x bird(x . 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. category. a. All the beings that have wings can fly. Birds except penguins can fly 2. Specify what variables you are using for each ATOMIC predicate, and then translate the following statements into predicate logic expressions [3 marks] a. F(x) : x can fly. The logic of propositions (also called propositional logic) is an alternative form of knowledge representation, which overcomes some of the weakness of production systems. For example , Ex.1: All birds fly. "Fly" is a verb, not a plural noun. First-order logic is also known as Predicate logic or First-order predicate logic. Subject Predicate Sentence 3.8: Only birds fly. A/--,4}) and let E be Th({--,E}) (the set of all predicate logic formulas derivable from ---A). Example: birds b such that b can fly. - Some birds can't fly. All things that do not travel at the speed of light are nonphotons. (some birds can fly) Negation: birds b, b cannot fly. The predicate in this question is "fly(bird)." Because all birds are able to fly, it will be portrayed as follows. 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. 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. No nonelms are things that are not red oaks. 4 Negation of Universal Conditional . - All dogs are mammals. Some automobiles are not Fords. Predicate logic is an extension of Propositional logic. Represent statement into predicate calculus forms : "Not all birds can fly". Not all birds can y. Propositional logic cannot capture the detailed semantics of these sentences. In this question, the predicate is "respect(x, y)," where x=man, and y= parent. "A except B" in English normally implies that there are at least some instances of the exception. Determine if the following logical and arithmetic statement is true or false and justify [3 marks] This problem has been solved! Penguins can only survive at places with cold temperature. (a) Translate the following sentences into the language of predicate logic, by choosing the indicated symbols for predicates. 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 Type E - Universal Negative proposition None of the subject will be distributed in the class defined by the predicate. Solution: Preconditions (a set of uents that have to be true for the ope rator to be x bird(x) fly(x). Modularity sacrificed. The set of premises in each argument are actually consistent. Bow-Yaw Wang (Academia Sinica) Predicate Logic October 13, 202116/156. (BI), F(x)) (iii) There is no student in this class who speaks both Greek and Italian. 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. using predicates penguin (), fly (), and bird () . 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! Cumbersome control information. F(x) ="x can y". L What are the \meaning" of these sentences? This paper establishes a general scheme for . "Not all integers are even" is equivalent to "Some integers are not even". Every man respects his parent. Propositional logic and Predicate logic are fundamental to all logic. B(x) = x is a bird. Chapter 1b Propositional Logic II (SAT Solving and Application) Discrete Mathematics II BK TPHCM. 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). 5 Predicates x > 3 value of propositional function P at x P(x) denotes predicate All the beings that have wings can fly. Later we might discover that Fred is an emu. 6. 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. All the triangles are above all the circles. Almost all species of birds can fly. Statement 3.8: Only birds fly. Once a value has been assigned to the variable , the statement becomes a proposition and has a truth or false(tf) value. Not in general valid *7. C. Therefore, all birds can fly. "Not all birds fly" is equivalent to "Some birds don't fly". 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. James has a friend named Sean, a penguin. Not all birds can fly. Recall that inferences with modus ponens for KB in the Horn normal form are both sound and b. 1.4 pg. \Not all birds can y.":(8xBird(x) )Fly(x)) ; which is the same as In this question the predicate is "fly(bird)." And since there are all birds who fly so it will be represented as follows. One is of the form "All birds can fly exceptb 1,b 2,, andb m (m1)", and the other "All birds can fly, but there exist exceptions". In general, a statement involving n variables can be denoted by . John's father loves Mary's mother 3. Be sure to define all predicates, constants, and variables. Let us assume the following predicates bird(x): "x is bird" fly(x): "x can fly". The first type of defaults is readily formalized but the other, as some researchers have noticed, is difficult to deal with. Valid 8. All birds can fly . Every man respects his parent. (D(), L(x)) (ii) Every bird can fly. F and G, as always, are predicate letters. e.g. . 1. Changes in knowledge base might have far-reaching effects. Every man respects his parent. If a bird cannot fly, then not all birds can fly. Propositional logic is not enough to express the meaning of all statements in mathematics and natural language. Predicate Logic Predicate Logic Propositional logic is rather limited in its expressive power. All of the subject will be distributed in the class defined by the predicate. 2,569. Not all birds can y . For dinner I can have potato or rice but not both. (the subject of a sentence), can be substituted with an element from a . Hey!! All the triangles are blue. Ak B, that is, all the statements are in the Horn form. Use predicate logic to state the following sentences. . First-order logic is also known as Predicate logic or First-order predicate logic. It says that, X is a bird if X can fly (or, if X can fly, then X must be a bird ). b. 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. 