Today s Lecture 4/13/ WFFs/ Free and Bound Variables 9.3 Proofs for Pred. Logic (4 new implicational rules)!

Transcription

1 Today s Lecture 4/13/ WFFs/ Free and Bound Variables 9.3 Proofs for Pred. Logic (4 new implicational rules)!

2 Announcements Welcome Back! Answers to latest symbolizations HW are posted on-line Homework: --Ex 9.1 pg. 431 Part B (1-15 All) --Ex 9.3 pgs Part A (1-15 All).

3 Well Formed Formulas (WFFs) What counts as a well formed symbolic expression in predicate logic?

4 WFFs Any capital letter by itself (i.e. a statement letter representing an atomic statement) counts as a WFF.

5 WFFs Any capital letter by itself (i.e. a statement letter representing an atomic statement) counts as a WFF. Any capital letter coupled with either an individual constant or an individual variable (or both) counts as a WFF. Examples: Ca, Bx, Dxb.

6 WFFs Any capital letter by itself (i.e. a statement letter representing an atomic statement) counts as a WFF. Any capital letter coupled with either an individual constant or an individual variable (or both) counts as a WFF. Examples: Ca, Bx, Dxb. If a WFF is negated, the result is also a WFF. Examples: ~A, ~Bx, ~Ha

7 WFFs If two WFFs are connected by a,!, ", or a #, the result is also a WFF. Examples: (Fa Gb); (Hx # Kd); (Sy " Ry); (Ab! Bc).

8 WFFs If two WFFs are connected by a,!, ", or a #, the result is also a WFF. Examples: (Fa Gb); (Hx # Kd); (Sy " Ry); (Ab! Bc). If a WFF is connected with a (x) or an ($x), the result is also a WFF. Examples: (x)nx, ($z)ma, (x)(sx" Ra), and (x)(y)gxy.

9 WFFs If two WFFs are connected by a,!, ", or a #, the result is also a WFF. Examples: (Fa Gb); (Hx # Kd); (Sy " Ry); (Ab! Bc). If a WFF is connected with a (x) or an ($x), the result is also a WFF. Examples: (x)nx, ($z)ma, (x)(sx" Ra), and (x)(y)gxy. Nothing counts as a WFF unless it meets the aforementioned criteria.

10 WFFs For convenience, we ll continue to permit the omission of parentheses when no ambiguity in meaning results (in practice this means dropping the outermost parentheses). We ll also permit the use of brackets when doing do so can make for easier reading of a symbolic statement.

11 Scope of a Quantifier The scope of any quantifier is the shortest WFF immediately to the right of the quantifier.

12 Scope of a Quantifier The scope of any quantifier is the shortest WFF immediately to the right of the quantifier. Recall that (x) and ($x), where x can replaced by any variable, are the two quantifiers.

13 Scope of a Quantifier The scope of any quantifier is the shortest WFF immediately to the right of the quantifier. Recall that (x) and ($x), where x can replaced by any variable, are the two quantifiers. Examples: Consider: (x)(hx " Mx). Here the scope of the quantifier (x) is Hx " Mx

14 Scope of a Quantifier The scope of any quantifier is the shortest WFF immediately to the right of the quantifier. Recall that (x) and ($x), where x can replaced by any variable, are the two quantifiers. Examples: Consider: (x)(hx " Mx). Here the scope of the quantifier (x) is Hx " Mx In (y)hy " My, the scope of the quantifier (y) is Hy

15 Free and Bound Variables The occurrence of a variable x is bound if and only if x lies within the scope of an x-quantifier.

16 Free and Bound Variables The occurrence of a variable x is bound if and only if x lies within the scope of an x-quantifier. To Clarify: z quantifiers [e.g. (z) and ($z)] cannot bind occurrences of any other variable but the variable z.

17 Free and Bound Variables The occurrence of a variable x is bound if and only if x lies within the scope of an x-quantifier. To Clarify: z quantifiers [e.g. (z) and ($z)] cannot bind occurrences of any other variable but the variable z. y quantifiers [e.g. (y) and ($y)] cannot bind occurrences of any other variable but the variable y

18 Free and Bound Variables The occurrence of a variable x is bound if and only if x lies within the scope of an x-quantifier. To Clarify: z quantifiers [e.g. (z) and ($z)] cannot bind occurrences of any other variable but the variable z. y quantifiers [e.g. (y) and ($y)] cannot bind occurrences of any other variable but the variable y The same can be said of all the other variables. A variable is free if it s not bound

