All the beings that have wings 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". 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. Determine if the following logical and arithmetic statement is true or false and justify [3 marks] This problem has been solved! Predicate Logic x Variables: T, U, V, etc. All birds have wings. 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. For the rst sentence, propositional logic might help us encode it with a single proposition but . 1. Consistency — not all deductions may be correct. Every child is younger than its mother. • 1. Ans : - P ( x ) : x is a bird . Domain : !X!≠!φ Predicates: In common sense reasoning two typical types of defaults are encountered. Not all birds can fly. ∃x∀y is not similar to ∀y∃x. The predicate in this question is "respect(x, y)," where x=man, and y= parent. b. The negation of some are is all are not. Penguins can only survive at places with cold temperature. 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. (BI), F(x)) (iii) There is no student in this class who speaks both Greek and Italian. (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 . 6. . 0. a. Not all birds can fly x ( B(x) F(x) ) x ( (B(x) F(x) ) B(x) : x is a bird. Therefore, a crow can fly. 5 Predicates x > 3 value of propositional function P at x P(x) denotes predicate could be written symbolically as (x(B(x) ( F(x) where. Not in general valid *7. - Some birds can't fly. 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 > Do \not all birds can y" and \some bird cannot y" have the same meaning? All birds fly. . C. Therefore, all birds can fly. The predicate is "fly(bird)." And since there are all birds . Represent statement into predicate calculus forms : "Not all birds can fly". 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. 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. . Some dogs are not collies. Statement 3.8: Only birds fly. All birds can fly (1) Penguin is a bird (2) Then you may conclude Penguin can fly. • But logical aspects of natural and artificial languages are much . Specify what variables you are using for each ATOMIC predicate, and then translate the following statements into predicate logic expressions [3 marks] a. Hey!! First-order logic is also known as Predicate logic or First-order predicate logic. Once a value has been assigned to the variable , the statement becomes a proposition and has a truth or false(tf) value. 2. Type I - Particular Affirmative proposition Tải xuống (.pdf) 0 (73 trang) Lịch sử tải xuống. For instance, it can join simple sentences or clauses by logical connectives to represent more complex sentences. FMSE lecture 06. It says that, X is a bird if X can fly (or, if X can fly, then X must be a bird ). And since there are all birds who fly so it will be represented as follows. "Not all integers are even" is equivalent to "Some integers are not even". Some automobiles are not Fords. Title: . "Not all cars are expensive" is equivalent to "Some cars are not expensive", . What Donald cannot do, can noone do. In this section we look at two operations that generalize the and and or operations to predicates. All birds have wings. The logic of propositions (also called propositional logic) is an alternative form of knowledge representation, which overcomes some of the weakness of production systems. Chapter 1b Propositional Logic II (SAT Solving and Application) Discrete Mathematics II BK TPHCM. Modularity sacrificed. "Not all birds fly" is equivalent to "Some birds don't fly". 2. Later we might discover that Fred is an emu. Example: No birds have gills Type I and O proposition. domain the set of real numbers . Cumbersome control information. All birds fly. (a) Translate the following sentences into the language of predicate logic, by choosing the indicated symbols for predicates. . First-order logic is another way of knowledge representation in artificial intelligence. x Predicates: 2 : T ;, 3 : T ;, etc. 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. 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. No nonelms are things that are not red oaks. F(x) : x can fly. 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". 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 "Fly" is a verb, not a plural noun. cEvery bird can fly. 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. C. Therefore, all birds can fly. a particular kind of argument containing three categorical propositions, two of them premises, one a conclusion. 1. It is an extension to propositional logic. 1.4 pg. . c. Mary and Sue have the same paternal grandfather. (some birds can fly) Negation: birds b, b cannot fly. Valid 8. FOL is sufficiently expressive to represent the natural language statements in a concise way. 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. It overcame some of the problems in representing logical issues using propositional logic. 