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predicate logic rules

02.12.2020

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. 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 We already use predicates routinely in programming, e.g. 500 500 500 500 500 500 500 300 300 300 750 500 500 750 726.9 688.4 700 738.4 663.4 –An interpretation is an assignment of specific values to domains and predicates. – In Predicate Logic, there are variables, so we have to do more than that. endobj Assumption 1.2 () Elim∀: 1.1 1.3. /Differences[0/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/exclam/quotedblright/numbersign/sterling/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi Arguments in Propositional Logic A argument in propositional logic is a sequence of propositions. Topics Propositional logic proofs A brief review of . /Subtype/Type1 Inference Rules and Proofs for Predicate Logic Emina Torlak and Kevin Zatloukal 1. But with the approach of predicate logic, we can integrate the two levels of analysis, and say: 1. 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 Intro ∃: 1.2. 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 (2) A statement with variable has two parts: x is greater than 9 The first part, the … • We extend propositional logic with domains (sets of values), variables whose values range over these domains, and operations on values (e.g. Consider the following famous argument: All men are mortal. Existential quantifier states that the statements within its scope are true for some values of the specific variable. The empha- sis of this chapter is being put on an introduction of rules for proving in predicate logic. /Widths[300 500 800 755.2 800 750 300 400 400 500 750 300 350 300 500 500 500 500 These rules should be helpful for both checking the correctness of given proofs and for generating correct proofs on one’s own. It is denoted by the symbol $\exists $. /Flags 4 Eliminate all implications Þ 2. Predicate calculus, also called Logic Of Quantifiers, ... by the rules of the calculus. >> Eliminate Universal Quantifiers * 7. Eliminate Universal Quantifiers * 7. >> 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. Knowledge representation using predicate logic in artificial intelligence. 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 The smallest English sentence is formed by combining a verb with a subject. /LastChar 196 Ture notes on knowledge representation describes computational methods of these dierent types. In Predicate Logic, the smallest proposition is formed by combining a predicate with an individual. Cp. /Type/Encoding endobj Eliminate Existential Quantifiers * 6. 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? /Name/F4 << Subjects to be Learned. /Descent -200 /Subtype/Type1 But with the approach of predicate logic, we can integrate the two levels of analysis, and say: 1. /Type/Font Well-Formed Formula for First Order Predicate Logic --- Syntax Rules. We already use predicates routinely in programming, e.g. 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 Inference rules for propositional logic plus additional inference rules to handle variables and quantifiers. • There is often a choice of how to represent knowledge. 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)’ /ProcSet[/PDF/Text/ImageC] 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 The well-formed formulas of predicate logic are interpreted with respect to a domain of objects called universe of discourse, which we denote by “ D ”. The following are some examples of predicates. Predicate Logic 4. Consider the following two statements: Every SCE student must study discrete mathematics. •Knowledgeis a general term. 255/dieresis] 777.8 777.8 500 500 833.3 500 555.6 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 The type of logic that uses predicates is called predicate logic, or, when the emphasis is on manipulating and reasoning with predicates, predicate calculus. 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. The Interpretation Function This handout is a continuation of the previous handout and deals exclusively with the semantics of Predicate Logic. 1. /FontDescriptor 19 0 R Basically, propositional logic is limited to infer statements from general rules. It is denoted by the symbol $\forall$. 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 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. There are two types of quantifier in predicate logic − Universal Quantifier and Existential Quantifier. Knowledge representation issues predicate logic rules how do we represent what we know. Reduce the scope of all Ø to single term. The ex-ceptions to this rule are the names for binary relations in mathematics: for greater than, and so on. 