1. They love to eat fish. Not all students like both Mathematics and Aristotelian Logic, also known as Categorical Syllogism or Term Logic, may well be the earliest works of Formal Logic. e.g If we know that Fred is a bird we might deduce that Fred can fly. Here is also referred to as n-place predicate or a n-ary predicate. Every bird can fly. It overcame some of the problems in representing logical issues using propositional logic. Rule 3 Penguins are carnivorous birds that cannot fly. Each of those propositions is treated independently of the others in propositional logic. 1. (c) move(x,y,z) (move x from y to z) consist of? All birds fly. If an object is to the right of all the squares, then it is above all the circles. The statement " If a predicate p ( n) holds for n, then p ( n + 1) also holds ", or. Assuming that birds usually fly, and tweety is a bird, when can we conclude that tweety flies? Domain : !X!! Predicates: What Donald cannot do, can noone do. 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. Tweety is a penguin. Let us assume the following predicates bird(x): "x is bird" fly(x): "x can fly". 2. Prof.) Ans:- P(x): x is an integer. All entities that do not have IQs of at . NB: Evaluating an argument often calls for subjecting a critical The negation of some are is all are not. It is an extension to propositional logic. 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". Syntax of Predicate Logic Terms: a reference to an object variables, constants, functional expressions (can be arguments to predicates) . 0. "Not all integers are . Every child is younger than its mother. Some birds can't fly. John's father loves . Because there is every man so will use , and it will be portrayed as follows: 2, then x2! Every student is younger than some instructor. Be sure to define all predicates, constants, and variables. CS 561, Session 12-13 17 Semantics Referring to individuals Jackie son-of(Jackie), Sam All birds have wings. The method for writing a Instead, they walk. In this question, the predicate is "respect(x, y)," where x=man, and y= parent. Our convention will be to capitalize at least the rst letter of constant symbols and use lowercase for variables. xy is not similar to yx. 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. In this question the predicate is "fly(bird)." And since there are all birds who fly so it will be represented as follows. | 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. All birds fly. Every child is younger than its mother. (Jan-2015-win-new)[3], (June-2017-sum-new)[3] Q(x): x is a rational number. 3 birds can't 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". - 3 birds can't fly. Ti liu lin quan. Propositional Logic (PL) : A proposition is a statement, which in English would be a declarative sentence. Example: No birds have gills Type I and O proposition. 1. Unit-1 Predicate Logic 9 All birds can fly. At least one bird can fly and swim. 2. When you add Penguin cannot fly, then that theorem cannot be proved anymore. Express the following sentence in Predicate Logic(Define Ontology first and use it.) Predicate Logic Question 3 (10 points) Write out the following statements in first order logic: All birds can fly. - 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. 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 FMSE lecture 06. 1. . Title: USING PREDICATE LOGIC Representation of Simple Facts in Logic Use predicate logic to state the following sentences. 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. It adds the concept of predicates and quantifiers to better capture the meaning of statements that cannot be adequately expressed by propositional logic. 1. Chapter 2 Predicate Logic Discrete Mathematics II BKTPHCM. 4.2).4 Evidently the formalization of the flight attributes of penguins is insufficient. Every man respects his parent. Modularity sacrificed. Translating an English sentence into predicate logic can be tricky. Rats cannot fly. First, the higher the frequency, the stronger the logic can be. But logical aspects of natural and artificial languages are much . In this section we look at two operations that generalize the and and or operations to predicates. Prove that p (q r) = (p q) (p r) a. using a truth table. Valid 6. x bird (x) fly (x). 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 Predicate Logic Anvesh Komuravelli 1 Why Predicate Logic? What is a predicate? Conclusion: . 4 Predicates x > 3 Variable: subject of the statement Predicate: property that the subject of the statement can have. Consider the premises: P1: Nothing intelligible puzzles me. The predicate is "fly(bird)." And since there are all birds . x Predicates: 2 : T ;, 3 : T ;, etc. 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. And since there are all birds who fly so it will be represented as follows. Since there is every man so will use , and it will be . 4. Rule 2 Eagles are carnivorous birds that can fly. Consider the statement, " is greater than 3. (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. . 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." 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. All birds have wings. Use mathematical induction to prove that, for n1, 12 + 22 + 32 + .. + n2 = n (n+1) (n+2)/6 4. John likes everyone, that is older than $22$ years old and that doesn't like those who are younger than $22$. NB: Evaluating an argument often calls for subjecting a critical. FMSE lecture 06. 