19 Exercises 9.1 pg. 431 Part B (1-15 all):

20 # s x in Hx is bound by (x); y in Gy is free. 2. x in Ax is bound by (x), y in By by (y), z is free 3. y in Cy bound by ($y) 4. both z s are bound by (z); the x is free. 5. x in Kx bound by ($x); x in Lx bound by (x) 6. x in Mx bound by (x); y in ~Ny bound by the (y). 7. x and y in Pxy are bound by (x) and (y) respectively. The x in Qx is free. 8. y in Sy bound by ($y)

21 # s all of the x s bound by (x) 10. x in Bx bound by (x); y in Dy bound by (y); z in Ez bound by (z) 11. x in Fxz bound by (x); z in Fxz bound by ($z); y in Fyz bound by ($y); the z in Fyz bound by (z). 12. z in Hz bound by (z); the y in Ky is free. 13. x in Lx bound by the first ($x); x in Nx bound by the second ($x) 14. x in Ox is free; x in Px bound by (x); y in Wy bound by (y) 15. z in Abz is free; y in By bound by (y); y in Cyx is free; x in Cyx bound by (x).

22 Implicational Rules for Predicate Logic Before examining the four implicational rules for predicate logic, we need to note that: --Our past implicational and equivalence rules still apply (e.g. MP, MT, DS, ME, MI, etc). We also need an understanding of: --What it is for a statement to be an instance of a quantified statement.

23 The Concept of an Instance What it is for a statement to be an instance of a quantified statement?

24 The Concept of an Instance What it is for a statement to be an instance of a quantified statement? The process of instantiation takes us from a quantified statement containing variables to a non-quantified statement that has at least some of its variables replaced by individual constants.

25 The Concept of an Instance What it is for a statement to be an instance of a quantified statement? The process of instantiation takes us from a quantified statement containing variables to a non-quantified statement that has at least some of its variables replaced by individual constants. Instantiation always takes us from individual variables to individual constants.

26 The Concept of an Instance The process of instantiation (the process whereby we arrive at an instance of a quantified statement) has two steps:

27 The Concept of an Instance The process of instantiation (the process whereby we arrive at an instance of a quantified statement) has two steps: Step 1: Remove the initial quantifier (the one that governs the entire statement), (x) or ($x).

28 The Concept of an Instance The process of instantiation (the process whereby we arrive at an instance of a quantified statement) has two steps: Step 1: Remove the initial quantifier (the one that governs the entire statement), (x) or ($x). Step 2: In the remaining WFF, uniformly replace all the free occurrences of the variable x (the variable that appeared in the initial quantifier) with an individual constant.

29 The Concept of an Instance Example: (x)[ax " ($y)(gy # Fx)]

30 The Concept of an Instance Example: (x)[ax " ($y)(gy # Fx)] Step 1: Ax " ($y)(gy # Fx)

31 The Concept of an Instance Example: (x)[ax " ($y)(gy # Fx)] Step 1: Ax " ($y)(gy # Fx) Step 2: Aa " ($y)(gy # Fa)

32 The Concept of an Instance Example: (x)[ax " ($y)(gy # Fx)] Step 1: Ax " ($y)(gy # Fx) Step 2: Aa " ($y)(gy # Fa) With the concept of an instance intact, we can move to our first implicational rule.

33 1 st Rule: Universal Instantiation (UI) This rule permits us to instantiate a universally quantified statement. This rule basically permits us to drop universal quantifiers and introduce individual constants to a proof.

34 1 st Rule: Universal Instantiation (UI) Consider the following universally quantified statement: (1) (x)(hx " Mx). This can be read as: Everything is such that if it is human then it is mortal. Well, Obama is a thing, so if Obama is human, then Obama is mortal. Assuming that o= Obama, we can infer the following statement from (1) via UI: Ho " Mo.

35 1 st Rule: Universal Instantiation (UI) (1) (x)(hx " Mx). This can be read as: Everything is such that if it is human then it is mortal. Gizmo is a thing, so if Gizmo is human, then Gizmo is mortal. Let g=gizmo. We can infer the following statement from (1) via UI: Hg " Mg

36 1 st Rule: Universal Instantiation (UI) (1) (x)(hx " Mx). This can be read as: Everything is such that if it is human then it is mortal. Gizmo is a thing, so if Gizmo is human, then Gizmo is mortal. Let g=gizmo. We can infer the following statement from (1) via UI: Hg " Mg UI allows us to move from any universally quantified statement to any instance of that statement.

37 1 st Rule: Universal Instantiation (UI) A brief summary of Universal Instantiation (UI): A line in a proof follows from some previous line via UI if and only if the previous line is a universally quantified statement and the line that follows it is an instance of the universally quantified statement. Memorize this!