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 with variables can be made a proposition by either authorizing a value to the variable or by quantifying the variable. Predicate Logic if inference rules are added to it. Proof Rules for Predicate Logic 2.1 Introduction Mathematical activity can be classified mainly as œprovingł, œsolvingł, or œsimplifyingł. << Predicate Logic and CNF • Converting to CNF is harder - we need to worry about variables and quantifiers. 9 0 obj 17 0 obj 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. /CapHeight 850 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 Equivalence Rules for Sentential Logic. 761.6 272 489.6] << in conditional statements of the form A predicate is an expression of one or more variables determined on some specific domain. 5 Predicate Logic - Derived Theorems Theorem 5.1 [Definition of ∃] (m≥ n) ⇒ ∃i : m�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�? KR using Logic – predicate logic, propositional logic, statements, variables, symbols, connective, truth value, contingencies, tautologies, contradictions, antecedent, consequent, argument, expressions, quantifiers, formula, representing “IsA” and “Instance” relationships. endobj stream Large amount of knowledge 2. 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 Semantic networks are alternative of predicate logic for knowledge representation. * 3. $\forall x P(x)$ is read as for every value of x, P(x) is true. A predicate is an expression of one or more variables determined on some specific domain. endstream To interpret a formula as a sentence (a statement or an open sentence) from the natural language, we need to interpret the … What is type inference in C++? << 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. << /Ascent 850 Consider, for example, the first-order formula "if a is a philosopher, then a is a scholar". /Name/F1 See also propositional calculus. This is part of the courseware on Artificial Intelligence, by R C Chakraborty, at JUET. As we have already mentioned, a predicate is just a function with a range of two values, say false and true. An answer to the question, "how to represent knowledge", requires an analysis to distinguish between knowledge “how” and knowledge “that”. /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 �R8�r��C(��L����VJ7Kh�'J����Ba5>����w�D�k@z��vݝ[����i�8�sHd��nC��a����O�i�C��R�n�^�ɼ��lC��]5�턨��G5�W� ��W�kaFu��z)�ڂ��1&⛝��))�I�]�~j _�w�}q�nX�(!�{�z=OQ���H�� However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. Predicate Logic deals with predicates, which are propositions containing variables. Predicate Logic deals with predicates, which are propositions, consist of variables. /BaseFont/JTTKIG+MSAM10 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 It is different from propositional logic which lacks quantifiers. The argument is valid if the premises imply the conclusion. 23 0 obj Consider the following two statements: Every SCE student must study discrete mathematics. >> >> (Bp . (x) [(Cx . Well-Formed Formula for First Order Predicate Logic --- Syntax Rules. /ItalicAngle 0 << Example 21. The type of logic that uses predicates is called predicate logic, or, when the emphasis is on manipulating and reasoning with predicates, predicate calculus. CSI2101 Discrete Structures Winter 2010: Predicate LogicLucia Moura. Notice carefully, that five of the rules are inference rules (upward-oriented rules), but one of them (universal derivation) is a show-rule (downward-oriented rule), much like conditional derivation. /LastChar 196 /F3 16 0 R /Type/Font 777.8 777.8 0 0 1000 1000 777.8 722.2 888.9 611.1 1000 1000 1000 1000 833.3 833.3 The law of variable substitution is an inference rule for use in proofs in predicate logic.. /Subtype/Type1 8 0 obj The standard in predicate logic is to write the predicate first, then the objects. /Type/Font 611.1 611.1 722.2 722.2 722.2 777.8 777.8 777.8 777.8 777.8 666.7 666.7 760.4 760.4 With sentential logic, you use the following equivalence rules to make those comparisons: Identity and Quantifier Rules for Quantifier Logic. Let us start with a motivating example. Issues, Predicate Logic, Rules How do we represent what we know ? The smallest English sentence is formed by combining a verb with a subject. /Name/F3 E.g., for the integers we add the set ℤ, –An interpretationis an assignment of specific values to domains and predicates. •Knowledgeis a general term. /Length1 714 The standard in predicate logic is to write the predicate first, then the objects. The most well-known FDA regulations are the GMP regulations. What’s new is moving from a strict universal statement (x), to a case of that statement. 10. If we use a quantifier that appears within the scope of another quantifier, it is called nested quantifier. 