn new tricks. F(x) ="x can fly". • But logical aspects of natural and artificial languages are much . USING PREDICATE LOGIC Representation of Simple Facts in Logic òEvery bird can fly. Tải xuống (.pdf) 0 (73 trang) Lịch sử tải xuống. All things that do not travel at the speed of light are nonphotons. A Categorical Syllogism is modernly defined as. Rule 4 Ostriches are granivorous birds that can fly. If a bird cannot fly, then not all birds can fly. This is equivalent to demonstrating that A is not a subset of B. A second-order logic can also quantify over formulas of the first order, and a third-order logic can quantify over formulas of the second order. All entities that do not have IQs of at . 4 Predicates x > 3 Variable: subject of the statement Predicate: property that the subject of the statement can have. f lies(x ) - X can fly in the bird(x ) - x is a bird Functions: NONE Connectives: ¬ - not ∧ - and Quantifiers: ∃x - there exists an x Restricted: ∃bird(X ) Restricted formula: ∃bird(X ) ¬flies(X) Logic formula: ∃X (bird(X ) ∧ ¬f lies(X )) Every person has something that they love. In this question the predicate is "fly(bird)." And since there are all birds who fly so it will be represented as follows. To make this work, we need a formula inside the ∀ that says F ( x) if x is a bird but says nothing extra about x if x is not a bird. All the beings that have wings can fly. (Jan-2015-win-new)[3], (June-2017-sum-new)[3] Q(x): x is a rational number. "A except B" in English normally implies that there are at least some instances of the exception. 1.4 pg. . 1. Be sure to define all predicates, constants, and variables. Rule 3 Penguins are carnivorous birds that cannot fly. All the beings that have wings can fly. It is an extension to propositional logic. The values are taken from the domain of the predicate variables: the domain of x is the set of all students, and the domain of y is the set of all colleges. Every man respects his parent. e.g If we know that Fred is a bird we might deduce that Fred can fly. ∀x bird(x) →fly(x). Express the following sentence in Predicate Logic(Define Ontology first and use it.) No nonelms are things that are not red oaks. Cumbersome control information. Not all birds can y . Penguins can only survive at places with cold temperature. Domain : !X!≠!φ Predicates: Predicate Logic Outline • Predicate logic • Predicate logic as formal language • Quantifiers • Parse Trees • Replacing free variables • Scope of quantifiers • Mixing quantifiers • Order of quantifiers Propositional logic • It deals with sentence components like not, and, or and if… then. One is of the form "All birds can fly exceptb 1,b 2,…, andb m (m≥1)", and the other "All birds can fly, but there exist exceptions". Like all birds, seagulls have two wings and can fly. C. Therefore, all birds can fly. • First-order logic is another way of knowledge representation in artificial intelligence. Predicate Calculus. Predicate Logic Predicate Logic Propositional logic is rather limited in its expressive power. First-order logic is also known as Predicate logic or First-order predicate logic. a particular kind of argument containing three categorical propositions, two of them premises, one a conclusion. ó 3. But we can easily turn it into a plural noun. Every man respects his parent. Therefore, a crow can fly. (D(), L(x)) (ii) Every bird can fly. Every child is younger than its mother. For example, the assertion "x is greater than 1", where x is a variable, is not a proposition because you can not tell whether it is true or false unless you . Statement 3.8: Only birds fly. The negation of some are is all are not. It tells the truth value of the statement at . Aristotelian Logic, also known as Categorical Syllogism or Term Logic, may well be the earliest works of Formal Logic. Not all birds can fly. Valid 5. Represent statement into predicate calculus forms : "Some men are not giants." Let us assume the following predicatesman(x): "x is Man" giant(x): "x is giant". Prove that p (q r) = (p q) (p r) a. using a truth table. For dinner I can have potato or rice but not both. domain the set of real numbers . In this question, the predicate is "respect(x, y)," where x=man, and y= parent. L What are the \meaning" of these sentences? All birds fly. Can you identify problem(s) in the example? Rule 2 Eagles are carnivorous birds that can fly. Represent statement into predicate calculus forms : "Some men are not giants." Let us assume the following predicatesman(x): "x is Man" giant(x): "x is giant". . If an object is to the right of all the squares, then it is above all the circles. All birds fly. Every student is younger than some instructor. Type I - Particular Affirmative proposition - 3 birds can't fly. • Some birds can't fly. The exclusivity of only would occur due to the absence of any other predicate that says some other creature can fly, such as: bee (X) :- fly (X). (a) Translate the following sentences into the language of predicate logic, by choosing the indicated symbols for predicates. 55 # 35 Hey!! For the rst sentence, propositional logic might help us encode it with a single proposition but . b. Every man respects his parent.
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