H‰b```f``ÚÇÀÂÀÀúƒ�—x�b¬@Ì‘À€Lhq+°zøGøO‰R­�fwûñ….|=+1 –Eliminate all implications → –Reduce the scope of all ¬to single term –Make all variable names unique –Move quantifiers left (prenex normal form) –Eliminate … 1. A predicate P describes a relation or property. First order formulas are built up from ations: negation ¬, conjunction , disjunction , implication alence , existential quantifier , and ersal quantifier arentheses . Example ! p/ø¢3S0hi#÷»jF&C³�=g:ö¥(ªÎî÷¬.2š!Ã…‡˜g‹KDXZ‘å>O{W(èZ¤˜ØÖ„;l;” ¢Ÿ¥³®¬a¸¸»’àe™Áü¬&lªdÍèeñÇxÓ4\$ØØ…8.VˆäOéÔÁ ?#•'R~®œê�,…+�ĞX:Iò˜’L“ä •Ş‰yòv Using inference rules one can derive new formula using the existing ones. Example 21. Still have two truth values for statements (T and F) ! Consider the following two statements: Every SCE student must study discrete mathematics. 153 0 obj << /Linearized 1 /O 155 /H [ 702 540 ] /L 117456 /E 7532 /N 42 /T 114277 >> endobj xref 153 13 0000000016 00000 n 0000000611 00000 n 0000001242 00000 n 0000001400 00000 n 0000001562 00000 n 0000001669 00000 n 0000001772 00000 n 0000001880 00000 n 0000001921 00000 n 0000004159 00000 n 0000004299 00000 n 0000000702 00000 n 0000001220 00000 n trailer << /Size 166 /Info 136 0 R /Root 154 0 R /Prev 114266 /ID[<81cb04b88d9590646861629a461ef395><81cb04b88d9590646861629a461ef395>] >> startxref 0 %%EOF 154 0 obj << /Type /Catalog /Pages 138 0 R /JT 152 0 R /PageLabels 135 0 R >> endobj 164 0 obj << /S 525 /L 581 /Filter /FlateDecode /Length 165 0 R >> stream ! Predicate Logic if inference rules are added to it. (Details about tense and aspect can be ignored) (7)a.Charles is nice, but Elsa isn’t. We already use predicates routinely in programming, e.g. Predicate logic vs. set theory A Predicate is a declarative sentence whose true/false value depends on one or more variables. 3. %PDF-1.3 %âãÏÓ When most people say ‘logic’, they mean either propositional logic or ﬁrst-order predicate logic. Why • Reviewing/overviewing logic is necessary because we’ll be using it in the course. MATHEMATICAL LOGIC EXERCISES Chiara Ghidini and Luciano Seraﬁni Anno Accademico 2013-2014 We thank Annapaola Marconi for her work in previous editions of this booklet. Let Q(x,y) denote “x=y+3”. All animals like honey. Interpretations of Formulae in Predicate Logic – In propositional logic, an interpretation is simply an assignment of truth values to the atoms. A simple proposition is an unanalyzed whole which is either true or false. Imagination will take you every-where." 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. • Note: Although these lecture notes mirror the discussion from the supplement, they do not replace it. • Predicates and programs have meanings relative to states of memory, which give values to variables. Any ‘formal system’ can be considered a logic if it has: No animal likes honey. Statements in Predicate Logic P(x,y) ! 1. – In Predicate Logic, there are variables, so we have to do more than that. A. Einstein In the previous chapter, we studied propositional logic. ! Jackson is an SCE student. b.If peter didn’t hear the news from Charles, he heard it from Elsa. (Alternative names: predicate logic, rst order logic, elementary logic, restricted predicate calculus, restricted functional calculus, relational calculus, theory of quanti cation, theory of quanti cation with equality, etc.) 2. Worksheet 3: Predicate Logic Given the definitions: F the set of foods. o Väänänen: Predicate logic ulas Quantifiers are the final elements that first order (i.e. Lecture 13 Predicate Logic: Syntax Preable • Before going over these notes: Read carefully Section 6.3 from Chap-ter 6 in the textbook (’Predicate Logic’). Translating English into Predicate Logic Let the domain be the set of animals. The statement \x is greater than 3" has two parts: the subject: x is the subject of the statement the predicate: \is greater than 3" (a property that the subject can have). B. Outcomes At the end of this class, you should Practice in 1st-order predicate logic – with answers. R¢sÚj[ûbáÄ 1 What is a Logic? ôñ ŒİÊ\$­–gx õ/Nm÷>åè~. However, the precise deﬁnition is quite broad, and literally hundreds of logics have been studied by philosophers, computer scientists and mathematicians. This chapter is dedicated to another type of logic, called predicate logic. Variables (x,y) can take arbitrary values from some domain. Predicate Logic \Logic will get you from A to B. ∀is like AND Let x range over set {1, 2, 3, 4, 5, 6} P(x) ::= [x > 3] ∀x P(x) same as P(1) AND P(2) AND P(3) AND P(4) AND P(5) AND P(6) At least one animal likes honey. g Alice grows. Predicate logic is also used in rea-soning systems or “expert” systems, such as automatic medical diagnosis programs predicate logic) formulas are built up from. I. )ÛS¤³ÁYyu ñVJÙiŠ�áÖƒ2�¼’}¯ ühŸÅÀ\$!áĞÒVbhÈŞ²'¦HOzÍ^3e…SÌşï'Ì™àÄ!m1Û!ÕkZ…wˆ«³…KGkDEì‹v—²–v�é.R~J*�•Z¶yiZ²\\¦3ñ„Æ&w…G'9�L‰P\òRáœ¯§õ]‘! Propositional and Predicate logic \Logic will get you from a to B Einstein in the course but! From CPSC 121 at University of British Columbia do not replace it from... University of British Columbia we assign values to x and y, then P has a truth.. Quantifiers are the final elements that first order ( i.e already use predicates routinely programming... 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