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 As we have already mentioned, a predicate is just a function with a range of two values, say falseand true. endobj >> /BaseFont/XZECJH+CMR12 /FontDescriptor 15 0 R 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 Quantifier logic encompasses the rules of sentential logic and expands upon them so that you can write whole statements with logic symbols. The following are some examples of predicates. Viele übersetzte Beispielsätze mit "predicate rules" – Deutsch-Englisch Wörterbuch und Suchmaschine für Millionen von Deutsch-Übersetzungen. Various restricted forms of the higher-order calculi have been shown, however, to be susceptible to routine decision procedures for all of their formulae. Predicate Logic and CNF • Converting to CNF is harder - we need to worry about variables and quantifiers. This chapter is dedicated to another type of logic, called predicate logic. /FontBBox[-34 -251 988 750] /FirstChar 33 Issues, Predicate Logic, Rules How do we represent what we know ? My thoughts: I am quite good at translating predicate logic expressions, but here I struggled to come up with formula for Horses' tails. /Filter[/FlateDecode] endobj Predicate logic builds heavily upon the ideas of proposition logic to provide a more powerful system for expression and reasoning. qt�����I�Kijgk�2���������������p kk��?��1����@�=����������3�8���U�/6y�)���߻��`k�����5��/ �$u��*A�M,@f`k'�?u���C���?��t�Ee���J��TCm���֬���;G�;H�����������W��������)�����5;����ߡ�|�s�bd� 1�q��xyx@ܜ,_�W��-��"-�daa�����j����u��W��y��6����1�g�Aa ?�0��tϓk��/(: Reduce the scope of all Ø to single term. /F4 20 0 R Predicate Logic - Definition. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 /FirstChar 33 (Bx v Ax)) > Px] / Pp. Predicate Logic - Definition. 25 0 obj 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. Would be welcomed to hear your ideas about this task. 10 0 obj 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 For example, when a theory defines the concept of a relation, a predicate simply becomes the … /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 The variable of predicates is quantified by quantifiers. >> /Differences[33/exclam/quotedblright/numbersign/dollar/percent/ampersand/quoteright/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash/zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon/exclamdown/equal/questiondown/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N/O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/quotedblleft/bracketright/circumflex/dotaccent/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m/n/o/p/q/r/s/t/u/v/w/x/y/z/endash/emdash/hungarumlaut/tilde/dieresis/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/Gamma/Delta/Theta/Lambda/Xi/Pi/Sigma/Upsilon/Phi/Psi 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 /Widths[1388.9 1000 1000 777.8 777.8 777.8 777.8 1111.1 666.7 666.7 777.8 777.8 777.8 << 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 The topics are : 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 Interpretations of Formulae in Predicate Logic – In propositional logic, an interpretation is simply an assignment of truth values to the atoms. The rules of identity are shown here: And, when talking about identities, you can quantify statements, using the rules in […] >> 450 500 300 300 450 250 800 550 500 500 450 412.5 400 325 525 450 650 450 475 400 Ap) 2. When you feel comfortable with the syntax of Predicate Logic, I urge you to read these notes carefully. 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. A predicate is a kind of incomplete proposition, which becomes a proposition when it is applied to some entity (or, as we’ll see later, to several entities). In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called a predicate on X. /BaseFont/VPJGFJ+CMMI12 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. 1. Techniques for solving heavily depend on the structure of the formulae under consideration and will be discussed in many special lectures on systems of linear equations, differential equations, or integral equations. ~� Eliminate Existential Quantifiers * 6. While first-order logic allows for the use of predicates, such as "is a philosopher" in this example, propositional logic does not. 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. /LastChar 196 Predicate Logic 10.1 Introduction Predicate logic builds heavily upon the ideas of proposition logic to provide a more powerful system for expression and reasoning. (2) /StemV 65 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 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�) �� All but the final proposition are called premises. 7 0 obj Eliminate all implications Þ 2. 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 /F1 10 0 R Artificial Intelligence – Knowledge Representation, Issues, Predicate Logic, Rules. Handout 5 – The Semantics of Predicate Logic LX 502 – Semantics I October 17, 2008 1. 