Homework. Turn in Homework #4. The fifth homework assignment has been posted on the course website. It is due on Monday, March 2.
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1 Homework Turn in Homework #4. The fifth homework assignment has been posted on the course website. It is due on Monday, March 2.
2 First-Order Logic Formation Rules and Translations
3 First Order Logic
4 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,...
5 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,...
6 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Constants
7 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Constants
8 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Constants Variables
9 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,...
10 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Predicates: G, H, L,...
11 Well-Formed Formulas First-order logic (sometimes called predicate logic) adds three basic elements to our zerothorder formal language: Terms: a, b, c, and x, y, z,... Predicates: G, H, L,... Quantifiers: and
12 Well-Formed Formulas We add two formation rules to the rules we already had for zeroth-order logic. As before, the grammatically correct strings are called well-formed formulas.
13 Well-Formed Formulas New Formation Rules: 3. If Φ is a predicate and α 1, α 2,, α n are terms, then Φα 1 α 2 α n is a well-formed formula. 4. Let ϕ be a well-formed formula, and let β be a variable. Then ( β)ϕ and ( β)ϕ are both well-formed formulas.
14 Well-Formed Formulas Positive Examples: Ga (Ga Hbc) (~Lcxy ᴧ Gb) ( x)gx ( x)( y)hxy (( x)gx P)
15 Well-Formed Formulas Positive Examples: Ga (Ga Hbc) (~Lcxy ᴧ Gb) ( x)gx ( x)( y)hxy (( x)gx P) Negative Examples: GaH alx ( a) (( x) ᴧ Gx) ( B)Bc ( P)(P Ga)
16 Translations Predicates let us represent sentences like: Kermit is green. Kermit is a Muppet. Kermit plays the banjo.
17 First-Order Logic If P is a predicate symbol and a is a term, then Pa is a well-formed formula. The basic idea is to think of a simple sentence as composed of a logical subject a constant term, like Kermit and a logical predicate like is green.
18 First-Order Logic Predicates may be interpreted in many different ways. A predicate is a kind of truth function. assigns a property to an individual. indicates membership in a kind.
19 First-Order Logic Kermit is green. Constant Predicate
20 First-Order Logic Kermit is green. TRUE Constant Predicate
21 First-Order Logic Gonzo is green. Constant Predicate
22 First-Order Logic Gonzo is green. FALSE Constant Predicate
23 Quantifiers Add quantifiers and we can also represent sentences like: All Muppets are silly. Some Muppets wear hats.
24 Quantifiers Suppose ( β)ϕ and ( β)ϕ are well-formed formulas. Then we know that β is a variable term. Occurrences of β in ϕ are bound by the quantifiers. The variable β is also bound in the quantifier expressions ( β) and ( β).
25 Quantifiers If a variable in some formula is not bound, then it is free. If a well-formed formula contains no free variables, then it is called a sentence.
26 Quantifiers If a variable in some formula is not bound, then it is free. If a well-formed formula contains no free variables, then it is called a sentence. P (Q v R) Mxy Ga ( x)gx ( x)(mxy ᴧ Gy)
27 Quantifiers If a variable in some formula is not bound, then it is free. If a well-formed formula contains no free variables, then it is called a sentence. P (Q v R) Mxy Ga ( x)gx ( x)(mxy ᴧ Gy) Which of these are sentences?
28 Quantifiers Previously, we had no need to distinguish between well-formed formulas and sentences because every well-formed formula stood for a sentence in ordinary language. But a well-formed formula that contains a free variable does not stand for any sentence in ordinary language.
29 Quantifiers What do the quantifiers do though? Quantifiers let us talk about all of the constant terms without needing to actually list them all. That is convenient, since there are infinitely many constant terms.
30 Quantifiers What do the quantifiers do though? Quantifiers let us talk about all of the constant terms without needing to actually list them all. That is convenient, since there are infinitely many constant terms. We would get tired trying to write them all down!
31 Quantifiers Quantified formulas tell us something about all of the constant terms. The formula ( β)ϕ says that for every constant c, if you substitute c for β everywhere β appears in ϕ, then ϕ will come out true.
32 Quantifiers Let G be the predicate is green. Then the formula ( x)gx says that everything is green. The formula ( x)gx says that for every constant c, if you substitute c for x in Gx, then Gx will come out true.
33 Quantifiers Let G be the predicate is green. Then the formula ( x)gx says that everything is green. In other words, ( x)gx says that for every constant c, Gc is true. The universal quantifier works like an infinite conjunction.
34 Quantifiers Again, quantified formulas tell us something about all of the constant terms. The formula ( β)ϕ says that there is at least one constant c such that if you substitute c for β everywhere it appears in ϕ, then ϕ will be true.
