predicate logic rules

Subjects to be Learned. –An interpretationis an assignment of specific values to domains and predicates. ?�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�? >> /Name/F4 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 /LastChar 196 It is different from propositional logic which lacks quantifiers. Consider the following two statements: Every SCE student must study discrete mathematics. The type of logic that uses predicates is called predicate logic, or, when the emphasis is on manipulating and reasoning with predicates, predicate calculus. << Convert to conjunction of disjuncts 8. A statement with variable has two parts: x is greater than 9 The first part, the … Consider, for example, the first-order formula "if a is a philosopher, then a is a scholar". /Subtype/Type1 The last statement is the conclusion. If we use a quantifier that appears within the scope of another quantifier, it is called nested quantifier. /FontDescriptor 19 0 R The ex-ceptions to this rule are the names for binary relations in mathematics: for greater than, and so on. endstream /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 /Type/Encoding /Encoding 7 0 R Such calculi are, in the precise sense, incomplete. endobj Assumption 1.2 () Elim∀: 1.1 1.3. >> Issues, Predicate Logic, Rules How do we represent what we know ? •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. /StemV 65 Reduce the scope of all Ø to single term. 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. It is denoted by the symbol $\exists $. 255/dieresis] To interpret a formula as a sentence (a statement or an open sentence) from the natural language, we need to interpret the … /Filter[/FlateDecode] – In Predicate Logic, there are variables, so we have to do more than that. 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. Predicate Logic 4. Inference Rules and Proofs for Predicate Logic Emina Torlak and Kevin Zatloukal 1. What’s new is moving from a strict universal statement (x), to a case of that statement. • Obvious information may be necessary for reasoning • We may not know in advance which statements to deduce (P or P). /Length2 8798 /Name/F3 17 0 obj 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. Predicate calculus, also called Logic Of Quantifiers, ... by the rules of the calculus. Make all variable names unique 4. Eliminate Existential Quantifiers * 6. /FirstChar 33 /Subtype/Type1 • We extend propositional logic with domains (sets of values), variables whose values range over these domains, and operations on values (e.g. They are basically promulgated under the authority of the Food Drug and Cosmetic Act or under the authority of the Public Health Service Act. Consider the following two statements: Every SCE student must study discrete mathematics. For example, when a theory defines the concept of a relation, a predicate simply becomes the … 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 /LastChar 196 Thus, predicate logic employs six rules, in addition to all of the rules of sen-tential logic. 1. 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] /Type/FontDescriptor /FirstChar 33 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. /Widths[300 500 800 755.2 800 750 300 400 400 500 750 300 350 300 500 500 500 500 10 0 obj Quantifier logic encompasses the rules of sentential logic and expands upon them so that you can write whole statements with logic symbols. 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. /F5 23 0 R 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. Last Class: Predicate Logic Proof Prove ∀x P(x)→ ∃x P(x) 1. 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. To interpret a formula as a sentence (a statement or an open sentence) from the natural language, we need to interpret the … 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. The empha- sis of this chapter is being put on an introduction of rules for proving in predicate logic. /Descent -200 in conditional statements of the form 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. 20 0 obj Lecture 07 2. With sentential logic, you use the following equivalence rules to make those comparisons: Identity and Quantifier Rules for Quantifier Logic. 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. • There is often a choice of how to represent knowledge. /FontBBox[-34 -251 988 750] 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. 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 /ItalicAngle 0 777.8 777.8 777.8 777.8 777.8 777.8 1333.3 1333.3 500 500 946.7 902.2 666.7 777.8 82 Using Predicate Logic • Many English sentences are ambiguous. As we have already mentioned, a predicate is just a function with a range of two values, say falseand true. Example 21. 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. – 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. 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 The smallest English sentence is formed by combining a verb with a subject. >> 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 << The rules of identity are shown here: And, when talking about identities, you can quantify statements, using the rules in […] wff (well formed formula) atomic formula syntax of wff Contents Not all strings can represent propositions of the predicate logic. 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 $\exists x P(x)$ is read as for some values of x, P(x) is true. /FirstChar 33 173/Omega/ff/fi/fl/ffi/ffl/dotlessi/dotlessj/grave/acute/caron/breve/macron/ring/cedilla/germandbls/ae/oe/oslash/AE/OE/Oslash/suppress/dieresis Eliminate all implications Þ 2. See also propositional calculus. In Predicate Logic, the smallest proposition is formed by combining a predicate with an individual. * 3. endobj 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 (2) /LastChar 196 Subjects to be Learned. Predicate Logic deals with predicates, which are propositions, consist of variables. stream An in-depth look at predicate logic proofs Understanding rules for quantifiers through more advanced examples. >> 450 500 300 300 450 250 800 550 500 500 450 412.5 400 325 525 450 650 450 475 400 �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�� /BaseFont/JTTKIG+MSAM10 /F4 20 0 R 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 Would be welcomed to hear your ideas about this task. Basically, propositional logic is limited to infer statements from general rules. This is part of the courseware on Artificial Intelligence, by R C Chakraborty, at JUET. 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 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 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. Predicate Logic and CNF • Converting to CNF is harder - we need to worry about variables and quantifiers. Predicate Logic - Definition. stream 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 The Interpretation Function This handout is a continuation of the previous handout and deals exclusively with the semantics of Predicate Logic. Example 21. 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. (Bp . wff (well formed formula) atomic formula syntax of wff Contents Not all strings can represent propositions of the predicate logic. 