�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�� The Interpretation Function This handout is a continuation of the previous handout and deals exclusively with the semantics of Predicate Logic. /FontDescriptor 22 0 R << Basically, propositional logic is limited to infer statements from general rules. Predicate rules are the requirements that can be found in 21 CFR Food and Drugs regulations. 27 0 obj /FontBBox[-34 -251 988 750] Equivalence Rules for Sentential Logic. –An interpretation is an assignment of specific values to domains and predicates. << Predicate logic, first-order logic or quantified logic is a formal language in which propositions are expressed in terms of predicates, variables and quantifiers. /Subtype/Type1 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 /Type/Font Move Quantifiers Left * 5. 17 0 obj /BaseFont/JTTKIG+MSAM10 It is different from propositional logic which lacks quantifiers. For example, when a theory defines the concept of a relation, a predicate simply becomes the … 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. What is type inference in C++? 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? 25 0 obj 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 /Encoding 17 0 R ���#lu@��>h 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 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. /LastChar 196 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 ”. Eliminate Universal Quantifiers * 7. Predicate logic builds heavily upon the ideas of proposition logic to provide a more powerful system for expression and reasoning. Ap) 2. The variable of predicates is quantified by quantifiers. 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��/(: All but the final proposition are called premises. 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 >> endobj Intro ∃: 1.2. 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. Working with sentential logic means working with a language designed to express logical arguments with precision and clarity. Consider the following two statements: Every SCE student must study discrete mathematics. The following are some examples of predicates. endobj These rules should be helpful for both checking the correctness of given proofs and for generating correct proofs on one’s own. /FontFile 8 0 R >> 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. Predicate Logic and CNF • Converting to CNF is harder - we need to worry about variables and quantifiers. Using inference rules one can derive new formula using the existing ones. Quantifier logic encompasses the rules of sentential logic and expands upon them so that you can write whole statements with logic symbols. Predicate Logic deals with predicates, which are propositions, consist of variables. The ex-ceptions to this rule are the names for binary relations in mathematics: for greater than, and so on. Predicate Logic - Definition. /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 /FirstChar 33 Arguments in Propositional Logic A argument in propositional logic is a sequence of propositions. 82 Using Predicate Logic • Many English sentences are ambiguous. 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 20 0 obj endobj /LastChar 196 /Filter[/FlateDecode] The type of logic that uses predicates is called predicate logic, or, when the emphasis is on manipulating and reasoning with predicates, predicate calculus. Predicate Logic allows to make propositions from statements with variables. 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 While first-order logic allows for the use of predicates, such as "is a philosopher" in this example, propositional logic does not. 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 It consists eight hours of lectures. Predicate Logic if inference rules are added to it. Predicate Logic \Logic will get you from A to B. 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 Knowledge representation using predicate logic in artificial intelligence. (Bp . Issues, Predicate Logic, Rules How do we represent what we know ? •Knowledgeis a general term. /Subtype/Type1 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 Ture notes on knowledge representation describes computational methods of these dierent types. In mathematical logic, a predicate is commonly understood to be a Boolean-valued function P: X→ {true, false}, called a predicate on X. /FirstChar 33 << To interpret a formula as a sentence (a statement or an open sentence) from the natural language, we need to interpret the … /Subtype/Type1 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 The standard in predicate logic is to write the predicate first, then the objects. Predicate Logic and CNF • Converting to CNF is harder - we need to worry about variables and quantifiers. 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 following are some examples of predicates. Universal quantifier states that the statements within its scope are true for every value of the specific variable. addition). Existential quantifier states that the statements within its scope are true for some values of the specific variable. Interpretations of Formulae in Predicate Logic – In propositional logic, an interpretation is simply an assignment of truth values to the atoms. • Obvious information may be necessary for reasoning • We may not know in advance which statements to deduce (P or P). The type of logic that uses predicates is called predicate logic, or, when the emphasis is on manipulating and reasoning with predicates, predicate calculus. /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 endobj It is denoted by the symbol $\exists $. • There is often a choice of how to represent knowledge. >> 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. 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 As we have already mentioned, a predicate is just a function with a range of two values, say falseand true. /Subtype/Type1 •Knowledgeis a general term. Example 21. A predicate is an expression of one or more variables defined on some specific domain. >> Well-Formed Formula for First Order Predicate Logic --- Syntax Rules. The ex-ceptions to this rule are the names for binary relations in mathematics: for greater than, and so on. A. Einstein In the previous chapter, we studied propositional logic. 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. Predicate Logic deals with predicates, which are propositions, consist of variables. 23 0 obj /FirstChar 33 /CapHeight 850 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 /ProcSet[/PDF/Text/ImageC] Reduce the scope of all Ø to single term. My thoughts: I am quite good at translating predicate logic expressions, but here I struggled to come up with formula for Horses' tails. /Descent -200 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. /Flags 4 If we use a quantifier that appears within the scope of another quantifier, it is called nested quantifier. We already use predicates routinely in programming, e.g. Knowledge representation issues predicate logic rules how do we represent what we know. Imagination will take you every-where." /Ascent 850 The general rule is for uniformity, and it takes getting used to. Artificial Intelligence – Knowledge Representation, Issues, Predicate Logic, Rules. /BaseFont/XZECJH+CMR12 416.7 416.7 416.7 416.7 1111.1 1111.1 1000 1000 500 500 1000 777.8] Eliminate Existential Quantifiers * 6. The standard in predicate logic is to write the predicate first, then the objects. stream Relationships between predicates can be stated using logical connectives. in conditional statements of the form %PDF-1.2 It is denoted by the symbol $\forall$. Predicate calculus, also called Logic Of Quantifiers, ... by the rules of the calculus. 16 0 obj 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. << Consider the following two statements: Every SCE student must study discrete mathematics. 8 0 obj 726.9 726.9 976.9 726.9 726.9 600 300 500 300 500 300 300 500 450 450 500 450 300 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). Predicate Logic deals with predicates, which are propositions containing variables. It is possible to use a similar approach for predicate logic (although, of course, there are no truth tables in predicate logic). 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. peculiar to predicate logic, i.e., rules that do not arise in sentential logic. • We extend propositional logic with domains (sets of values), variables whose values range over these domains, and operations on values (e.g. CSI2101 Discrete Structures Winter 2010: Predicate LogicLucia Moura. 255/dieresis] 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 >> << 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. • There is often a choice of how to represent knowledge. 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. In any logic system, you compare statements to prove or disprove their validity. The topics are : /Type/Font Let us start with a motivating example. 6 0 obj wff (well formed formula) atomic formula syntax of wff Contents Not all strings can represent propositions of the predicate logic. /Name/F5 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 /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 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). Visit my website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW Hello, welcome to TheTrevTutor. << An answer to the question, "how to represent knowledge", requires an analysis to distinguish between knowledge “how” and knowledge “that”. 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 Move Quantifiers Left * 5. Predicate Logic Statements involving variables (e.g. wff (well formed formula) atomic formula syntax of wff Contents Not all strings can represent propositions of the predicate logic. /BaseFont/VPJGFJ+CMMI12 endobj /Type/Encoding /FontDescriptor 15 0 R Predicate Logic - Definition. Consider, for example, the first-order formula "if a is a philosopher, then a is a scholar". endobj 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. This is part of the courseware on Artificial Intelligence, by R C Chakraborty, at JUET. 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 Predicate Logic Rules. 5 Predicate Logic - Derived Theorems Theorem 5.1 [Definition of ∃] (m≥ n) ⇒ ∃i : m> 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
2020 predicate logic rules