# 9.1 Intro to Predicate Logic Practice with symbolizations. Today s Lecture 3/30/10

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1 9.1 Intro to Predicate Logic Practice with symbolizations Today s Lecture 3/30/10

2 Announcements Tests back today Homework: --Ex 9.1 pgs Part C (1-25)

3 Predicate Logic Consider the argument: All humans are mortal. Socrates is human. So, Socrates is mortal.

4 Predicate Logic Consider the argument: All humans are mortal. Socrates is human. So, Socrates is mortal. This argument is intuitively valid: if the premises are true, then the conclusion must be true.

5 Predicate Logic Consider the argument: All humans are mortal. Socrates is human. So, Socrates is mortal. This argument is intuitively valid: if the premises are true, then the conclusion must be true. The premises and the conclusion are atomic statements, so we could symbolize the first premise by replacing it with H; the second premise we could replace with S; and the conclusion we could replace with M. Thus we would have H S! M

6 Predicate Logic When we go to construct a truth table, however, the symbolic argument H S! M comes out invalid. The bottom line is that in order to translate the just mentioned argument (and others like it) in a way that allows us to demonstrate its validity, we need to expand upon (and in some cases adjust) our current symbolic language.

7 Predicate Logic Put differently, given the current state of our symbolic language, there are some English sentences whose meaning we cannot capture when symbolized. We need to make our symbolic language more sophisticated so that we can symbolize these English statements so as to retain their meaning. Once we do this, we'll be in a position to show (e.g. via proofs) that arguments that consist in such statements are valid.

8 Predicate Logic We can call this more sophisticated symbolic language predicate logic or quantification. For the English statements we ll be working with claim that things (oftentimes a certain number of things) have attributes or properties.

9 Predicate Logic We can call this more sophisticated symbolic language predicate logic or quantification. For the English statements we ll be working with claim that things (oftentimes a certain number of things) have attributes or properties.

10 Predicate Logic We can call this more sophisticated symbolic language predicate logic or quantification. For the English statements we ll be working with claim that things (oftentimes a certain number of things) have attributes or properties. We will now begin learning how to symbolize these kinds of English statements.

11 Symbolizing Consider the statement: 'Gizmo is a cat'. It claims that a particular thing (Gizmo) has a certain property (being a cat).

12 Symbolizing Consider the statement: 'Gizmo is a cat'. It claims that a particular thing (Gizmo) has a certain property (being a cat). We'll have the lowercase g stand for the particular thing (or individual) Gizmo; and we'll have the capital letter C stand for "is a cat". Thus the English statement can be symbolized as follows: Cg

13 Symbolizing We will still make use of our logical operators. Consider the statement: Gizmo and Mrs. Nibbles are Cats". If we have the lowercase letter n to stand for Mrs. Nibbles, and have the capital letter C stand for "is a cat", we can symbolize the above English statement as: Cg Cn

14 Symbolizing Capital letters A through Z will be used to designate properties (e.g. being a cat, being a human, being mortal, being silly, etc). We will call capital letters that stand for properties (i.e. replace predicates in English sentences), predicate letters.

15 Symbolizing Capital letters A through Z will be used to designate properties (e.g. being a cat, being a human, being mortal, being silly, etc). We will call capital letters that stand for properties (i.e. replace predicates in English sentences), predicate letters. [Note: we can still use capital letters as replacements for English atomic sentences if need be. Here we are saying they can also be used to stand for properties].

16 Symbolizing Lowercase letters a through u will be used to stand for particular individual things (e.g. Gizmo, Mrs. Nibbles, Barak Obama, Mount Everest). Put differently a through u will replace any proper nouns (names) found in English sentences. We ll call a through u individual constants

17 Symbolizing Oftentimes an English statement will claim that things in general have a property (or properties) without specifying the name of the things. We'll need to somehow capture this fact in our symbolic translation.

18 Symbolizing Oftentimes an English statement will claim that things in general have a property (or properties) without specifying the name of the things. We'll need to somehow capture this fact in our symbolic translation. We ll do this by utilizing the lower case letters v through z (called individual variables). So we ll use individual variables (e.g. x) to talk (symbolically) about a non-named thing (or things); and we ll say of the variable that it has the property in question.

19 Symbolizing Consider the statement: Everything is human (Hx: x is human). We can view the statement as saying that the predicate 'is human' applies to every individual thing. As a first approach to symbolizing, we can say any one of the following:!! Every x is such that x is human!! For all x, x is human!! For any individual thing x, x is human.

20 Symbolizing Everything is human (Hx: x is human). As a first approach to symbolizing, we can say any one of the following:!! Every x is such that x is human!! For all x, x is human!! For any individual thing x, x is human. Logicians have a symbol, called a universal quantifier, to stand for the phrases: 'Every x is such that', 'for all x', 'for any individual thing x': it is (x). So we have: (x), x is human.

