*> The general rule is for uniformity, and it takes getting used to. Predicate Logic 4. Convert to conjunction of disjuncts 8. An in-depth look at predicate logic proofs Understanding rules for quantifiers through more advanced examples. >> /Name/F1 * 3. 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 endobj Semantic networks are alternative of predicate logic for knowledge representation. 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 See also propositional calculus. They are basically promulgated under the authority of the Food Drug and Cosmetic Act or under the authority of the Public Health Service Act. Eliminate Universal Quantifiers * 7. Consider the following famous argument: All men are mortal. Predicate Logic \Logic will get you from A to B. The following are some examples of predicates −, Well Formed Formula (wff) is a predicate holding any of the following −, All propositional constants and propositional variables are wffs, If x is a variable and Y is a wff, $\forall x Y$ and $\exists x Y$ are also wff. The law of variable substitution is an inference rule for use in proofs in predicate logic.. /Filter[/FlateDecode] The difference between these logics is that the basic building blocks of Predicate Logic are much like the building blocks of a sentence in a language like English. /Type/Encoding Thus, predicate logic employs six rules, in addition to all of the rules of sen-tential logic. 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 Example 21. Predicate calculus, also called Logic Of Quantifiers, ... by the rules of the calculus. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 In Predicate Logic, the smallest proposition is formed by combining a predicate with an individual. But with the approach of predicate logic, we can integrate the two levels of analysis, and say: 1. /Name/F3 $\forall\ a\: \exists b\: P (x, y)$ where $P (a, b)$ denotes $a + b = 0$, $\forall\ a\: \forall\: b\: \forall\: c\: P (a, b, c)$ where $P (a, b)$ denotes $a + (b + c) = (a + b) + c$, Note − $\forall\: a\: \exists b\: P (x, y) \ne \exists a\: \forall b\: P (x, y)$, Let X(a, b, c) denote "a + b + c = 0". As we have already mentioned, a predicate is just a function with a range of two values, say false and true. Viele übersetzte Beispielsätze mit "predicate rules" – Deutsch-Englisch Wörterbuch und Suchmaschine für Millionen von Deutsch-Übersetzungen. 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 10. To make use of this language of logic, you need to know what operators to use, the input-output tables for those operators, and the implication rules. /Subtype/Type1 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 944.4 500 722.2 777.8 777.8 This chapter is dedicated to another type of logic, called predicate logic. Inference Rules and Proofs for Predicate Logic Emina Torlak and Kevin Zatloukal 1. With sentential logic, you use the following equivalence rules to make those comparisons: Identity and Quantifier Rules for Quantifier Logic. ?�5����]p���ϧ��Rā*K` ��bɣ�3#�2g��=���&�� �4�`���m���q�K�Mqst\�[�uv�h5 ہ͏;`s�B��]��[��O�z_����:.��r��ڊ1�j�Ǚ�ƴ� q}sC���}����ݘ�nl�'�m��-%�M)n;��OHm����Vl��'r�N6��J]w%���!�ʪw����`��G��>�6����2�'��I�*� "��YMkU�"r���Y�}��+5�d#Dq�!�]�Z#4/� ��y��0��f��~�����L�'EK�BKܗ�����Ad�W�-�w�3ӓI����u�J@� �T��*�AY��ȊlHY�L�RV=S��)�hV?��թ�c�;��b�? 777.8 777.8 0 0 1000 1000 777.8 722.2 888.9 611.1 1000 1000 1000 1000 833.3 833.3 endobj Since predicate logic adopts all the derivation rules of sentential logic, it is a good idea to review the salient features of sentential logic derivations. A predicate rule is any FDA regulation that requires a company to maintain certain records and submit specific information to the agency as part of compliance. We already use predicates routinely in programming, e.g. Assumption 1.2 () Elim∀: 1.1 1.3. The smallest English sentence is formed by combining a verb with a subject. What is type inference in C++? Would be welcomed to hear your ideas about this task. 300 325 500 500 500 500 500 814.8 450 525 700 700 500 863.4 963.4 750 250 500] Last Class: Predicate Logic Proof Prove ∀x P(x)→ ∃x P(x) 1. 