35 Quantifiers Let H be the predicate is happy. Then the formula ( x)hx says that something is happy. The formula ( x)hx says that there is a constant c such that Hc true.
36 Quantifiers Let H be the predicate is happy. Then the formula ( x)hx says that something is happy. The formula ( x)hx says that there is a constant c such that Hc true. The existential quantifier works like an infinite disjunction.
37 Categoricals We ve seen that if G is the predicate is green, then the formula ( x)gx says that everything is green. What if we want to say that every green thing is colorful?
38 Categoricals We ve seen that if G is the predicate is green, then the formula ( x)gx says that everything is green. What if we want to say that every green thing is colorful? That would be a categorical sentence!
39 Categoricals There are four kinds of categorical sentence: All Ss are Ps. Some Ss are Ps. No Ss are Ps. Some Ss are not Ps. All Muppets are silly. Some Muppets wear hats. No Muppets wear hats. Some Muppets are not silly.
40 Categoricals There are four kinds of categorical sentence: All Ss are Ps. Some Ss are Ps. No Ss are Ps. Some Ss are not Ps. All Muppets are silly. Some Muppets wear hats. No Muppets wear hats. Some Muppets are not silly.
41 Categoricals There are four kinds of categorical sentence: All Ss are Ps. Some Ss are Ps. No Ss are Ps. Some Ss are not Ps. All Muppets are silly. Some Muppets wear hats. No Muppets wear hats. Some Muppets are not silly.
42 Categoricals Let s translate the categorical sentence, All Muppets are silly. Start with the predicates: M = is a Muppet S = is silly
43 Categoricals Suppose b stands for Beaker.
44 Categoricals Suppose b stands for Beaker.
45 Categoricals Suppose b stands for Beaker. Then
46 Categoricals Suppose b stands for Beaker. Then Mb = Beaker is a Muppet. Sb = Beaker is silly.
47 Categoricals Suppose b stands for Beaker. Then Mb = Beaker is a Muppet. Sb = Beaker is silly. (Mb ᴧ Sb) = Beaker is a Muppet and Beaker is silly.
48 Categoricals But we want to say that every Muppet is silly.
49 Categoricals But we want to say that every Muppet is silly.
50 Categoricals But we want to say that every Muppet is silly.
51 Categoricals But we want to say that every Muppet is silly.
52 Categoricals But we want to say that every Muppet is silly.
53 Categoricals But we want to say that every Muppet is silly. How do we make sure to cover all of the Muppets?
54 Categoricals We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb).
55 Categoricals We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb). (Mh ᴧ Sh)
56 Categoricals We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb). (Mh ᴧ Sh) (Mf ᴧ Sf)
57 Categoricals We could try listing formulas like the one we used for Beaker: (Mb ᴧ Sb). (Mh ᴧ Sh) (Mf ᴧ Sf) (Mk ᴧ Sk)
58 Categoricals Could we use the formula ( x)(mx ᴧ Sx) to translate the categorical All Muppets are silly?
59 Categoricals Could we use the formula ( x)(mx ᴧ Sx) to translate the categorical All Muppets are silly? NO
60 Categoricals Could we use the formula ( x)(mx ᴧ Sx) to translate the categorical All Muppets are silly? NO Why not?
61 Categoricals The formula ( x)(mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists.
62 Categoricals The formula ( x)(mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists. But lots of things aren t Muppets.
63 Categoricals The formula ( x)(mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists. And some things aren t silly either.
64 Categoricals The formula ( x)(mx ᴧ Sx) says that everything is both a Muppet and silly. Everything that exists. And some things aren t silly either.
65 Categoricals So what do we mean when we say that all Muppets are silly?
66 Categoricals So what do we mean when we say that all Muppets are silly? For any object you want to pick, if the object is a Muppet, then it is silly.
67 Categoricals All Muppets are silly. ( x)(mx Sx)
68 Categoricals All Muppets are silly. ( x)(mx Sx) For any x you want
69 Categoricals All Muppets are silly. ( x)(mx Sx) For any x you want x is a Muppet
70 Categoricals All Muppets are silly. ( x)(mx Sx) For any x you want x is a Muppet only if
71 Categoricals All Muppets are silly. ( x)(mx Sx) x is silly For any x you want x is a Muppet only if
72 Categoricals All Muppets are silly. ( x)(mx Sx) x is silly For any x you want x is a Muppet only if
73 Categoricals All Muppets are silly. (Ma Sa) ᴧ (Mb Sb) ᴧ (Mc Sc) ᴧ : One way to think of a universal quantifier is as a big conjunction.
74 Next Time We will talk about the class of particular categorical sentences, and then we will talk about relations.
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