Assemble the relevant knowledge 3. -"$ -p v (q ^ r) -p + (q * r) Propositional logic is a weak language Hard to identify "individuals" (e.g., Mary, 3) Can't directly talk about properties of individuals or relations between individuals (e.g., "Bill is tall") Generalizations, patterns, regularities can't easily be represented (e.g., "all triangles have 3 sides") First-Order . - x y Likes(x, y) "Everyone has someone that they like." - x y Likes(x, y) "There is someone who likes every person." Pros and cons of propositional logic . - x y Likes(x, y) "Everyone has someone that they like." - x y Likes(x, y) "There is someone who likes every person." Pros and cons of propositional logic . In fact, the FOL sentence x y x = y is a logical truth! Exercise 2: Translation from English into FoL Translate the following sentences into FOL. 1.Everything is bitter or sweet 2.Either everything is bitter or everything is sweet 3.There is somebody who is loved by everyone 4.Nobody is loved by no one 5.If someone is noisy, everybody is annoyed 1 It is an extension to propositional logic. FOL syntax Sentence: T/F expression Atom Complex sentence using connectives: . FOL is sufficiently expressive to represent the natural language statements in a concise way. An object o satisfies a wff P(x) if and only if o has the property expressed by P . But wouldn't that y and z in the predicate husband are free variables. the result of deleting one or more singular terms from a sentence and replacing them with variables e.g. x. A well-formed formula (wff) is a sentence containing no "free" variables. Everyone is a friend of someone. Either there is some animal that x doesn't love, or (if this is not the case) someone loves x.-----Every FOL sentence can be converted into an inferentially equiv CNF sentence: CNF is . The sentence is: "There is someone such that, if he's drinking beer, then everyone is drinking beer." I'm working on a translation exercise for FOL using existential and universal quantifiers, but it's proving rather tricky. Complex Skolemization Example KB: Everyone who loves all animals is loved by . . Properties and . . - If the sentence is false, then there is no guarantee that a procedure will ever determine this-i.e., it may never halt. means "Everyone is at CSU and everyone is smart" October 27, 2014 15 Existential quantification <variables> <sentence> Someone at CSU is smart: x At(x, CSU) Smart(x) $ x P(x) is true iff P is true for some object x $ Roughly speaking, equivalent to the disjunction of instantiations of P At(KingJohn,CSU) Smart(KingJohn) Original sentences are satisfiable if and only if skolemized sentences are. Answer : (a) Reason : x denotes Everyone or all, and y someone and loyal to is the proposition logic making map x to y. Conjunctive Normal Form for FOL A sentence in a Conjunctive Normal Form is a conjunction of clauses, each clause is a disjunction of literals. In the first step we will convert all the given statements into its first order logic. 3. Lucy* is a professor 7. or y. Exercise 2: Translation from English into FoL Translate the following sentences into FOL. First-order logic is also known as Predicate logic or First-order predicate logic. Like BC of PL, BC here is also an AND/OR search. Everyone likes ice cream - there is no one who does not like ice cream; Connections Between \(\forall . People only criticize people that are not their friends. everyone has someone whom they love. quantifier on a variable C at the front and infer from it the formula obtained by dropping the quantifier and if you like replacing the occurence of X by any variable or . Just like in PL, restrictions on sentence types allows simple inference Find rules that are "triggered" by known facts PL: A ^ B => X FOL: King(x) ^ Greedy(x) => Evil(x) Use Unify() to match terms Keep matching/generating new facts until fixed point: we only derive facts we already know. Below I'll attach the expressions and the question. So could I say something like that. Good(x)) and Good(jack). \item There are four deuces. $\begingroup$ @New_Coder, I am not sure about the second FOL sentence. <variables > < sentence > Everyone at Pitt is smart: x At(x,Pitt) Smart(x) . Now consider the following statement taken from the OP: AxEy(Likes( man(x), woman(y) ) -> Likes(alex, man(x) )) This statement is from a different language. "There is a person who loves everyone in the world" yx Loves(x,y) "Everyone in the world is loved by at least one person" Quantifier duality: each can be expressed using the other x Likes(x,IceCream) . Example.. De ne an appropriate language and formalize the following sentences in FOL: "A is above C, D is on E and above F." "A is green while C is not." "Everything is on something." "Everything that has nothing on it, is free." "There is a person who loves everyone in the world" - y x Loves(x,y) (Ax) S(x) v M(x) 2. FOL has variables, universal and existential quantification (infinite AND and OR), predicates that assert properties of things, and functions that map between things. "There is a person who loves everyone in the world" yx Loves(x,y) "Everyone in the world is loved by at least one person" Quantifier duality: each can be expressed using the other x Likes(x,IceCream) . If someone is noisy, everybody is annoyed 6. 