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. . Every man respects his parent. F(x) = x can fly . Valid 6. Solution: Preconditions (a set of fluents that have to be true for the ope rator to be Only two students took both French and Greek in spring 2010 4. F and G, as always, are predicate letters. FMSE lecture 06. | 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. Even adding only the induction axiom for the natural numbers makes the logic incomplete. Bow-Yaw Wang (Academia Sinica) Predicate Logic October 13, 202116/156. 2. The statement " If a predicate p ( n) holds for n, then p ( n + 1) also holds ", or. If a bird cannot fly, then not all birds can fly. ∀x bird(x) →fly(x). Predicate Logic Question 3 (10 points) Write out the following statements in first order logic: All birds can fly. If an object is not to the right of all the squares, then it is not blue. 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 . It adds the concept of predicates and quantifiers to better capture the meaning of statements that cannot be adequately expressed by propositional logic. They love to eat fish. (c) move(x,y,z) (move x from y to z) consist of? John's father loves… USING PREDICATE LOGIC Representation of Simple Facts in Logic Exs: Some Examples of FOL using quantifier: 1. 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. - 3 birds can't fly. Not all students like both Mathematics and The more direct translation to Prolog would then be: bird (X) :- fly (X). (D(), L(x)) (ii) Every bird can fly. Predicate Logic Predicate logic have the following features to express propositions: Variables: x;y;z, etc. ∀x bird(x) fly(x).→ • 2. Predicate Logic Predicate Logic Propositional logic is rather limited in its expressive power. Conclusion: . Valid 5. Even though penguins are also birds, they cannot fly. Not all birds can y. Propositional logic cannot capture the detailed semantics of these sentences. category. 3. 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". cont'd It has two parts. Subject Predicate Sentence 3.8: Only birds fly. Tài liệu liên quan. 1. 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. • First-order logic is another way of knowledge representation in artificial intelligence. Some Examples of FOL using quantifier: All birds fly. Penguins are birds 3. We cannot say it in propositional logic. Use predicate logic to state the following sentences. All things that do not travel at the speed of light are nonphotons. Every man respects his parent. ∧Ak → B, that is, all the statements are in the Horn form. 4. E.g., "For every x, x > 0" is true if x is a positive integer. Rule 2 Eagles are carnivorous birds that can fly. Predicate logic is an extension of Propositional logic. 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." James has a friend named Sean, a penguin. All birds fly. Consider the premises: P1: Nothing intelligible puzzles me. All penguins are birds. All Germans speak at least two languages e.g. • First-order logic is also known as Predicate logic or First-order predicate logic. Prove that p (q r) = (p q) (p r) a. using a truth table. Our convention will be to capitalize at least the rst letter of constant symbols and use lowercase for variables. 2. a. Represent statement into predicate calculus forms : "Not all birds can fly". Consider the following statements. Example: All birds have wings Type E proposition. b. "Not all integers are . Recall that inferences with modus ponens for KB in the Horn normal form are both sound and E is not grounded in the sense above: If we take E as a belief set (relevant for the . It is an extension to . Because there is every man so will use ∀, and it will be portrayed as follows: Predicate Logic More powerful Express a wide range of statements in mathematics and computer science. 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 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. Domain for x is all birds. But we can easily turn it into a plural noun. A Categorical Syllogism is modernly defined as. Aristotle contemplating a bust of Homer by Rembrandt van Rijn. B(x) = x is a bird. In this question, the predicate is "respect(x, y)," where x=man, and y= parent. - All dogs are mammals. Valid 9. (Jan-2012-win-old)[3] A crow is a bird. 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. In general, a statement involving n variables can be denoted by . 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 . "Not all birds fly" is equivalent to "Some birds don't 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! (i) Some old dogs can learn new tricks. Express the following sentence in Predicate Logic(Define Ontology first and use it.) . Let us assume the following predicates bird(x): "x is bird" fly(x): "x can fly". Aristotelian Logic, also known as Categorical Syllogism or Term Logic, may well be the earliest works of Formal Logic. Modularity sacrificed. 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". Bhavin B. Joshi (Asst. | 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. Convert your first order logic sentences to canonical form. John's father loves Mary's mother 3. The predicate in this question is " fly (bird) ." Because all birds are able to fly, it will be portrayed as follows.
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