21 Symbolizing Everything is human (Hx: x is human). Somewhat symbolically: Every x is such that x is human Logicians have a symbol to stand for phrases like: 'Every x is such that : the phrase is (x). So we have: (x), x is human. Given our scheme of abbreviation, we are ready to translate our statement in full: (x)hx

22 Symbolizing Univ. Affirmative Consider the Universal Affirmative (All S are P) statement: All humans are mortal (Hx: x is human; Mx: x is mortal). This can be viewed as saying: the predicate 'is mortal' applies to every individual thing that is human.

23 Symbolizing --Univ. Affirmative All humans are mortal (Hx: x is human; Mx: x is mortal). This can be viewed as saying: the predicate 'is mortal' applies to every individual thing that is human. Notice how we can't view this as saying every x is such that x is mortal: (x)mx That would be to view the statement as saying everything is mortal. But the statement says that everything that is human is mortal.

24 Symbolizing --Univ. Affirmative All humans are mortal (Hx: x is human; Mx: x is mortal). In order to capture the meaning of the English statement in symbols, we need to note the fact that the statement is equivalent to a certain conditional. (Note Well: every universal statement, affirmative or negative, will be symbolized using the ") We could view the statement as saying: every x is such that, if x is human, then x is mortal.

25 Symbolizing --Univ. Affirmative All humans are mortal (Hx: x is human; Mx: x is mortal). Every x is such that, if x is human, then x is mortal. Given our scheme of abbreviation, we are now ready to symbolize in full: (x)(x is human " x is mortal) (x)(hx " Mx)

26 Symbolizing --Univ. Affirmative Some stylistic variants of the Universal Affirmative, as illustrated by the statement: all humans are mortal. -- Every human is mortal -- Each human is mortal -- Humans are mortal -- A thing is human only if it is mortal * -- Only mortals are humans * All are (x)(hx " Mx)

27 Symbolizing --Univ. Affirmative A noteworthy stylistic variant of : all humans are mortal, is: A thing is human only if it is mortal. Don t let the A trip you up; the statement makes a claim about all things. Consider a slightly more concrete example: A dog is a mammal. This statement makes a claim about all dogs.

28 Symbolizing --Univ. Affirmative Another noteworthy stylistic variant of: All humans are mortal, is: Only mortals are humans. All is not synonymous with Only ; that s why the order of the terms mortal and human are reversed.

29 Symbolizing --Univ. Affirmative More on the term only (a slightly more concrete example). All dogs are mammals and Only dogs are mammals are not equivalent. The former is true. The latter however, is false. Surely there are other things besides dogs that are mammals. This shows that all and only are not synonymous.

30 Symbolizing --Univ. Affirmative Back to the (humans/mortal) example: a way to read and translate a statement with the term only as a Univ. Affirmative. Only mortals are humans could read as: Nothing besides mortals are humans. That is, if anything is not mortal, then it s not human. By contraposition we can say: if anything is human, then it is mortal. As we have seen (c.f. pg. 423), this latter conditional can be symbolized as: (x)(hx " Mx)

31 Symbolizing Univ. Negative Consider the Univ. Negative Statement (No S are P): No trees are animals (Tx: x is a tree; Ax: x is an animal) Because this is universal statement (all trees are such that they are not animals), it will be symbolized utilizing (x) and the ". As we will see, we will also utilize the ~ We can view the statement as saying: for all x, if x is a tree, then x is not an animal. Thus we have: (x) (Tx " ~Ax)

32 Symbolizing Particular Type Statements Consider: Something is mortal (Mx: x is mortal.) We can view this as saying: the predicate 'is mortal' applies to at least one thing (i.e. something). As a first approach at symbolizing, we can say:!! Some x is such that x is mortal!! There is an x such that x is mortal!! There exists an x such that x is mortal

33 Symbolizing Particular Type Statements Consider: Something is mortal (Mx: x is mortal.)!! Some x is such that x is mortal!! There is an x such that x is mortal!! There exists an x such that x is mortal Logicians have a symbol, called an existential quantifier, to stand for the phrases that end in such that : the symbol is (#x). Thus we have (#x), x is mortal.

34 Symbolizing Particular Type Statements Consider: Something is mortal (Mx: x is mortal.)!! Some x is such that x is mortal!! There is an x such that x is mortal!! There exists an x such that x is mortal Thus we have (#x) x is mortal. Given our scheme of abbreviation, we are ready to translate our English statement in full: (#x)mx

35 Symbolizing -- Particular Affirmative Consider the following Particular Affirmative statement (Some S are P): Some dogs are pugs (Dx: x is a dog; Px: x is a pug). This can be viewed as saying: The property of being a pug (the predicate 'is a pug') applies to at least one thing that is a dog. To be even more precise we can say: there's at least one thing that is both a dog and a pug.

36 Symbolizing -- Particular Affirmative Some dogs are pugs (Dx: x is a dog; Px: x is a pug). There's at least one thing that is both a dog and a pug. Thus as a first approach at symbolizing: There is an x such that, x is a dog and x is a pug With our scheme of abbreviation, we can symbolize as: (#x)(dx Px)

37 Symbolizing -- Particular Affirmative Some stylistic variants of Particular Affirmatives as illustrated by the statement: Some dogs are pugs (Dx: x is a dog; Px: x is a pug). --At least one dog is a pug -- There are dogs that are pugs --Something is both a dog and a pug --There exits a dog that is a pug All symbolized as: (#x)(dx Px)

38 Symbolizing -- Particular Affirmative Note that Particular Affirmatives should symbolized using the (#x) and the. See pg. 425 for an argument on why the is used and not the " when symbolizing Particular Affirmatives. (Basically since there s a possible state of affairs that makes the statement (#x)(dx " Px) true but doesn t make the other statement (#x)(dx Px) true, that shows that both statements don t mean the same thing).