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. /CapHeight 850 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)’ 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 Equivalence Rules for Sentential Logic. An answer to the question, "how to represent knowledge", requires an analysis to distinguish between knowledge “how” and knowledge “that”. endobj https://www.tutorialspoint.com/.../discrete_mathematics_predicate_logic.htm /Length1 714 /BaseFont/RXUMZP+CMTI12 The variable of predicates is quantified by quantifiers. 9 0 obj ��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�� /Subtype/Type1 << Sentential Logic Operators, Input–Output Tables, and Implication Rules. 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) 16 0 obj Issues, Predicate Logic, Rules How do we represent what we know ? %PDF-1.2 –An interpretation is an assignment of specific values to domains and predicates. << Universal quantifier states that the statements within its scope are true for every value of the specific variable. We'll illustrate this with an example. addition). For example: x>9; x=y+9; x+y=z; Predicate Logic allows to make propositions from statements with variables. peculiar to predicate logic, i.e., rules that do not arise in sentential logic. Knowledge representation using predicate logic in artificial intelligence. 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)’ A. Einstein In the previous chapter, we studied propositional logic. In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called a predicate on X. Predicate Logic if inference rules are added to it. 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. CSI2101 Discrete Structures Winter 2010: Predicate LogicLucia Moura. 1. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 /BaseFont/LZVMXX+CMSY10 The smallest English sentence is formed by combining a verb with a subject. /Font 27 0 R 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. A predicate with variables can be made a proposition by either authorizing a value to the variable or by quantifying the variable. /Ascent 850 • Obvious information may be necessary for reasoning • We may not know in advance which statements to deduce (P or P). Eliminate all implications Þ 2. Substitution Rule. 761.6 272 489.6] /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 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. Such calculi are, in the precise sense, incomplete. We already use predicates routinely in programming, e.g. It is possible to use a similar approach for predicate logic (although, of course, there are no truth tables in predicate logic). >> Let us start with a motivating example. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] << 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 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? endobj /BaseFont/VPJGFJ+CMMI12 Natural deduction for predicate logic Readings: Section 2.3. 2.1.1 Proof Situations and Proofs E.g., for the integers we add the set ℤ, Knowledge representation issues predicate logic rules how do we represent what we know. (x) [(Cx . • There is often a choice of how to represent knowledge. Topics Propositional logic proofs A brief review of . A predicate is an expression of one or more variables determined on some specific domain. 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 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. 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 ”. 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 5 Predicate Logic - Derived Theorems Theorem 5.1 [Deﬁnition of ∃] (m≥ n) ⇒ ∃i : m> Handout 5 – The Semantics of Predicate Logic LX 502 – Semantics I October 17, 2008 1. When you feel comfortable with the syntax of Predicate Logic, I urge you to read these notes carefully. Cp. Cp. /Encoding 17 0 R /Length 1188 1 The Language PLE Vocabulary The vocabulary of PLE consists in the following: 1. Let us start with a motivating example. 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 The main things we have to deal with are equality, and the two quantiﬁers (existential and universal). endobj 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 25 0 obj Make all variable names unique 4. >> * 3. My thoughts: I am quite good at translating predicate logic expressions, but here I struggled to come up with formula for Horses' tails. What’s new is moving from a strict universal statement (x), to a case of that statement. Predicate Logic allows to make propositions from statements with variables. Predicate Logic deals with predicates, which are propositions containing variables. 27 0 obj << << Eliminate Universal Quantifiers * 7. Predicate calculus, also called Logic Of Quantifiers, ... by the rules of the calculus. Predicate logic, first-order logic or quantified logic is a formal language in which propositions are expressed in terms of predicates, variables and quantifiers. 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|�. There are two types of quantifier in predicate logic − Universal Quantifier and Existential Quantifier. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable. 726.9 726.9 976.9 726.9 726.9 600 300 500 300 500 300 300 500 450 450 500 450 300 /Type/Font Imagination will take you every-where." 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 Various restricted forms of the higher-order calculi have been shown, however, to be susceptible to routine decision procedures for all of their formulae. >> A predicate is an expression of one or more variables determined on some specific domain. 7 0 obj /F2 13 0 R The ex-ceptions to this rule are the names for binary relations in mathematics: for greater than, and so on. See also propositional calculus. /FontDescriptor 22 0 R Inference rules for propositional logic plus additional inference rules to handle variables and quantifiers. 255/dieresis] /ProcSet[/PDF/Text/ImageC] The following are some examples of predicates. 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 ”. endobj 416.7 416.7 416.7 416.7 1111.1 1111.1 1000 1000 500 500 1000 777.8] Various restricted forms of the higher-order calculi have been shown, however, to be susceptible to routine decision procedures for all of their formulae. 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 /F3 16 0 R Well-Formed Formula for First Order Predicate Logic --- Syntax Rules. Interpretations of Formulae in Predicate Logic – In propositional logic, an interpretation is simply an assignment of truth values to the atoms. Intro ∃: 1.2. /F1 10 0 R Predicate Logic and CNF • Converting to CNF is harder - we need to worry about variables and quantifiers. � �oy�_�Rv��Ɉ� ��3 �m ��'�֐܅�m��#�:Y3��b�&C��kkJs�M,��[Oū%�3�j]��)M��ru��=,�u&R� ��o��? But with the approach of predicate logic, we can integrate the two levels of analysis, and say: 1. 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. /Length 9354 10. Move Quantifiers Left * 5. 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. Those symbols come into play when you work with identities, or interchangeable constants. /FontDescriptor 15 0 R Imagination will take you every-where." Consider the following famous argument: All men are mortal. The argument is valid if the premises imply the conclusion. Large amount of knowledge 2. Ture notes on knowledge representation describes computational methods of these dierent types. $\forall\ a\: \exists b\: P (x, y)$ 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. This chapter is dedicated to another type of logic, called predicate logic. 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