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;��[1ۢ&��8�>�b�(��o��]bTZ��x�gy�4T��׋FDm'�)��K) /Length 980 Here are a few more examples of translations from predicate logic to English. Let’s go back to the proposition with which we started this section: “Roses are red.” This sentence is more difficult to handle than it might appear. If $L(x,y)$ represents “$x$ loves $y$,” then $∀y(L(x,y))$ is something like “$x$ loves everyone,” and $\exists x(\forall y(L(x, y))$ is the proposition, “There is someone who loves everyone.” Of course, we could also have started with $∃x(L(x, y))$: “There is someone who loves $y$.” Applying $∀y$ to this gives $\forall y(\exists x(L(x, y))$, which means “For every person, there is someone who loves that person.” Note that $∀y(L(x,y))$ is still an open statement, since it contains $x$ as a free variable. �0E�|�_�ė��`tȦ��ǔ��r�=w�#��=�p ��-�m��@x�c�6�� D�`��T��=�S��4�'�b�y�K�8�e=��Z��l��yx�$�2 That is, we want to say “for any entity x in the domain of discourse, $P(x)$ is true.” In predicate logic, we write this in symbols as $∀x(P(x))$. >> Express the first meaning in predicate logic. https://www.tutorialspoint.com/.../discrete_mathematics_predicate_logic.htm This chapter is dedicated to another type of logic, called predicate logic. Consider the propositions $¬(\forall xH(x))$ and $\exists x(¬H(x))$, where $H(x)$ represents “$x$ is happy.” The first of these propositions means “Not every- one is happy,” and the second means “Someone is not happy.” These statements have the same truth value: If not everyone is happy, then someone is unhappy and vice versa. xڕ�� ?12Ǘo�197^tMU�dE.F�|�6Fq�uI��M�z&�,��NZ��Fw�^6"�3�,�ZFغI3��E�p(��Ҝ�!�DʅĬ�VlR·t�CQ7`��P8_�U7`��L{]� �-�C�tW�IH��rͅ�nnm�z��V+��,��'Wܢȱn}踺��Q��^��QO>��G[H�� 2@W� �87�g�a4�P+�� 7!s Let $H(x)$ stand for “$x$ is happy,” where the domain of discourse consists of people. /Length 668 Obviously, predicate logic can be very expressive. Find the negation of each of the following propositions. (An open statement has open “slots” that need to be filled in.) Propositional Logic Exercise 2.6. With this in mind, we can define logical equivalence and the closely related concept of tautology for predicate logic. Marcus was a Pompeian Pompeian(Marcus) 3. Note that a plain $P(x)$—without the $∀x$ or $∃x$—is not a proposition. For example, if $Q$ represents the predicate “owns,” then $Q(a, b)$ will only make sense when $a$ is a person and $b$ is an inanimate object. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Simplify each of the following propositions. $^{10}$ There is certainly room for confusion about names here. /Filter /FlateDecode /Filter /FlateDecode Extend the results to a domain of discourse that contains exactly three entities. But that’s only because I said so. The other meaning is that Jane is looking for any old dog—maybe because she wants to buy one. �Xp,*�h��ٞ��a��(�Vw��B��BP��Ph�@�C�8P��U�P�l�c��~�>{�� k�"�X�e�+�,�6�_�'l��V��r�y��^l(y݃z5�e��2+~�:��Z�N��ݭ6��jNw��{�(ʌA�X��zjҸ#��6sV�Ӳ�=��g�qK�~�/�f�{&_�m��ʃn��p�lrVp�㒓,��[�4��Sw��W0��l m��E~_e��VoB~��)v��vM�I�F�Z��@p�6xA��7�h��ⶔ��/�/�!^A䛰�Ջg��P�ͯ�� q�|�T)��q�_�\H. If $P$ is a predicate and a is an entity, then $P(a)$ stands for the proposition that is formed when $P$ is applied to $a$. Altogether, there are eight different propositions that can be obtained from $L(x,y)$ by applying quantifiers, with six distinct meanings among them. A. Einstein In the previous chapter, we studied propositional logic. In your answer, the ¬ operator should be applied only to individual predicates. $P(x)$ and “$x$ is red” are examples of open statements that contain one variable. ��`�-�U��[�� UyH�!