6. In order to infer new knowledge from these sentences, we need to process these sentences by using inference methods. single predicates) sentences P and Q and returns a substitution that makes P and Q identical. All professors consider the dean a friend or don't know him. Answer 5.0 /5 2 Brainly User Answer: in the form of a single formula of FOL, which says that there are exactly two llamas. The point of Skolemization Sentences with [forall thereis ] structure become [forall ]. All professors are people. Everything is bitter or sweet 2. Nobody is loved by no one 5. FOL sentences have structure, like programs In particular, the variables in a sentence have a scope For example, suppose we want to say "everyone who is alive loves someone" ( x) alive(x) ( y) loves(x,y) Here's how we scope the variables ( x) alive(x) ( y) loves(x,y) Scope of x Scope of y Given the following two FOL sentences: yx(Loves(x,y)) Says there is someone who is loved by everyone in the universe. FOL has practical advantages, especially for automation. Answer : (d) Reason : Quantity structure is not a FOL structure while all other are. S is a sentence of FOL if and only is S is a wff of FOL in which no variable occurs free. Identify the problem/task you want to solve 2. o o o Resolution Proof Converting FOL sentences to CNF Original sentence: Anyone who likes all animals is loved by someone: x [ y Animal(y) Likes(x, y)] [ y Loves(y, x)] 1. Chiara Ghidini ghidini@fbk.eu Mathematical Logic Translation into FOL Sentences Let S(x) mean x is a skier, M(x) mean x is a mountain climber, and L(x,y) mean x likes y, where the domain of the first variable is Hoofers Club members, and the domain of the second variable is snow and rain. Comment: I am reading this as `there are \emph { at least } four \ldots '. xhates y) (a) Alice likes everyone that hates Bob. 1.Everything is bitter or sweet 2.Either everything is bitter or everything is sweet 3.There is somebody who is loved by everyone 4.Nobody is loved by no one 5.If someone is noisy, everybody is annoyed 1 Complex Skolemization Example KB: Everyone who loves all animals is loved by . Everyone likes someone. View the full answer. Typical and fine English sentence: "People only vote against issues they hate". Someone likes all kinds of food 4. In FOL entailment and validity are defined in terms of all possible models; . nobody loves Bob but Bob loves Mary. (d) There is someone who likes everyone that Alice hates. if someone loves David, then he (someone) loves also Mary. Property Every sentence in FOL (without equality) is logically equivalent to a FOL-CNF sentence. Pose queries to the inference procedure and get answers. 3. - A common mistake is to represent this English sentence as the FOLsentence: ( x) student (x) => smart (x) It also holds if there no student exists in the domain because student (x) => smart (x) holds for any individual who is not astudent. x y Loves(x,y) "There is a person who loves everyone in the world" y x Loves(x,y) "Everyone in the world is loved by at least one person" Quantifier duality: each can be expressed using the other x Likes(x,IceCream) x Likes(x,IceCream) x Likes(x,Broccoli) x Likes(x,Broccoli) - "There is a person who loves everyone in the world" y x Loves(x,y) - "Everyone in the world is loved by at least one person" Quantifier duality: each can be expressed using the other xLikes(x,IceCream) x Likes(x,IceCream) x Likes(x,Broccoli) x Likes(x,Broccoli) "Everyone who loves all animals is loved by . The motivation comes from an intelligent tutoring system teaching . . Our model satisfies this specification. If the suggestion is that there are \emph { exactly } four, then we should offer instead: \\. First-order logic is a logical system for reasoning about properties of objects. sentences and wffs a term (denoting a real-world individual) is a constant symbol, avariable symbol, or an n-place function of n terms. $\endgroup$ - Everyone likes someone: (Ax)(Ey)likes(x,y) Someone is liked by everyone: (Ey)(Ax)likes(x,y) Example 7. there existsyallxLikes(x, y) Someone likes everyone. Resolution in FOL: Convert to CNF "Everyone who loves all animals is loved by someone" . Godel's Completeness Theorem says that FOL entailment is only semidecidable: - If a sentence is true given a set of axioms, there is a procedure that will determine this. Y x Likes(x, IceCream) ax Likes(x,Broccoli) Likes(x, IceCream)) If the suggestion was that there are \emph { exactly } two, then a different FOL sentence would be required, namely: \\. 12. Just "smash" clauses until empty clause or no more new clauses. "There is a person who loves everyone in the world" x y Loves(x, y) "Everyone in the world is loved by at least one person" y x Loves(x, y) Quantifier Duality - Each of the following sentences can be expressed using the other x Likes(x, IceCream) x Likes(x, IceCream)