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? A. Einstein In the previous chapter, we studied propositional logic. endobj 2.1.1 Proof Situations and Proofs [�]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� ��۝ͣ. Predicate rules are the requirements that can be found in 21 CFR Food and Drugs regulations. /LastChar 196 $\exists x P(x)$ is read as for some values of x, P(x) is true. Convert to conjunction of disjuncts 8. 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)’ ��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��� It is possible to use a similar approach for predicate logic (although, of course, there are no truth tables in predicate logic). See also propositional calculus. 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. Relationships between predicates can be stated using logical connectives. << Let us start with a motivating example. Those symbols come into play when you work with identities, or interchangeable constants. This chapter is dedicated to another type of logic, called predicate logic. What’s new is moving from a strict universal statement (x), to a case of that statement. ���#lu@��>h We'll illustrate this with an example. A predicate with variables can be made a proposition by either authorizing a value to the variable or by quantifying the variable. /FontName/XZECJH+CMR12 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 762 689.7 1200.9 https://www.tutorialspoint.com/.../discrete_mathematics_predicate_logic.htm /FontDescriptor 12 0 R /Name/F2 >> A predicate is a kind of incomplete proposition, which becomes a proposition when it is applied to some entity (or, as we’ll see later, to several entities). /FirstChar 33 >> /Length 9354 Thus, predicate logic employs six rules, in addition to all of the rules of sen-tential logic. 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 16 0 obj 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 /Type/FontDescriptor Predicate logic is superior to propositional logic in the sense that it is able to capture the structure of several arguments in a formal sense which propositional logic cannot. •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. Predicate calculus: area of logic dealing with predicates and quanti ers. $\forall\ a\: \exists b\: P (x, y)$ 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". 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 In this module, we will extend our previous system of natural deduction for propositional logic, to be able to deal with predicate logic. (Bx v Ax)) > Px] / Pp. 416.7 416.7 416.7 416.7 1111.1 1111.1 1000 1000 500 500 1000 777.8] 726.9 726.9 976.9 726.9 726.9 600 300 500 300 500 300 300 500 450 450 500 450 300 /Type/Encoding Move Quantifiers Left * 5. 300 325 500 500 500 500 500 814.8 450 525 700 700 500 863.4 963.4 750 250 500] 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|�. /Font 27 0 R /Subtype/Type1 /Filter[/FlateDecode] << peculiar to predicate logic, i.e., rules that do not arise in sentential logic. They are basically promulgated under the authority of the Food Drug and Cosmetic Act or under the authority of the Public Health Service Act. It consists eight hours of lectures. << • Obvious information may be necessary for reasoning • We may not know in advance which statements to deduce (P or P). The well-formed formulas of predicate logic are interpreted with respect to a domain of objects called universe of discourse, which we denote by “ D ”. Imagination will take you every-where." /Type/Font Predicate Logic Statements involving variables (e.g. /FirstChar 33 >> 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 6 0 obj /Subtype/Type1 endobj A quick look at predicate logic proofs Inference rules for quantifiers and a “hello” world example. >> /Name/F5 %PDF-1.2 /Length3 533 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. * 3. An answer to the question, "how to represent knowledge", requires an analysis to distinguish between knowledge “how” and knowledge “that”. With the propositional rules, the rules themselves were motivated by truth-tables and considered what was needed to 'picture' the truth of the formula being extended. /FontDescriptor 22 0 R endobj 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 275 500 777.8 777.8 777.8 Working with sentential logic means working with a language designed to express logical arguments with precision and clarity. 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. My initial idea was to consider similar sentence such as "w is a tail of a horse" to form required inference, but it was not successful. Make all variable names unique 4. >> 20 0 obj 13 0 obj << For example: x>9; x=y+9; x+y=z; Predicate Logic allows to make propositions from statements with variables. /LastChar 196 27 0 obj /FontFile 8 0 R 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 Move Quantifiers Left * 5. 