Rules Inference for Quantifiers Modern Symbolic Logic
Week 9: Predicate Logic: Proofs Reading 8.2-8.3 A useful link: http://www.earlham.edu/~peters/ courses/log/transtip.htm#tip27
The 18 rules for propositional logic can all be used in predicate logic. But we need new rules to deal with the quantifiers. Four new rules allow us to take off quantifiers and put them back on.
Universal instantiation Universal generalization Existential instantiation Existential generalization
General strategy is to remove quantifiers with the instantiation rules, use the rules of propositional logic to get to our goal, then put the quantifiers back on with the generalization rules.
Universal Instantiation
If everything is F, it clearly follows that some particular thing is F. Universal instantiation allows us to represent this inference: 1. (x)Fx 2. Fa 1 UI Here, we have instantiated the universal statement with the instantial letter "a".
Universal Instantiation
Illustration: 1. (x)(CxDx) 2. ~Db 3. CbDb 4. ~Cb
/~Cb 1 UI 2,3 MT
Universal Instantiation over Variables
Universal instantiation also allows us to instantiate a universal statement with a variable: 1. (x)Fx 2. Fy 1 UI The expression "Fy" is a statement function it "stands in" for any of the particular statements "Fa", Fb", etc. Any variable x, y, or z could be used to instantiate the universal statement. Thus, any further expression we can derive using the variable "y" is fully general we could have derived it for any particular object.
This feature is utilized in Universal generalization.
Universal Generalization
Suppose we are given: 1. (x)(CxDx) 2. (x)~Dx Using Universal Instantiation twice, we can obtain: 3. CyDy 1 UI 4. ~Dy 2 UI 5. ~Cy 3,4 MT Our conclusion so far applies to whatever we might put in the place of the "y". So we are entitled to generalize our conclusion: 6. (x)~Cx 5 UG
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Universal Generalization
Universal generalization allows us to move from a statement function containing a free variable to a universal statement in which that variable is bound to a universal quantifier. 1. Fx 2. (x)Fx 1 UG All occurrences of the variable in the statement function must be replaced when generalizing. Universal generalization cannot be used to replace individual constants (a,b,c) with bound variables: 1. Fa 2. (x)Fx 1 UG not allowed
Existential Generalization
If I know that a is an F, then it is certainly true that there is something that is an F. This inference is an existential generalization: 1. Fa 2. (x)Fx 1 EG Any statement containing an individual constant can be existentially generalized. Any statement function containing a free variable may also be existentially generalized: 1. (x)Fx 2. Fy 1 UI 3. (x)Fx 2 EG
Existential Instantiation
1. 2. 3. Consider the following argument: All students are lazy. There is at least one student. Therefore, someone is lazy. To show the validity of this argument, I might reason as follows:
There is at least one student. Let's call him "Bob". All students are lazy, and Bob is a student, so Bob is lazy. Since Bob is lazy, it follows that someone is lazy.
Existential Instantiation
In the foregoing reasoning, we introduced a hypothetical name "Bob" to stand for a student asserted to exist in an existential claim. Existential instantiation allows a similar move in a formal proof: 1. (x)(SxLx) 2. (x)Sx / (x)Lx 3. Sb 2 EI This step allows us to complete the proof: 4. SbLb 1 UI 5. Lb 3,4 MP 6. (x)Lx 5 EG
Existential Instantiation
Great care must be taken when using existential instantiation:
Since the name introduced is a hypothetical (or existential) name, we are not entitled to use that name in our final We conclusion. cannot use any names already in use in our proof when using existential introduction. Otherwise, the following would be a valid argument: Bob is a student. Some students are lazy. Therefore Bob is lazy. For this reason, existential instantiation should generally be performed before universal instantiation in a proof.
Existential Instantiation
Illustration: 1. (x)(SxLx) 2. (x)Sx / (x)Lx Suppose we use UI first: 3. SbLb 1 UI Now we try to use EI: 4. Sb 2 EI But "b" has already been used in our proof, so this is not allowed!
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Summary of Quantifier Inference Rules
Universal Instantiation 1. (x)Fx 2. Fa 1 UI 3. Fy 1 UI
Restrictions: none
An Easy Error
The four quantifier rules, EG, UG, EI, and UI, are all inference rules. Therefore, they can only be used when the quantifier ranges over the entire sentence. Thus, the following step is invalid: 1. (x)PxQa 2. PbQa UI Since (x)PxQa is a conditional sentence, not a universal generalization, only the inference rules involving conditionals may be used on it.
Universal Generalization 1. Fx 2. (x)Fx UG
Restrictions: Cannot replace constants (a,b,c etc) All occurrences of variable letter must be replaced.
Existential Instantiation 1. (x)Fx 2. Fa 1 EI
Restrictions: Instantial letter "a" must not appear on any earlier line of proof.
Existential Generalization 1. Fa or Fy 2. (x)Fx 1 EG
Restrictions: none
Change of Quantifier Rules
If a quantified statement is negated (e.g. ~(x)Fx)), then we cannot apply the quantifier inference rules to it. Quantifier exchange rules allow us to move the negation signs inside the quantifier, to allow the use of quantifier inference rules. The change of quantifier rules work a lot like DeMorgan's rules for conjunctions and disjunctions.
Change of Quantifier Rules
Consider the statement "Not everything is a banana." For this statement to be true, it must be the case that there is something which is not a banana. The statements "Not everything is a banana" and "Something is not a banana" are equivalent. Thus, in symbolic terms: ~(x)Fx (x)~Fx (CQ)
Change of Quantifier Rules
Consider now the statement "Nothing is a banana." For this statement to be true, it cannot be ...