Logic, educational
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Predicate Logic Quantification --- Forming Propositions from Predicates Subjects to be Learned universe universal quantifier existential quantifier free variable bound variable scope of quantifier order of quantifiers Contents A predicate with variables is not a proposition. For example, the statement x > 1 with variable x over the universe of real numbers is neither true nor false since we don't know what x is. It can be true or false depending on the value of x. For x > 1 to be a proposition either we substitute a specific number for x or change it to something like "There is a number x for which x > 1 holds", or "For every number x, x > 1 holds". More generally, a predicate with variables (called an atomic formula ) can be made a proposition by applying one of the following two operations to each of its variables: 1. assign a value to the variable 2. quantify the variable using a quantifier (see below). For example, x > 1 becomes 3 > 1 if 3 is assigned to x, and it becomes a true statement, hence a proposition. In general, a quantification is performed on formulas of predicate logic (called wff ), such as x > 1 or P( x) , by using quantifiers on variables. There are two types of quantifiers: universal quantifier and existential quantifier. The universal quantifier turns, for example, the statement x > 1 to "for every object x in the universe, x > 1", which is expressed as " x x > 1". This new statement is true or false in the universe of discourse. Hence it is a proposition once the universe is specified. Similarly the existential quantifier turns, for example, the statement x > 1 to "for some object x in the universe, x > 1", which is expressed as " x x > 1." Again, it is true or false in the universe of discourse, and hence it is a proposition once the universe is specified. Universe of Discourse The universe of discourse, also called universe, is the set of objects of interest. The propositions in the predicate logic are statements on objects of a universe. The universe is thus the domain of the (individual) variables. It can be the set of real numbers, the set of integers, the set of all cars on a parking lot, the set of all students in a classroom etc. The universe is often left implicit in practice. But it should be obvious from the context. The Universal Quantifier The expression: x P(x) , denotes the universal quantification of the atomic formula P(x). Translated into the English language, the expression is understood as: " For all x, P(x) holds", " for each x, P(x) holds" or " for every x, P(x) holds". is called the universal quantifier , and x means all the objects x in the universe. If this is followed by P(x) then the meaning is that P(x) is true for every object x in the universe. For example, "All cars have wheels" could be transformed into the propositional form, x P(x) , where: P(x) is the predicate denoting: x has wheels, and the universe of discourse is only populated by cars. Universal Quantifier and Connective AND If all the elements in the universe of discourse can be listed then the universal quantification x P(x) is equivalent to the conjunction: P(x 1 ) ) P(x 2 ) P(x 3 ) ... P(x n ) . For example, in the above example of x P(x) , if we knew that there were only 4 cars in our universe of discourse (c1, c2, c3 and c4) then we could also translate the statement as: P(c1) P(c2) P(c3) P(c4) converted by Web2PDFConvert.com
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Transcript of Logic, educational
Predicate Logic
Quantification --- Forming Propositions from Predicates
Subjects to be Learned
universeuniversal quantifierexistential quantifierfree variablebound variablescope of quantifierorder of quantifiers
Contents
A predicate with variables is not a proposition. For example, the statement x > 1 with variable x over the universe of real numbers is neither true nor false sincewe don't know what x is. It can be true or false depending on the value of x. For x > 1 to be a proposition either we substitute a specific number for x or change it to something like "There is a number x for which x > 1 holds", or "Forevery number x, x > 1 holds".
More generally, a predicate with variables (called an atomic formula) can be made a proposition by applying one of the following two operations to each of itsvariables:
1. assign a value to the variable2. quantify the variable using a quantifier (see below).
For example, x > 1 becomes 3 > 1 if 3 is assigned to x, and it becomes a true statement, hence a proposition.
In general, a quantification is performed on formulas of predicate logic (called wff ), such as x > 1 or P(x), by using quantifiers on variables. There are two typesof quantifiers: universal quantifier and existential quantifier.
The universal quantifier turns, for example, the statement x > 1 to "for every object x in the universe, x > 1", which is expressed as " x x > 1". This newstatement is true or false in the universe of discourse. Hence it is a proposition once the universe is specified.
Similarly the existential quantifier turns, for example, the statement x > 1 to "for some object x in the universe, x > 1", which is expressed as " x x > 1."Again, it is true or false in the universe of discourse, and hence it is a proposition once the universe is specified.
Universe of Discourse
The universe of discourse, also called universe, is the set of objects of interest. The propositions in the predicate logic are statements on objects of a universe.The universe is thus the domain of the (individual) variables. It can be the set of real numbers, the set of integers, the set of all cars on a parking lot, the set of allstudents in a classroom etc. The universe is often left implicit in practice. But it should be obvious from the context.
The Universal Quantifier
The expression: x P(x), denotes the universal quantification of the atomic formula P(x). Translated into the English language, the expression is understood as:"For all x, P(x) holds", "for each x, P(x) holds" or "for every x, P(x) holds". is called the universal quantifier, and x means all the objects x in theuniverse. If this is followed by P(x) then the meaning is that P(x) is true for every object x in the universe. For example, "All cars have wheels" could betransformed into the propositional form, x P(x), where:
P(x) is the predicate denoting: x has wheels, andthe universe of discourse is only populated by cars.
Universal Quantifier and Connective AND If all the elements in the universe of discourse can be listed then the universal quantification x P(x) is equivalent to the conjunction: P(x1)) P(x2) P(x3) ... P(xn) .