38 On Incorrectly and Correctly Applying our Rules If one misapplies an implicational rule, it s highly likely that one could come up with substitution instance (English content) to uniformly replace the symbolized lines in the proof such that the inferred line is made false and the line from which it was inferred is made true (or one could make the substitution instance such that it s easily conceivable that the inferred line can be false while the relevant previous line can be true). To do this is to demonstrate that one hasn t shown that the inferred symbolized line in the proof must be true (i.e. impossible to be false) given the truth of the relevant previous line, which is just to say that one hasn t shown that the inferred line logically follows.

39 On Incorrectly and Correctly Applying our Rules Now when one correctly applies an implicational rule, no matter what substitution instance one comes up with, the inferred line will always come out true, on the assumption that the relevant previous line is true.

40 Some Incorrect Applications of UI 1. (y)(dy " My) 2. Dl " Me. 1, incorrect use of UI. Here the variable is not uniformly replaced, thus 2 is not an instance of 1.

41 Some Incorrect Applications of UI 1. (y)(dy " My) 2. Dl " Me. 1, incorrect use of UI. Here the variable is not uniformly replaced, thus 2 is not an instance of 1. In the above inference, 1 could say: everything is such that if it is a dog, then it s a mammal. Consequently, 2 could be read as: if Lassie is a dog, then Mt Everest is a mammal. If so, 2 is false, thus it doesn t follow from 1.

42 Some Incorrect Applications of UI 1. (y)(dy " My) 2. Dl " Me. 1, incorrect use of UI. Here the variable is not uniformly replaced, thus 2 is not an instance of 1. In the above inference, 1 could say: everything is such that if it is a dog, then it s a mammal. Consequently, 2 could be read as: if Lassie is a dog, then Mt Everest is a mammal. If so, 2 is false, thus it doesn t follow from 1. If the variable y is uniformly replaced with a constant, however, it doesn t matter what English name we replace the constant with -- nor does it matter what the predicate letters stand for -- 2 must be true given the assumption that 1 is true.

43 Some Incorrect Applications of UI 1. (y)(dy " My) 2. Dl " Me. 1, incorrect use of UI. In the above inference, for all we know, 1 says: everything is such that if it is a dog, then it s a mammal. Consequently, 2 could be read as: if Lassie is a dog, then Mt Everest is a mammal. If so, 2 is false, thus it doesn t follow from 1. If the variable y is uniformly replaced with a constant, however, it doesn t matter what English name we replace the constant with -- nor does it matter what the predicate letters stand for -- 2 must be true given the assumption that 1 is true.

44 Some Incorrect Applications of UI 1. ~(y)gy 2. ~Gh 1, incorrect use of UI 1 is not a universally quantified statement; it s rather a negation of a universally quantified statement. (UI is an implicational rule and as such cannot be applied to parts of lines).

45 Some Incorrect Applications of UI 1. ~(y)gy 2. ~Gh 1, incorrect use of UI 1 is not a universally quantified statement; it s rather a negation of a universally quantified statement. (UI is an implicational rule and as such cannot be applied to parts of lines). Let Gy = y is German and let h = Heidi Klum. The above inference could read: not everything is German (true), so Heidi Klum is not German (false). The latter doesn t follow from the former

46 Some Incorrect Applications of UI 1. (x)mx " (y)wy 2. Mg " (y)wy 1, incorrect application of UI 1 is not a universally quantified statement; it s rather a conditional. Say the inference reads: (1) if everything is a mammal, then everything is warm-blooded, so (2) if gizmo is a mammal, then everything is warm-blooded. (2) is false. We certainly haven t made a valid inference.

47 2 nd Rule: Existential Generalization (EG) This rule permits us to introduce existential quantifiers and replace individual constants with individual variables. We are able to move from an instance of an existentially quantified statement to an existentially quantified statement.

48 2 nd Rule: Existential Generalization (EG) Consider: (1) Fs. This can be read as: Seinfeld is funny. Well, if it s true that Seinfeld is funny, then it follows that something (or someone) is funny. The person Seinfeld is afterall a thing, in the broadest sense of that term. Thus we can infer the following statement from (1) via EG: ($x)fx (this says that something i.e. at least one thing is funny)

49 2 nd Rule: Existential Generalization (EG) A Brief Summary of Existential Generalization (EG): A line in a proof follows from some previous line via EG if and only if the line that follows is an existentially quantified statement, and the line from which it follows is an instance of it (the existentially quantified statement). Memorize this!

50 An Incorrect Application of EG 1. Ph " Ch 2. ($y)py " Ch 1, incorrect use of EG When we make an inference such as this, for all we know the line we have inferred is false on the assumption that the line from which the inference was made is true.