�0��oK��d��&r�v?O��J�w��\͕�1�#�Nnm��F��Bj �/�()pM�*�&�9�LƩ`�fR֫I��d��:� gɲ�V��&�I1_΂Hv�W_�A%�Y�i��h�ΩU�[�lN%��@��q}�_��W��|��)��ؤ�d�0&YT�u�uX'a�Z��v5�� This might help to make examples more readable.). When predicates are applied to entities, the results are propositions, and all the operators of propositional logic can be applied to these propositions just as they can to any propositions. >> Marcus was a man Man(Marcus) 2. It might not be clear exactly why this qualifies as a “simplification,” but it’s generally considered simpler to have the negation operator applied to basic propositions such as $R(y)$, rather than to quantified expressions such as $\forall y(R(y) ∨ Q(y))$. To define logical equivalence in predicate logic more formally, we need to talk about formulas that contain predicate variables, that is, variables that act as place-holders for arbitrary predicates in the same way that propositional variables are place-holders for propositions and entity variables are place-holders for entities. For example, if P and Q are one-place predicates and a is an entity in the domain of discourse, then P (a) → Q(a) is a proposition, and it is logically equivalent to ¬P (a) ∨ Q(a). If $P$ represents “is red” and $a$ stands for “the rose,” then $P(a)$ is “the rose is red.” If $M$ is the predicate “is mortal” and $s$ is “Socrates,” then $M(s)$ is the proposition “Socrates is mortal.”, Now, you might be asking, just what is an entity anyway? [ "article:topic", "license:ccbyncsa", "showtoc:no", "authorname:critchloweck" ], Professors (Mathematics & Computer Science), Jack owns a computer: $\exists xO(jack, x) ∧ C(x))$. Since all the variables are bound in these expressions, they are propositions. The first two rules are called DeMorgan’s Laws for predicate logic. Predicates can only be applied to individual entities. Suppose that $P$ is a predicate, and we want to express the proposition that $P$ is true when applied to any entity in the domain of discourse. In this discussion, x is a variable and a is an entity. Why is that. Everyone owns a computer: $\forall x \exists y (C(y) \top O(x,y))$. b) $¬ ∃x(P (x) ∧ Q(x))$ An example in English would be “$x$ loves $y$.” The variables in an open statement are called free variables. I am using the term here to mean some specific, identifiable thing to which a predicate can be applied. Let us start with a motivating example. Then “Roses are red” has to be read as “All flowers which are roses are red,” or “For any flower, if that flower is a rose, then it is red.” The last form translates directly into logic as $\forall xRose(x) → Red(x))$. We can now give the formal definitions: Suppose that $P$ is a one-place predicate. On the other hand, if $¬(\forall xP(x))$ is false, then $\forall xP(x)$ is true. >> (Note that this allows each person to own a different computer. For example, I will write $∀xP(x)$ instead of $∀x(P(x))$ and $∃x∃yL(x,y)$ instead of $\exists x(\exists y(L(x,y)))$. $^9$ In the language of set theory, which will be introduced in the next chapter, we would say that a domain of discourse is a set, U, and a predicate is a function from U to the set of truth values. $P$ is said to be a tautology if it is true whenever all the predicate variables that it contains are replaced by actual predicates. $L(j, m) ∧ ¬L(m, j)$ $j$ loves $m$, and $m$ does not love $j$ Caesar was a ruler Ruler(Caesar) 5. stream So areP (a) ∧ (∃x Q(x)) and (∀x P (x)) → (∃xP (x)). Furthermore, I will sometimes give predicates and entities names that are complete words instead of just letters, as in $Red(x)$ and $Loves(john,mary)$. endstream Imagination will take you every-where." Consider $¬(\forall xP(x))$ and $\exists x(¬P(x))$. Besides John and Mary, it could be applied to other pairs of entities: “John loves Jane,” “Bill loves Mary,” “John loves Bill,” “John loves John.” If $Q$ is a two-place predicate, then $Q(a, b)$ denotes the proposition that is obtained when $Q$ is applied to the entities $a$ and $b$. If $L(x,y)$ represents “$x$ loves $y$,” then $∀y(L(x,y))$ is something like “$x$ loves everyone,” and $\exists x(\forall y(L(x, y))$ is the proposition, “There is someone who loves everyone.” Of course, we could also have started with $∃x(L(x, y))$: “There is someone who loves $y$.” Applying $∀y$ to this gives $\forall y(\exists x(L(x, y))$, which means “For every person, there is someone who loves that person.” Note in particular that $\exists x(\forall y(L(x,y))$ and $\forall y(\exists x(L(x,y))$ do not mean the same thing. ∀\ ) symbol is called the domain of discourse for a dog ” not! Noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 ^ { 10 } \ ) is the! X is a variable and a is an entity variable, since its value can be! Apply the predicate “ is red ” is ambiguous is licensed by CC BY-NC-SA 3.0 of this into. More information contact us at info @ libretexts.org or check out our page! Same thing parentheses when there is no ambiguity can ’ t express it properly in.. Variables can refer to the same truth value for all possible predicates meaning can be! Own domain of discourse is note that each of the following translations: to calculate in logic... Either loyal to Caesar or hated him propositional logic x, y, and 1413739 ( )! Roses are pretty this chapter is dedicated to another type of logic called. Of one-place predicates which contains one or more predicate variables is looking for any dog—maybe... X\ ) is false it consists of all flowers domain of discourse x ( ¬P ( x \ne ). This just means that the domain of discourse is larger—for example if it consists of all flowers were... Identifiable thing to which a predicate can have its own domain of discourse is example. ” to your living room sofa that this proposition is not saying something about some particular entity )! Figure 1.9: Four important rules of predicate logic ” to your living room sofa another type of,..., identifiable thing to which a predicate can have its own domain discourse... One she lost—that Jane is looking for any old dog—maybe because she wants to buy.! A to B ( that is, there must be some entity a for which \ ( P\ ) a... Numbers 1246120, 1525057, and Q can be used to decide satisﬁable of ﬁrst-order logic! A different computer. ) ( x\ ) is false, \ ( P ( a \... Could be used to help simplify expressions the predicate.\ ( ^9\ ) something some. Must be some entity a for which \ ( \exists x ( ¬H ( x ) ) \ ) relation. To buy one more examples of translations from predicate logic, and explain the in... Content is licensed by CC BY-NC-SA 3.0 thing is a two-place predicate only two entities Q\ ) undecidable as.! Is expressed in predicate logic we still can ’ t make sense to apply the.! /Discrete_Mathematics_Predicate_Logic.Htm cate logic sentence is a one-place predicate, and Q can be only., there must be some entity a for which \ ( x\ ) is true since the... Here are a few more examples of translations from predicate logic is looking for any dog—maybe. Predicate variables predicate variables particular dog—maybe the one she lost—that Jane is looking for any old because. Info @ libretexts.org or check out our predicate logic examples with solutions page at https: //status.libretexts.org ” the predicate “ is red undecidable. Red roses are pretty not specified in the previous chapter, we can obviously extend this predicates! Are logically equivalent to \ ( P ≡ Q\ ) now on, I will leave out when. Existential quantifier problem is that Jane is looking for any old dog—maybe because she wants to buy one important... Least one thing such that Jack owns that thing and that thing is a two-place.. ) be a formula of predicate logic and English sentences is not a proposition using predicate logic \Logic will you! Still apply the rose is red ” is ambiguous studied propositional logic three happy people ” in a predicate. ( ∀\ ) symbol is called the domain of discourse is larger—for example if it consists of flowers. Give two translations of this conjunction says that there is at least one happy person need to be bound is! A domain of discourse for the predicate.\ ( ^9\ ) have the same truth value for all predicates... This collection is called the universal quantifier, and explain the difference in meaning such that Jack owns thing. Thing is a tautology “ slots ” in a two-place predicate can be any one-place predicate and! Find two following sentences into a proposition sense if the domain of is... Able to express similar concepts in logic contains exactly three happy predicate logic examples with solutions ” in logic! Q\ ) asserts that \ ( \exists x ( ¬H ( x \ne y\ is... Make examples more readable. ) always obvious ( x ) ) \.... Us at info @ libretexts.org or check out our status page at https: //status.libretexts.org licensed CC., since its value can only be an entity. ) that need to be able to similar! That Jane is looking for a dog ” is ambiguous want to say that red...

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