Existential and Universal Quantifiers Unit 13 UNIT: 13 EXISTENTIAL AND UNIVERSAL QUANTIFIERS Unit Structure 13.1Learning objectives 13.2Introduction 13.2Predicate logic 13.3Symbols used in predicate logic 13.4Quantifiers and types of Quantifiers 13.5Difference between universal quantifier and existential quantifier. 13.6Symbolisation of categorical statements with quantifiers 13.7.Scope of quantifier 13.8Symbolisation 13.9Let us sum up 13.10Further Reading 13.11Answers to check your progress 13.12Model question 13.1 LEARNING OBJECTIVES After going through this unit, you will be able to: l define predicate logic, l explain the symbols used in predicate logic, l describe the distinction between universal quantifier and existential quantifier, l discuss the statements symbolised in predicate logic. 13.2 INTRODUCTION The three distinct branches of symbolic logic are propositional logic, predicate logic and logic of classes. Predicate logic deals with the internal structure of both the simple and compound propositions. It is the logic of predicates or properties, and things or objects to which the predicates may be ascribed. Now question arises, how many of the individuals have the property in question? In predicate logic, the expressions which state that, how many of the individuals have the property in question are known as Philosophy 47 PDF created with pdfFactory Pro trial version www.pdffactory.com Unit 13 Existential and Universal Quantifiers quantifiers. This logic uses only two kinds of quantifier symbols: Universal quantifier and Existential quantifier. Universal quantifier stands for "for all x" or "for any x" . The symbol used for universal quantifier is -(x). The existential quantifier stands for 'there is at least one x' and it has existential commitment. The symbol used for existential quantifier is -( $ x). These quantifiers give us the way to symbolise quantity terms such as 'all', 'some', 'every', 'none' which occur in statements about predications and thereby bring out the formal structure of general proposition. 13.3 PREDICATE LOGIC Predicate logic is that branch of symbolic logic which deals with the internal structure of both the simple and compound propositions. It is concerned with predicates or properties, and things or objects to which the predicates may be ascribed. Predicate logic on the other hand deals with the sub-structures of both simple and compound propositions. In more formal terms, it is also known Quantification Theory or the Predicate calculus. Predicate logic contains the entire propositional logic. In predicate logic, a distinction between singular proposition and general proposition is made. A singular proposition is one which states that a particular individual possesses or does not possess a particular property or attribute. 'Kalidasa is a poet'. On the other hand a general proposition asserts that some individuals or all individuals possess or do not possess a certain property. For example: 'All men are wise', 'some flowers are red', etc. 13.4 SYMBOLS USED IN PREDICATE LOGIC Symbols play a pivotal role in predicate logic. A symbol is an ideogram which represents logical concepts. Predicate logic contains entire propositional logic. So, it uses same symbols adopted in propositional logic. For example, logical constants and propositional variable are used both in propositional logic and predicate logic. In addition to it predicate logic uses a kind of symbol to signify the quantity term occur in statements that is called quantifier. Now, it is necessary to deal with the symbols used in propositional logic. Propositional logic uses two types of symbols namely, (1) logical constant or logical operator and (2) propositional variable. Logical constants: Logical constants are those words which are used to connect simple propositions to produce compound propositions and which express the form or structure of the compound propositions. They connect statements. Hence, they are also called the connectives. Propositional logic 48 Philosophy PDF created with pdfFactory Pro trial version www.pdffactory.com Existential and Universal Quantifiers Unit 13 uses five basic logical constants or connectives. They are- not, and, either- or, if-then, if and only if. These logical constants are represented by their respective symbols. The logical constants and their respective symbols are the following: Symbol Name Meaning ~ (Tilde) Not, It is not the case that . (Dot) and Ú (Wedge) or É (Horseshoe) if-then º (triple bar) if and only if Propositional variables: The lower case letters of the English alphabet p, q, r, s, t etc. which stand for simple propositions are known as propositional variables. For example: If p then q This argument can be symbolically represented in the following way: P É q P \ q Here, p and q are variables because they can be replaced by any simple proposition within a definite range. In addition to these symbols, predicate logic uses a kind of symbol to signify the quantity term occurring in statements what is called quantifier. Predicate logic uses only two kinds of quantifiers: Universal quantifier and Existential quantifier. 