51 An Incorrect Application of EG 1. Ph " Ch 2. ($y)py " Ch 1, incorrect use of EG Let the above inference read: (1) if Hillary Clinton is the President of the U.S., then Hillary Clinton is the Commander and Chief; so (2) if something is the President of the U.S., then Hillary Clinton is the Commander and Chief. This latter conditional is false thus we can t say that it logically follows from 1.

52 3 rd Rule: Existential Instantiation (EI) Consider: All baseball players are athletes. At least one baseball player takes steroids. So at least one athlete take steroids. Intuitively this is clearly a valid argument: If the premises are true, the conclusion must be true.

53 3 rd Rule: Existential Instantiation (EI) All baseball players are athletes. At least one baseball player takes steroids. So at least one athlete take steroids. How can we show that this is a valid argument via a proof?

54 3 rd Rule: Existential Instantiation (EI) All baseball players are athletes. At least one baseball player takes steroids. So at least one athlete take steroids. How can we show that this is a valid argument via a proof? It turns out that the fact that we can make up a name to talk about and refer to the non-named baseball player mentioned in premise two is what enables us to show that this argument is valid.

55 3 rd Rule: Existential Instantiation (EI) All baseball players are athletes. At least one baseball player takes steroids. So at least one athlete take steroids. Given that we re assuming it s true that someone indeed satisfies the description of being a baseball player and a taker of steroids (we just don t know who it is), we re justified in coming up with a name to refer to this someone say John Doe ; just so long as all we re claiming about John Doe is that he is a baseball player that takes steroids.

56 3 rd Rule: Existential Instantiation (EI) All baseball players are athletes. At least one baseball player takes steroids. So at least one athlete take steroids. Importantly, we re not doing anything unreasonable in coming up with the name John Doe to refer to (talk about) this one baseball player who takes steroids.

57 3 rd Rule: Existential Instantiation (EI) All baseball players are athletes. At least one baseball player takes steroids. So at least one athlete take steroids. Importantly, we re not doing anything unreasonable in coming up with the name John Doe to refer to (talk about) this one baseball player who takes steroids. Equally important, it s useful for us to do so in order to show the relevant argument to be valid.

58 3 rd Rule: Existential Instantiation (EI) All baseball players are athletes. At least one baseball player takes steroids. So at least one athlete take steroids. Importantly, we re not doing anything unreasonable in coming up with the name John Doe to refer to (talk about) this one baseball player who takes steroids. Equally important, it s useful for us to do so in order to show the relevant argument to be valid. Consider the above argument symbolically and its proof:

59 3 rd Rule: Existential Instantiation (EI) 1. (x)(bx " Ax) 2. ($x)(bx Sx) % ($x)(ax Sx) 3. Bd Sd 2, EI 4. Bd 3, Simp 5. Sd 3, Simp 6. Bd " Ad 1, UI 7. Ad 6,4 MP 8. Ad Sd 7, 5 Conj 9. ($x)(ax Sx) 8, EG

60 First Restriction on EI Restriction 1: the constant we instantiate to via EI cannot be a constant that already appears in the proof. To see this, assume that the following two statements are the only lines in a proof:

61 First Restriction on EI 1. ($x)(bx Mx) There s at least one married baseball player 2. ($x)(bx ~Mx) There s at least one unmarried baseball player

62 First Restriction on EI 1. ($x)(bx Mx) There s at least one married baseball player 2. ($x)(bx ~Mx) There s at least one unmarried baseball player From 1 we can claim the following via EI: 3. Bd Md John Doe is a baseball player that is married (the married baseball player alluded to in 1 we re calling John Doe)

63 First Restriction on EI 1. ($x)(bx Mx) There s at least one married baseball player 2. ($x)(bx ~Mx) There s at least one unmarried baseball player From 1 we can claim the following via EI: 3. Bd Md John Doe is a baseball player that is married [We re just picking out one baseball player that is married, alluded to in 1 -- who it is exactly we don t know -- and calling him Jon Doe. Jon Doe now refers to the individual -- whoever that is -- who fits the description in 1].