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 Predicate logic, first-order logic or quantified logic is a formal language in which propositions are expressed in terms of predicates, variables and quantifiers. Using inference rules one can derive new formula using the existing ones. To interpret a formula as a sentence (a statement or an open sentence) from the natural language, we need to interpret the … • There is often a choice of how to represent knowledge. addition). 82 Using Predicate Logic • Many English sentences are ambiguous. � �oy�_�Rv��Ɉ� ����3 �m ���'�֐܅�m����#�:Y3��b�&C���kkJs�M,�����[Oū%�3�j]���)M���ru��=,�u&R� ���o���? 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 wff (well formed formula) atomic formula syntax of wff Contents Not all strings can represent propositions of the predicate logic. wff (well formed formula) atomic formula syntax of wff Contents Not all strings can represent propositions of the predicate logic. Predicate calculus, also called Logic Of Quantifiers, ... by the rules of the calculus. /BaseFont/RXUMZP+CMTI12 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) 2��8��!�P[ �?��m��@���M]���� Sentential Logic Operators, Input–Output Tables, and Implication Rules. 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 The main things we have to deal with are equality, and the two quantifiers (existential and universal). /Type/Font Substitution Rule. Imagination will take you every-where." endobj Example − "Man is mortal" can be transformed into the propositional form $\forall x P(x)$ where P(x) is the predicate which denotes x is mortal and the universe of discourse is all men. What is type inference in C++? Convert to conjunction of disjuncts 8. stream Such calculi are, in the precise sense, incomplete. In any logic system, you compare statements to prove or disprove their validity. Ap) 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. /F2 13 0 R 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 The ex-ceptions to this rule are the names for binary relations in mathematics: for greater than, and so on. Visit my website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW Hello, welcome to TheTrevTutor. Example 21. Make all variable names unique 4. /FontDescriptor 9 0 R 500 500 722.2 722.2 722.2 777.8 777.8 777.8 777.8 777.8 750 1000 1000 833.3 611.1 Predicate Logic allows to make propositions from statements with variables. Predicate Logic \Logic will get you from A to B. Natural deduction for predicate logic Readings: Section 2.3. An in-depth look at predicate logic proofs Understanding rules for quantifiers through more advanced examples. /Encoding 7 0 R /FirstChar 33 endobj (Bp . Cp. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] 255/dieresis] Various restricted forms of the higher-order calculi have been shown, however, to be susceptible to routine decision procedures for all of their formulae. Direct Proof Rule 1.1. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable. 82 Using Predicate Logic • Many English sentences are ambiguous. Such calculi are, in the precise sense, incomplete. /Length2 8798 The Predicate Logic Rules. << (x) [(Cx . – Predicate logic inference rules whole formulas only – Predicate logic equivalences (De Morgan’s) even on subformulas – Propositional logic inference rules whole formulas only – Propositional logic equivalences even on subformulas. •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. /Encoding 17 0 R Predicate Logic \Logic will get you from A to B. Universal quantifier states that the statements within its scope are true for every value of the specific variable. 500 1000 500 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G. Predicate Logic • In propositional logic, we assert truths about boolean values; in predicate logic, we assert truths about values from one or more “domains of discourse” like the integers. 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. /F5 23 0 R /Length 1188 1 The Language PLE Vocabulary The vocabulary of PLE consists in the following: 1. A predicate is an expression of one or more variables defined on some specific domain. • Obvious information may be necessary for reasoning • We may not know in advance which statements to deduce (P or P). 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] The general rule is for uniformity, and it takes getting used to. A. Einstein In the previous chapter, we studied propositional logic. Subjects to be Learned. x, y) are neither true nor false when the values of the variables are not specified. endobj /BaseFont/LZVMXX+CMSY10 In Predicate Logic, the smallest proposition is formed by combining a predicate with an individual. The general rule is for uniformity, and it takes getting used to.

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