13.5 QUANTIFIERS AND TYPES OF QUANTIFIERS Quantifiers are expressions that state how many of the individuals or objects have the property in question. They do not state which one of the individuals have the property. Prof. Chhanda Chakraborti states that a quantifier consists of: l A left parenthesis or '(' l A quantifier symbol l One of the individual variable symbol Philosophy 49 PDF created with pdfFactory Pro trial version www.pdffactory.com Unit 13 Existential and Universal Quantifiers l A right parenthesis or ')' In predicate logic, we have two quantifier symbols: l The inverted 'A' or V l And the inverted 'E' or ' $ ' The quantifiers provide us a way to symbolise quantity terms such as 'all', 'some', 'every', 'none' which occur in the propositions. Predicate logic uses two types of quantifiers: Universal Quantifier and Existential Quantifier. Universal quantifier: The expression "given any x", or "for all x" or "for any x" or "for every x" is known as universal quantifier. Universal quantifier is used to assert that all entities have a certain property or properties. Thus it yields a universal proposition when prefixed to a propositional function. (We have already discussed about propositional function in the chapter: 11) The symbol used for universal quantifier is: '(Vx)'. The symbol '(Vx)' consists of all the components of a quantifier. First it is enclosed within a pair of parentheses. Secondly, it is expressed with an individual variable 'x'. Thirdly, it stands for 'for all x', or 'for any x', or 'for every x', or 'given any x'. But conventionally the symbols '(x)', '(y)', '(z)' are used for this purpose. Hence, universal quantifier is used to symbolise the statements containing the words 'all', 'every', 'each', 'any', and their equivalent expressions. Let us symbolise the proposition 'Everything is movable' in predicate logic step by step. The proposition can be paraphrased as: In the first step, we express it as: For all x, x has the property of being temporary or for all x, x is temporary or 'given any x, x is temporary'. In the second step, we use a predicate symbol to symbolise the property 'being movable'. Here, we can use 'Mx' to symbolise the property 'being movable'. In the third step, the expression 'for all x', which is called universal quantifier is expressed by the quantifier symbol '(x)'. Thus the general proposition 'everything is movable' can be symbolically expressed in predicate logic as follows: (x) Mx Existential quantifier: The expression "there is at least one x" is known as existential quantifier. It is used to assert that some entities or at least one 50 Philosophy PDF created with pdfFactory Pro trial version www.pdffactory.com Existential and Universal Quantifiers Unit 13 entity have a given property. Thus it yields an existential proposition when prefixed to a propositional function. The symbol '?' stands for particular quantity terms such as 'some', 'a few', etc. and their equivalent terms in English. As a quantifier, it is used as '(?x). The symbol '(?x)' stands for the expression "there is at least one x" which is known as the existential quantifier. Let us symbolise the proposition 'Something is solid' by as using the existential quantifier. The proposition may be paraphrased as -There is at least one x such that x is solid. Using the same process as shown above we may symbolise the statement as follows: ( $ x) Sx Similarly, 'Some things are not solid' may be rephrased as 'There is at least one x such that x is not temporary. We put negation sign (~) in the predication part and symbolise the proposition as follows: ( $ x) ~ Sx We have already discussed in the preceding chapter (11.15) that general propositions are different from singular propositions because general propositions contain words signifying quantity. So quantifiers are used to bring out the formal structure of general proposition. CHECK YOUR PROGRESS Q 1: What is quantifier? .................................................................................................................... .................................................................................................................... Q 2: What are the types of quantifier? ....................................................................................................................................................................................................................................