64 First Restriction on EI 1. ($x)(bx Mx) There s at least one married baseball player 2. ($x)(bx ~Mx) There s at least one unmarried baseball player 3. Bd Md John Doe is a baseball player that is married

65 First Restriction on EI 1. ($x)(bx Mx) There s at least one married baseball player 2. ($x)(bx ~Mx) There s at least one unmarried baseball player 3. Bd Md John Doe is a baseball player that is married From 2 we cannot claim the following via EI: 4. Bd ~Md John Doe is a baseball player that is not married

66 First Restriction on EI 1. ($x)(bx Mx) There s at least one married baseball player 2. ($x)(bx ~Mx) There s at least one unmarried baseball player 3. Bd Md John Doe is a baseball player that is married From 2 we cannot claim the following via EI: 4. Bd ~Md John Doe is a baseball player that is not married To claim this is to claim that the individual that fulfills the description in 1 (the one we re calling John Doe) also fulfills the description in 2. But we can t assume that one and the same individual fulfills the description in both lines

67 First Restriction on EI 1. ($x)(bx Mx) There s at least one married baseball player 2. ($x)(bx ~Mx) There s at least one unmarried baseball player 3. Bd Md John Doe is a baseball player that is married 4. Bd ~Md John Doe is a baseball player that is not married

68 First Restriction on EI 1. ($x)(bx Mx) There s at least one married baseball player 2. ($x)(bx ~Mx) There s at least one unmarried baseball player 3. Bd Md John Doe is a baseball player that is married 4. Bd ~Md John Doe is a baseball player that is not married 4 basically says: the baseball player who is married (the one we re calling John Doe) is a baseball player who is not married.

69 First Restriction on EI 1. ($x)(bx Mx) There s at least one married baseball player 2. ($x)(bx ~Mx) There s at least one unmarried baseball player 3. Bd Md John Doe is a baseball player that is married 4. Bd ~Md John Doe is a baseball player that is not married 4 basically says: the baseball player who is married (the one we re calling John Doe) is a baseball player who is not married. When we make an inference such as this (where we instantiate to a constant that already appears in the proof), we can t be guaranteed that the inferred line is true. The inference from 1 to 4 is an example. 4 is false because we ve already established that John Doe is married.

70 First Restriction on EI Put differently, if we re permitted to existentially instantiate to a variable that is already in use, then the following argument is valid, as evidenced by the proof we could construct for it (see pg. 452). But this argument is invalid. Thus we re not permitted to instantiate in this way. 1.! ($x)(bx Mx) 2.! ($x)(bx ~Mx) % ($x)(mx ~Mx)

71 First Restriction on EI The upshot from all this is that if we want to apply EI to an existentially quantified line, we need to ensure that the constant used (the name picked out) is not already present in the proof; i.e. that the named picked out to refer to something is not already referring to something else.

72 Second Restriction on EI Restriction 2: the constant we instantiate to via EI cannot be a constant that is found in the conclusion of the argument to be proved.

73 Second Restriction on EI Consider the following: 1. ($x)(bx Sx) %Sa 2. Ba Sa 1, incorrect use of EI 3. Sa 2, Simp If we re permitted to instantiate to a constant that appears in the conclusion of the argument to be proved when applying EI, then the above argument is valid, as evidenced by this proof (see pg 452). But the argument is clearly invalid, thus we re not permitted to instantiate to such a constant.

74 Summary of EI A Brief Summary of EI: A line in a proof follows from a previous line via EI if and only if the previous line is an existentially quantified statement and the line that follows from it is an instance of that existentially quantified statement and the constant in the instance neither appears earlier in the proof nor in the conclusion to be proved. Memorize this!

75 4 th Rule: Universal Generalization (UG) Consider the following: All trees are plants. All plants are living things. So all trees are living things. In symbolic form: (x)(tx " Px) (x)(px " Lx) % (x)(tx " Lx)

76 4 th Rule: Universal Generalization (UG) In symbolic form: (x)(tx " Px) (x)(px " Lx) % (x)(tx " Lx) Intuitively, this is clearly valid. But how can we show that it s valid by means of a proof? What enables us to show this argument to be valid is the fact that we re permitted to make a certain universal generalization (a certain claim about all things) under certain conditions. UG permits us to infer a universally quantified formula from an instance of a universally quantified formula, given certain conditions.

77 4 th Rule: Universal Generalization (UG) Consider the proof for our symbolic argument: 1. (x)(tx " Px) 2. (x)(px " Lx) % (x)(tx " Lx) 3. Ta " Pa 1, UI 4. Pa " La 2, UI 5. Ta " La 3,4 HS 6. (x)(tx " Lx) 5, UG

78 4 th Rule: Universal Generalization (UG) Line 6 is our universal generalization about all things. What justifies it? We instantiated to the constant a in 3 and 4, but nothing in the proof required us to instantiate to a. Given what 1 and 2 are claiming, we could have instantiated to any individual constant in 3 and 4, which would have resulted in any constant in 5. Because we could say of any constant (not just a ) that if it were a tree, then it would be a living thing (line 5), we are permitted to claim that everything is such that if it is a tree, then it s a living thing (line 6).