Please help asap.
This reading introduces you to basic ideas about thequantifiers. The are two basic facts about the quantifiers you needto understand and from which all of the logical properties of thequantifiers follow:
Basic Fact 1: A universal quantifier (x) Fx istruth-functionally equivalent to an infinite conjunction: Fa& Fb & Fc & Fd & ........
where a, b, c, d, are the names of objects in theuniverse picked out by the 'x' in the universal quantifier(x)
Basic Fact 2: An existential quantifier istruth-functionally equivalent to an infinite disjunction Fa v Fb v Fc v Fd v ......
where a, b, c, d ... are the names of objects in theuniverse picked out by the 'x' in the existential quantifier(Ex)
The 'x' in (x) and in (Ex) is not a free or boundvariable. It (i) names the universe over which the quantifierranges and (ii) names the bound variablesused in the matrix to its right. In (x) Fx, the term Fx is thematrix and x in Fx is the bound variable. If a variable in a matrixis the same variable that occurs in the quantifier to the left ofthe matrix and that quantifier has the matrix within its scope,then the variable in the matrix is bound (by thequantifier).
Consider (x) (Fx v Gy). The matrix is Fx v Gy. Thematrix is within the scope of the quantifier. x (in the matrix) isbound by that quantifier. y in the matrix is free--it is not boundby that quantifier, since the quantifier is (x). If the quantifierwere (y), then y would be bound by it.
Consider the wff (x) Fx ⊃ Gz. The matrix isFx. It is not Fx ⊃ Gz. Why? The universal quantifieronly has Fx within its scope. Why? By convention, where there areno parentheses to the right of a quantifier (that are not part ofanother quantifier), the scope of that quantifier is the first termto its right and excludes truth-functions. That is why ⊃ Gz is not within the scope of (x), but Fx is within its scope. Thesame is true for (Ex) Fx ⊃ Gx.
What is the difference between (Ex) Fx and Fa? Theformer has an existential quantifer and its matrix is Fx. It saysthat there exists some object in the universe that has the propertyF. In the latter, a is the name of some object in the universe. Ithas the property F. The name a is not a variable. The free/bounddistinction only applies to variables. It does not apply to names.Names are also called individual constants.
Notice that Fa ⊃ (Ex) Fx, but(Ex)Fx ⊅ Fa. Why? Look again at the basic fact aboutexistential quantifiers, such as (Ex)Fx. It is truth-functionallyequivalent to an infinite disjunction Fa v Fb v Fc .... Youknow only that one (or more) objects in the universe have F, butnot which ones have it. On the other hand, if you know Fa is true,then it must be true that something in the universe has theproperty F. Hence Fa ⊃ (Ex)Fx. But (Ex)Fx ⊅ Fais also true, since if all we know is that (Ex) Fx is true, we donot thereby know that Fa is true. (But notice that ((Ex) Fx &Fa) ⊃ Fa is true, by the inference rulesimplification.)
- Subject Intro To Logic Please Help Asap This Reading Introduces You To Basic Ideas About The Quantifiers The Are Two 1 (110.53 KiB) Viewed 20 times
- Subject Intro To Logic Please Help Asap This Reading Introduces You To Basic Ideas About The Quantifiers The Are Two 2 (92.44 KiB) Viewed 20 times
2. For the following wffs, indicate which variables are free and which are bound (you can use 'F' for free and 'B' for bound.) Make sure you draw a vertical line underneath each variable with the letters 'F' or 'B' at the bottom of each vertical line. Make sure you know what is and what is not a variable. Not every symbol in the language of predicate logic is a variable. The reading (Scope, binding, expansions) lists the symbols that are variables and the symbols that are names (i.e., individual constants). Names are not variables. You can write the wffs with the vertical lines on a piece of paper and submit a photo of it. Instead of using vertical lines and F/B, you can color the bound variables red and the free variables green. Example: (x) ((Fx & Gx) Kxyw) (a) (x) (y) (z) ((Fxy Gxy) v (Hxa Hzbu)) (b) (Ex) (Ey) Hxxy (z) (Gz v Hx) (c) (z) (Ex) (y) (Axayzw v Bxycuvz)