2.1.1 Proof Situations and Proofs endstream Topics Propositional logic proofs A brief review of . 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 •A predicate logic (or calculus) expression X logically follows from a set S of predicate calculus expressions if every interpretation and variable assignment that satisfies S also satisfies X. << The argument is valid if the premises imply the conclusion. 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 Issues, Predicate Logic, Rules How do we represent what we know ? 255/dieresis] /FontDescriptor 19 0 R 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 The general strategy for predicate logic derivations is to work through these three phases: (1) instantiate the premises, (2) work with what you have then, using the original 19 rules plus CP and IP, and (3) then generalize as needed to put the right quantifiers on the conclusion. The main things we have to deal with are equality, and the two quantiﬁers (existential and universal). /Type/Font << 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 777.8 777.8 777.8 777.8 777.8 277.8 666.7 666.7 Natural deduction for predicate logic Readings: Section 2.3. A. Einstein In the previous chapter, we studied propositional logic. Let us start with a motivating example. (2) /Length 1188 1. (Bx v Ax)) > Px] / Pp. 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis ~� /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 450 500 300 300 450 250 800 550 500 500 450 412.5 400 325 525 450 650 450 475 400 But with the approach of predicate logic, we can integrate the two levels of analysis, and say: 1. Subjects to be Learned. /LastChar 196 $\forall x P(x)$ is read as for every value of x, P(x) is true. 761.6 272 489.6] Well-Formed Formula for First Order Predicate Logic --- Syntax Rules. –An interpretationis an assignment of specific values to domains and predicates. A predicate with variables can be made a proposition by either authorizing a value to the variable or by quantifying the variable. /FirstChar 33 The Predicate Calculus; Inference Theory of the Predicate Logic; Rules for Java method overriding; Rules for operator overloading in C++; Type Inference in C++; E.F. Codd’s 12 Rules for RDBMS; Difference between Relational Algebra and Relational Calculus; What are the rules for the body of lambda expression in Java? Prerequisite : Predicates and Quantifiers Set 1, Propositional Equivalences Logical Equivalences involving Quantifiers Two logical statements involving predicates and quantifiers are considered equivalent if and only if they have the same truth value no matter which predicates are substituted into these statements irrespective of the domain used for the variables in the propositions. Reduce the scope of all Ø to single term. /F2 13 0 R 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 /FontName/XZECJH+CMR12 /F5 23 0 R endobj Substitution Rule. << /Name/F4 Cp. This chapter is dedicated to another type of logic, called predicate logic. endobj The rules of identity are shown here: And, when talking about identities, you can quantify statements, using the rules in […] /F3 16 0 R /BaseFont/RXUMZP+CMTI12 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 See also propositional calculus. * 3. Subjects to be Learned. •A predicate logic (or calculus) expression X logically follows from a set S of predicate calculus expressions if every interpretation and variable assignment that satisfies S also satisfies X. What’s new is moving from a strict universal statement (x), to a case of that statement. A predicate is an expression of one or more variables determined on some specific domain. 