39 Symbolizing Particular Negative Consider the Particular Negative statement (Some S are not P): Some dogs are not pugs (Dx: x is a dog; Px: x is a pug). We can view this as: there s at least one thing that is dog but is not a pug. A first approach at symbolizing will look like: There is at least one x such that x is a dog but x is not a pug

40 Symbolizing Particular Negative Some dogs are not pugs (Dx: x is a dog; Px: x is a pug). A first approach at symbolizing will look like: There is at least one x such that x is a dog but x is not a pug. Given our scheme of abbreviation: (#x)(dx ~Px)

41 Some Problems Ex 9.1 pgs Part C (1-25 All):

42 # 1 No Zoroastrians are Muslims

43 # 1 No Zoroastrians are Muslims (Universal Affirmative)

44 # 1 No Zoroastrians are Muslims (Universal Affirmative) Every x is such that if x is a Zoroastrian, then x is not a Muslim

45 # 1 No Zoroastrians are Muslims (Universal Affirmative) Every x is such that if x is a Zoroastrian, then x is not a Muslim (x)(zx " ~Mx)

46 # 2 All Kangaroos are Marsupials

47 # 2 All Kangaroos are Marsupials (Univ. Affirmative)

48 # 2 All Kangaroos are Marsupials (Univ. Affirmative) Every x is such that if x is a Kangaroo, then x is a Marsupial

49 # 2 All Kangaroos are Marsupials (Univ. Affirmative) Every x is such that if x is a Kangaroo, then x is a Marsupial (x)(kx " Mx)

50 #3 Peter Abelard is a logician, but Jacob Boehme is not

51 #3 Peter Abelard is a logician, but Jacob Boehme is not Only talking about particular individual things (proper (nouns) i.e. names are given), so need for variables in our symbolizations.

52 #3 Peter Abelard is a logician, but Jacob Boehme is not Only talking about particular individual things (proper (nouns) i.e. names are given), so need for variables in our symbolizations. [Don't let the x in the scheme trip you up. To be precise, we need the x beside the L to let us know that L is a predicate letter and not a statement letter (i.e. a letter standing for an entire atomic statement)]

53 #3 Peter Abelard is a logician, but Jacob Boehme is not Only talking about particular individual things (proper (nouns) i.e. names are given), so need for variables in our symbolizations. [Don't let the x in the scheme trip you up. To be precise, we need the x beside the L to let us know that L is a predicate letter and not a statement letter (i.e. a letter standing for an entire atomic statement)] La ~Lb

54 #4 Not every Marsupial is a Kangaroo

55 #4 Not every Marsupial is a Kangaroo (Negation of a Univ. Affirmative; can be a Particular Negative)

56 #4 Not every Marsupial is a Kangaroo (Negation of a Univ. Affirmative; can be a Particular Negative) It's not the case that every x is such that, if x is a Marsupial, then x is a kangaroo.

57 #4 Not every Marsupial is a Kangaroo (Negation of a Univ. Affirmative; can be a Particular Negative) It's not the case that every x is such that, if x is a Marsupial, then x is a kangaroo. ~(x)(mx " Kx) or (#x)(mx ~Kx)

58 #5 Nothing is right

59 #5 Nothing is right (Resembles a Univ. Negative)

60 #5 Nothing is right (Resembles a Univ. Negative) Everything is such that it is not right

61 #5 Nothing is right (Resembles a Univ. Negative) Everything is such that it is not right Every x is such that x is not right

62 #5 Nothing is right (Resembles a Univ. Negative) Everything (or all things) are such that they are not right Every x is such that x is not right (x)~rx

63 #6 Not everything is right

64 #6 Not everything is right It's not the case that all things are right.

65 #6 Not everything is right It's not the case that all things are right. It's not the case that every x is such that x is right

66 #6 Not everything is right It's not the case that all things are right. It's not the case that every x is such that x is right ~(x)rx or (#x)~rx

67 #7 Something is right

68 #7 Something is right There exists at least one thing that is right

69 #7 Something is right There exists at least one thing that is right There exists at least one x such that x is right

70 #7 Something is right There exists at least one thing that is right There exists at least one x such that x is right (#x)rx

71 #8 Something is not right There exists at least one thing that is not right There exists at least one x such that x is not right. (#x)~rx or ~(x)rx

72 #9 Only dogs are animals (Recall that while this is a Univ. Affirmative, the term 'only' is not synonymous with 'all'; thus this is not saying 'all dogs are animals'. Rather it's saying 'all animals are dogs ). (x)(ax " Dx)

73 #10 At least one mortal is human (Particular Affirmative) There exists at least one x such that x is mortal and x is human (#x)(mx Hx)

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