PK 1.5 & 1.6 Nested Quantifiers and Rules of Inference By: Emily B and David R What Are Nested ● They are quantifiers within the scope of another quantifier. Quantifiers?? ● ∀x∃y(x + y = 0) ⇔ ∀xQ(x), where Q(x) = ∃y(x + y = 0). ● They come, but are not limited, in the following flavors: ○ ∀x ∀yQ(x,y) ○ ∀x ∃yQ(x,y) ○ ∃x ∀yQ(x,y) ○ ∃x ∃yQ(x,y) ○ ∀x ∀y ∃zQ(x,y,z) ○ Yada yada yada Order Matters! ● In all cases unless the type of quantification are the same ○ ∀x ∀yQ(x,y) ⇔ ∀y ∀xQ(x,y) ○ ∃x ∃yQ(x,y) ⇔ ∃y ∃xQ(x,y) ● Other than that, the order of quantifiers will affect the truth of the quantification ○ ∀x ∃yQ(x,y) != ∃y ∀xQ(x,y) ● Can be thought of as loops with the left most quantifier being the outer loop until the rightmost inner loop. ○ For example, in ∀x∃y(x + y = 0), you would loop through ALL the elements in the domain for x, searching for AT LEAST one element for y that satisfies the statement. ○ In ∃x∀yQ(x,y), you would loop through the domain, testing Work? every x until you find ONE x that satisfies the statement for ALL y How Do They A neat observation: ● After going through some observations, the book concludes that: ○ if ∃y∀xP(x, y) is true, then ∀x∃yP(x, y) must also be true ○ However, if ∀x∃yP(x, y) is true, it is not necessary for ∃y∀xP(x, y) to be true Table on how to read these quantifications Turning Sentences into Logical Expressions ● “The sum of two positive integers is always positive” x & y ϵ Z. ○ ∀x∀y((x > 0) ∧ (y > 0) → (x + y > 0)) ● “Every real number except zero has a multiplicative inverse” x & y ϵ Z. ○ ∀x((x ≠ 0) → ∃y(xy = 1)) Negating Nested Quantifiers ● Nested Quantifiers can be negated by using the same De Morgan’s laws for negating statements involving a single quantifier. ○ ¬∀xP(x) → ∃x¬P(x) ○ ¬∃xP(x) → ∀x¬P(x) Example ● ∀x∃y(xy = 1) negated is: ○ ¬∀x∃y(xy = 1) ⇔ ∃x¬∃y(xy = 1) ∃x¬∃y(xy = 1) ⇔ ∃x∀y[¬(xy = 1)] ∃x∀y[¬(xy = 1)] ⇔ ∃x∀y(xy ≠ 1) Swap the quantifier types! Wrap-up of Nested Quantifiers ● They are essentially just sequential single quantifiers that can be read as if they were nested loops ● Order does matter for the quantifiers ● Negation rules are the same as single quantifiers, just that this one chains one after another. ● Arguments ○ Sequence of statements ● Valid Proofs! ○ When statements lead to a true conclusion ● Fallacies ○ Incorrect reasoning Valid arguments Valid Arguments in Propositional Logic ∴ Denotes “Therefore” ∴ is used in the Argument Form Example: ( (p → q) ∧ p ) → q p → q p ∴ q Rules of Inference in Propositional Logic Rules of Inference ● Used as building blocks ● Tautology is the basis Using the Rules to Build Arguments Premises- p→q, ᅟᅠᆨp→r, and r→s and conclusion q→s Steps Reason 1. p→q Premise 2. ᅟᅠᆨq→ᅟᅠᆨp Contrapositive 3. ᅟᅠᆨp→r Premise 4. ᅟᅠᆨq→r Hypothetical... 5. r→s Premise 6. ᅟᅠᆨq→ᅟᅠᆨs Hypothetical... Fallacies Fallacy of Affirming the Conclusion Fallacy of Denying the Hypothesis Example: Example: ((p→q) ∧ q ) → p ((p→q) ∧ ᅟᅠᆨp ) → ᅟᅠᆨq False when p = False & q = True False when p = False & q = True Watch out for these fallacies! Blurry Table... Quantified Statements Universal Instantiation Universal Generalization ● P(c) is true, where c is the ● ∀xP(c) is true, given the premise particular member of the domain, that P(c) is true for all elements c given the premise ∀xP(c) in the domain Even More Quantified Statements! Existential Instantiation Existential Generalization ● There’s an element c in the ● Used to conclude that ∃xP(x) is domain for which P(c) is true if we true when a particular element c know that ∃xP(x) is true with P(c) true is known How to Use Quantified Statements Take it step by step! Steps Reason Example: 1. ∀x( D(x)→C(x) ) Premise 2. D(y)→C(y) Universal Instantiation Premises-∀x( D(x)→C(x) ) and D(y) 3. D(y) Premise conclusion C(y) 4. C(y) Modus Ponens Let’s Recall... ● There are many rules to know! ○ 8 total rules in Rules of Inference ○ 4 total rules in Rules of Inference for Quantitative Statements ● These rules are quite useful when finding the solution ○ Use the premises to lead to the desired conclusion! ○ 1 step at a time!.