10 0 obj Those symbols come into play when you work with identities, or interchangeable constants. To interpret a formula as a sentence (a statement or an open sentence) from the natural language, we need to interpret the … We can express the premises (above the line) and the conclusion (below the line) in predicate logic as an argument: We will see shortly that this is a valid argument. The smallest English sentence is formed by combining a verb with a subject. $\exists x P(x)$ is read as for some values of x, P(x) is true. Prerequisite : Predicates and Quantifiers Set 1, Propositional Equivalences Logical Equivalences involving Quantifiers Two logical statements involving predicates and quantifiers are considered equivalent if and only if they have the same truth value no matter which predicates are substituted into these statements irrespective of the domain used for the variables in the propositions. Laws and Rules for Predicate Logic (1) Laws of Quantiﬁer Distribution Law 1:(8x) ’(x) (9x):’(x) Law 2 (8x)(’(x)^ˆ(x)) ((8x)’(x)^(8x)ˆ(x)) Law 3 (9x)(’(x)_ˆ(x)) ((9x)’(x)_(9x)ˆ(x)) Law 4 ((8x)’(x)_(8x)ˆ(x)) =) (8x)(’(x)_ˆ(x)) Law 5 (9x)(’(x)^ˆ(x)) =) ((9x)’(x)^(9x)ˆ(x)) (2) Laws of Quantiﬁer (In)Dependence Law 6 (8x)(8y)’(x;y) (8y)(8x)’(x;y) Law 7 (9x)(9y)’ Imagination will take you every-where." 777.8 777.8 777.8 777.8 777.8 777.8 1333.3 1333.3 500 500 946.7 902.2 666.7 777.8 (Bp . 1 The Language PLE Vocabulary The vocabulary of PLE consists in the following: 1. 1. Eliminate all implications Þ 2. There are two types of quantifier in predicate logic − Universal Quantifier and Existential Quantifier. x, y) are neither true nor false when the values of the variables are not specified. • Obvious information may be necessary for reasoning • We may not know in advance which statements to deduce (P or P). A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable. Cp. Example − "Some people are dishonest" can be transformed into the propositional form $\exists x P(x)$ where P(x) is the predicate which denotes x is dishonest and the universe of discourse is some people. � �oy�_�Rv��Ɉ� ����3 �m ���'�܅�m����#�:Y3��b�&C���kkJs�M,�����[Oū%�3�j]���)M���ru��=,�u&R� ���o���? /Length2 8798 The general strategy for predicate logic derivations is to work through these three phases: (1) instantiate the premises, (2) work with what you have then, using the original 19 rules plus CP and IP, and (3) then generalize as needed to put the right quantifiers on the conclusion. /LastChar 196 The empha- sis of this chapter is being put on an introduction of rules for proving in predicate logic. /StemV 65 /Length3 533 Laws and Rules for Predicate Logic (1) Laws of Quantiﬁer Distribution Law 1:(8x) ’(x) (9x):’(x) Law 2 (8x)(’(x)^ˆ(x)) ((8x)’(x)^(8x)ˆ(x)) Law 3 (9x)(’(x)_ˆ(x)) ((9x)’(x)_(9x)ˆ(x)) Law 4 ((8x)’(x)_(8x)ˆ(x)) =) (8x)(’(x)_ˆ(x)) Law 5 (9x)(’(x)^ˆ(x)) =) ((9x)’(x)^(9x)ˆ(x)) (2) Laws of Quantiﬁer (In)Dependence Law 6 (8x)(8y)’(x;y) (8y)(8x)’(x;y) Law 7 (9x)(9y)’ E.g., for the integers we add the set ℤ, /Encoding 7 0 R << 9 0 obj – In Predicate Logic, there are variables, so we have to do more than that. Sentential Logic Operators, Input–Output Tables, and Implication Rules. The last statement is the conclusion. /Widths[300 500 800 755.2 800 750 300 400 400 500 750 300 350 300 500 500 500 500 Large amount of knowledge 2. /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 >> ��Iq���+��#�#\B~��hmC}�s�~��_y���8K��2��k����X^0��J_����R�`�6�RK�t{M��ly3�!�vh.��a���f>�F�� S \@� 0l��}�[���[ܳe\uKV��-���\[�/��u���x+�)"@/"����Mཎ΄��%"�nDp�;��#B ED����\'��N�a�1�����~�ZH�{�X�l��^O�#еGw�ofnb)uo��b��ʦ���H��e�1���ɭ��s��� /Name/F2 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 >> Various restricted forms of the higher-order calculi have been shown, however, to be susceptible to routine decision procedures for all of their formulae. In Predicate Logic, the smallest proposition is formed by combining a predicate with an individual. /Type/Font (Bx v Ax)) > Px] / Pp. We'll illustrate this with an example. 2��8��!�P[ �?