1. "Not all birds fly" is equivalent to "Some birds don't fly". Predicate Logic More powerful Express a wide range of statements in mathematics and computer science. (Jan-2012-win-old)[3] A crow is a bird. Not all birds can fly. | 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. Rule 4 Ostriches are granivorous birds that can fly. Ans : - P ( x ) : x is a bird . Solution: A predicate that can be true or false, depending on the situation/state [2 points] What does the denition of an operator (e.g. All penguins are birds. It tells the truth value of the statement at . Ti xung (.pdf) 0 (73 trang) Lch s ti xung. Therefore, a crow can fly. . Type I - Particular Affirmative proposition 2. Predicate Logic x Variables: T, U, V, etc. The predicate in this question is " respect (x, y)," where x=man, and y= parent. . Some boys play cricket. Represent statement into predicate calculus forms : "Not all birds can fly". could be written symbolically as (x(B(x) ( F(x) where. Some natural problem is not monotonic non-monotonic logic. 73. 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 . = Only birds are flying things. b. 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. Consistency not all deductions may be correct. Predicate Calculus. 1. It has two parts. Ans:- P(x): x is a bird. "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. Consider the following statements. "Not all cars are expensive" is equivalent to "Some cars are not expensive", . 55 # 35 a particular kind of argument containing three categorical propositions, two of them premises, one a conclusion. 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 . (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! E.g., "For every x, x > 0" is true if x is a positive integer. 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. P2: Logic puzzles me. All birds fly. Changes in knowledge base might have far-reaching effects. All noncats are things that cannot run at more than 50 miles an hour. . First-order logic is another way of knowledge representation in artificial intelligence. Consistency not all deductions may be correct. Bhavin B. Joshi (Asst. The predicate in this question is " fly (bird) ." Because all birds are able to fly, it will be portrayed as follows. It is an extension to . 2. . Do \not all birds can y" and \some bird cannot y" have the same meaning? For the rst sentence, propositional logic might help us encode it with a single proposition but . For instance, it can join simple sentences or clauses by logical connectives to represent more complex sentences. All Germans speak at least two languages cEvery bird can y. E is not grounded in the sense above: If we take E as a belief set (relevant for the . First-order logic is also known as Predicate logic or First-order predicate logic. domain the set of real numbers . Provide a resolution proof that tweety can fly. The predicate can be considered as a function. (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 . 4. Examples: Is T P1 True or False Is T is a great tennis player True or False? Regarding the second question: FOL is sufficiently expressive to represent the natural language statements in a concise way. e.g If we know that Fred is a bird we might deduce that Fred can fly. cont'd Example: All birds have wings Type E proposition. Not all birds can fly x ( B(x) F(x) ) x ( (B(x) F(x) ) B(x) : x is a bird. Valid 9. Can you identify problem(s) in the example? 3. Some dogs are not collies. Sentences - either TRUE or false but not both are called propositions. Every man respects his parent. But we can easily turn it into a plural noun. Even adding only the induction axiom for the natural numbers makes the logic incomplete. Chapter 2 Predicate Logic Discrete Mathematics II BKTPHCM. The predicate in this question is "respect(x, y)," where x=man, and y= parent. 3. Some Examples of FOL using quantifier: All birds fly. x bird(x) fly(x). F(x) = x can fly . Tweety is a penguin 2. In common sense reasoning two typical types of defaults are encountered. a. 2. C. Therefore, all birds can fly. 3. The logical operations and identities in the previous sections apply to both propositions and predicates. 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. Convert your first order logic sentences to canonical form. 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". Solution for Express the following sentence in Predicate Logic(Define Ontology first and use it.) :o I want to formulate the following statements into formulas of predicate logic. But logical aspects of natural and artificial languages are much . 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. . This is equivalent to demonstrating that A is not a subset of B. c. Mary and Sue have the same paternal grandfather. 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. (i) Some old dogs can learn new tricks. A Categorical Syllogism is modernly defined as. USING PREDICATE LOGIC Representation of Simple Facts in Logic If an object is not to the right of all the squares, then it is not blue. Later we might discover that Fred is an emu. All birds can fly (1) Penguin is a bird (2) Then you may conclude Penguin can fly. Only two students took both French and Greek in spring 2010 4. We cannot say it in propositional logic. The more direct translation to Prolog would then be: bird (X) :- fly (X). 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. Cumbersome control information. 3. 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 > 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. 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. 2. Not only is there at least one bird, but there is at least one penguin that cannot fly. . Aristotle contemplating a bust of Homer by Rembrandt van Rijn. Penguins are birds 3. . 8xF(x) 9x:F(x) There exists a bird who cannot y. Semantically equivalent formulas. 1.4 Predicate Logic. Domain for x is all birds. Like all birds, seagulls have two wings and can fly. x bird(x) fly(x). 2. 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. Even though penguins are also birds, they cannot fly. "Flying things" is a plural noun; we can count flying things. Valid 5. Predicate Logic Predicate logic have the following features to express propositions: Variables: x;y;z, etc.
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