For example, in the above example of x P(x), if we knew that there were only 4 cars in our universe of discourse (c1, c2, c3 and c4) then we could alsotranslate the statement as: P(c1) P(c2) P(c3) P(c4)
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Stephan
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MIT ch9 logic1b, said that a predicate with 0 terms is just a propsitional variable. See Apontamentos and also later in MIT slides that said when apply all terms, it becomes a proposition. However, another way to make the predicate with variables into a proposition besides assigning a value to variables is to quantify the variable using a quantifier.
Stephan
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Two examples with x >1 would be 1. assing a value so that x = 3 to get 3 > 1, which is a proposition and we can assign a truth value to it (like MIT ch9 1a on Interpretation I as assigning a truth value). Or 2. we can say For all x, x > 1, which becomes a proposition (becomes an affirmation), and we can assign an interpretation to it in terms of true or false (see MIT ch91a on this).
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Hence, notice that we always read the variable before reading the predicate, for all x, or for each x, or for every x. Then we say "P(x)" holds. The only difference are the words for all, vs for each, vs for every.
Stephan
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See he restricted U to cars instead of saying Car(x), but in this special case he only had one predicate P(x), whereas if he had 2 predicates, and they all have to have the same U (see ch 91b MIT logic), this may be not possible.
The Existential Quantifier
The expression: xP(x), denotes the existential quantification of P(x). Translated into the English language, the expression could also be understood as:"There exists an x such that P(x)" or "There is at least one x such that P(x)" is called the existential quantifier, and x means at least one object x in theuniverse. If this is followed by P(x) then the meaning is that P(x) is true for at least one object x of the universe. For example, "Someone loves you" could betransformed into the propositional form, x P(x), where:
P(x) is the predicate meaning: x loves you,The universe of discourse contains (but is not limited to) all living creatures.
Existential Quantifier and Connective OR If all the elements in the universe of discourse can be listed, then the existential quantification xP(x) is equivalent to the disjunction: P(x1) P(x2) P(x3) ...
P(xn).
For example, in the above example of x P(x), if we knew that there were only 5 living creatures in our universe of discourse (say: me, he, she, rex and fluff),then we could also write the statement as: P(me) P(he) P(she) P(rex) P(fluff)
An appearance of a variable in a wff is said to be bound if either a specific value is assigned to it or it is quantified. If an appearance of a variable is not bound, itis called free. The extent of the application(effect) of a quantifier, called the scope of the quantifier, is indicated by square brackets [ ]. If there are no squarebrackets, then the scope is understood to be the smallest wff following the quantification. For example, in x P(x, y), the variable x is bound while y is free. In x [ y P(x, y) Q(x, y) ] , x and the y in P(x, y) are bound, while y in Q(x, y) isfree, because the scope of y is P(x, y). The scope of x is [ y P(x, y) Q(x, y) ] .
How to read quantified formulas
When reading quantified formulas in English, read them from left to right. x can be read as "for every object x in the universe the following holds" and xcan be read as "there erxists an object x in the universe which satisfies the following" or "for some object x in the universe the following holds". Those do notnecessarily give us good English expressions. But they are where we can start. Get the correct reading first then polish your English without changing the truthvalues.
For example, let the universe be the set of airplanes and let F(x, y) denote "x flies faster than y". Then x y F(x, y) can be translated initially as "For every airplane x the following holds: x is faster than every (any) airplane y". In simpler English it means "Every
airplane is faster than every airplane (including itself !)". x y F(x, y) can be read initially as "For every airplane x the following holds: for some airplane y, x is faster than y". In simpler English it means "Every airplane
is faster than some airplane". x y F(x, y) represents "There exist an airplane x which satisfies the following: (or such that) for every airplane y, x is faster than y". In simpler English it says
"There is an airplane which is faster than every airplane" or "Some airplane is faster than every airplane". x y F(x, y) reads "For some airplane x there exists an airplane y such that x is faster than y", which means "Some airplane is faster than some airplane".
Order of Application of Quantifiers
When more than one variables are quantified in a wff such as y x P( x, y ), they are applied from the inside, that is, the one closest to the atomic formula isapplied first. Thus y x P( x, y ) reads y [ x P( x, y ) ] , and we say "there exists an y such that for every x, P( x, y ) holds" or "for some y, P( x, y ) holdsfor every x".
The positions of the same type of quantifiers can be switched without affecting the truth value as long as there are no quantifiers of the other type between theones to be interchanged. For example x y z P(x, y , z) is equivalent to y x z P(x, y , z), z y x P(x, y , z), etc. It is the same for the universal quantifier.
However, the positions of different types of quantifiers can not be switched. For example x y P( x, y ) is not equivalent to y x P( x, y ). For let P( x, y ) represent x < y for the set of numbers as the universe, for example. Then x y P( x, y ) reads "for every number x, there is a number y that is greater than x", which is true, while y x P( x, y ) reads "there is a number that is greaterthan every (any) number", which is not true.
Test Your Understanding of Quantification
Indicate which of the following statements are correct and which are not.
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Hence it seems that for finite sets U, the existential quantifier can be replaced by logical OR connectives. It could be seen as existential elimination, but is actually existential replacement with OR connectives.
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Good definition of bound variables. If either 1. or 2. holds on p 1/3 of this pdf. That is, if either value assigned to it or if it is quantified, the same things that make a predicate with terms a proposition.
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Great example, you would think that the existential quantifier would extend to Q(x,y) but it doesn't because the scope is the >smallest< wff, not the largest, that follows the quantification. Hence this means that in general, putting parentheses around wff's for quantifiers is in order to extend the scope of the quantifiers (make them larger).
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Notice that here it is saying it's faster than itself, since x and y could take on the same object in the universe of discourse U. What we want is to say x != y, but I'm not sure how we could do this in predicate logic??
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Click Yes or No , then Submit. There are two sets of questions.
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