��m��@���M]���� 500 500 500 500 500 500 500 300 300 300 750 500 500 750 726.9 688.4 700 738.4 663.4 Informally, this rule states that having established that a general fact (or expression) is true, we can assert that a specific instance of that general expression is also true. 0 0 0 613.4 800 750 676.9 650 726.9 700 750 700 750 0 0 700 600 550 575 862.5 875 Eliminate Existential Quantifiers * 6. A predicate is an expression of one or more variables determined on some specific domain. (x) [(Cx . (2) A predicate with variables can be made a proposition by either authorizing a value to the variable or by quantifying the variable. Ap) 2. (x) [(Cx . /BaseFont/LZVMXX+CMSY10 /FirstChar 33 Proof Rules for Predicate Logic 2.1 Introduction Mathematical activity can be classified mainly as œprovingł, œsolvingł, or œsimplifyingł. x��UTᶥ�۸m,��[p� ��]7��������%��ww'���7眾�G��/=��GW�Ԛk���ZU�S�)�2���C$�l�Y�X�@��*�l V& ��#���;C�@���� s�������� ����{8B�-�A��t�pq�Dl �P�-H�l��b��ڙ@!�L ���5H��8�T NGW�) �� Inference rules for propositional logic plus additional inference rules to handle variables and quantifiers. /FontDescriptor 12 0 R /Type/Font >> Direct Proof Rule 1.1. https://www.tutorialspoint.com/.../discrete_mathematics_predicate_logic.htm A statement with variable has two parts: x is greater than 9 The first part, the … When you feel comfortable with the syntax of Predicate Logic, I urge you to read these notes carefully. Such calculi are, in the precise sense, incomplete. /F4 20 0 R Handout 5 – The Semantics of Predicate Logic LX 502 – Semantics I October 17, 2008 1. 7 0 obj Such calculi are, in the precise sense, incomplete. x��[Ys�6~ϯ`�B>p��H'/;wҙ�u��&�Ȱ���H�����!��ٺƔ�D�X`w�o,`Bޭ��\x�^�~�=�As��ƣ�'^��}��G��]�H��")>G8���7�*`ڶd�X��]��?�N]3�B�5K�3��I��@��E�t&~�/s���:���nj�2����Yه���&��d���F���!F�B�A�t���GA�Y:�ȇ���&⏻q�ʓhD�4���j=���%�,N5�"�j�K˚�l.���m���Ҧo3��E^9�}��Ve���L5�*4��ʢ�U{���[���eJb}J�uJ�J���,c!V�*"�6����"�r�4�Z'Ƀ���J�.x� T����>�+-:h�}��=��䕟b1A��цh���Jlh��0q����Z�U�t���G��;םE���O �va���DP���t#��A�˰��E�/[W��� n� 8:�()��Ͱ��ӵ V�b�ܻ]�c;>�~=`Ў�q�Rw|�. [�]7���.-��[ک���+K�Hħ'������-$\O�3 GL/eqޔ���E�����y�$X_B�{���&�u(��%�?/G�j�-q���#���[���D���T�#T�Y9�ʬ��ǃ�Dx�����Ofr ב��_mvU�*h�,��4*,��u���w����ԕ��=�M�!y5�sk����Z�z��\(�ct��㟳M��Շ�/��Ӂ�������g���q2ڮ�p�q��D�Ҡ�D^Ɇ�o��k�����U�+d��"u$�ﺄegQ�2z2\Z���ߍ��~�|GS:���VFٛzåyழd�S�iD�����|UL�As�'��[�Voz4�$��>,%�ZhQrFً��q�� VIl� ��ͣ. Eliminate all implications Þ 2. Lecture 07 2. 13 0 obj 82 Using Predicate Logic • Many English sentences are ambiguous. /Length 9354 /Font 27 0 R << For example: x>9; x=y+9; x+y=z; Predicate Logic allows to make propositions from statements with variables. In this module, we will extend our previous system of natural deduction for propositional logic, to be able to deal with predicate logic. endobj Predicate calculus: area of logic dealing with predicates and quanti ers. An answer to the question, "how to represent knowledge", requires an analysis to distinguish between knowledge “how” and knowledge “that”. Chapter 5 10 Resolution in Predicate Logic Axioms in clause form: 1.man(Marcus) 2.Pompiean(Marcus) 3.- Pompiean(x1) ν Roman(x1) 4.ruler(Caesar ) 5.- Roman(x2) ν loyalto(x2,Caesar) ν hate(x2,Caesar) 6. loyal(x3,f(x3)) 7.- man(x4) ν - ruler(y1) ν - tryassassinate(x4,y1) ν loyalto(x4,y1) /ItalicAngle 0 A quick look at predicate logic proofs Inference rules for quantifiers and a “hello” world example. 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 /FontDescriptor 9 0 R stream 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] << /Type/FontDescriptor Predicate Logic PHI 201 Introductory Logic Spring 2011 This is a summary of deﬁnitions in Predicate Logic from the text The Logic Book by Bergmann et al. In predicate logic a logical expression is defined as follows: (1) If t 1, t 2,…, t n are terms and P is a predicate with n parameters, then P (t 1, t 2, …, t n) is an atomic formula and a logical expression. >> 638.4 756.7 726.9 376.9 513.4 751.9 613.4 876.9 726.9 750 663.4 750 713.4 550 700 The difference between these logics is that the basic building blocks of Predicate Logic are much like the building blocks of a sentence in a language like English. >> Convert to conjunction of disjuncts 8. Make all variable names unique 4. /F1 10 0 R What’s new is moving from a strict universal statement (x), to a case of that statement. /LastChar 196 /Length1 714*