First-Order Natural Deduction. Part 1: Universal Introduction and Elimination
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1 First-Order Natural Deduction Part 1: Universal Introduction and Elimination
2 Happy Birthday to Talan Two years old!
3 Business Homework #7 will be posted later today. It is due on Monday, October 21. The Midterm Exam is also on October 21.
4 Business The midterm exam will cover the first three chapters. You will not be asked to do first-order proofs!
5 Identity For the most part, we treat relations in a generic way. However, one relation is special. Identity gets its own symbol, =, and we write (a = b), rather than =ab.
6 Identity Identity is an equivalence relation: it is reflexive, symmetric, and transitive. In fact, identity is the smallest or most fine-grained equivalence relation. a b c
7 Identity Leibniz proposed two laws governing identity. Indiscernibility of Identicals: If a and b are identical, then everything true of a is true of b. Identity of Indiscernibles: If everything true of a is true of b, then a and b are identical.
8 Identity If whatever is true of includes relations, like identity itself, then Leibniz s Laws become trivial. However, Leibniz himself restricted his laws to monadic predicates. In its restricted form, the law of Identity of Indiscernibles is controversial.
9 Identity Leibniz himself pointed out the following odd consequence of his laws: There is no possible world containing two qualitatively identical, homogeneous spheres.
10 Identity Leibniz himself pointed out the following odd consequence of his laws: There is no possible world containing two qualitatively identical, homogeneous spheres. POOF!
11 Identity Worse, because of its symmetry properties, a world containing just a single homogeneous sphere is impossible!
12 Identity Such an object is indiscernible from a single point. POOF!
13 Identity When two constants a and b name the same individual, we write (a = b). Moreover, we can freely substitute a for occurrences of b in any non-intensional context in which it appears. What is an intensional context, and why does context matter?
14 Identity An intensional context is one where the sense, rather than the reference of a constant term matters to the meaning of the sentence in which it appears. An example of an intensional context is a sentence that uses the word believes.
15 Identity Harvey Dent believes that Bruce Wayne is Bruce Wayne, but he does not believe that Bruce Wayne is Batman.?
16 Identity Identity over time When I was younger, I was shorter.
17 Identity Identity over time When I was younger, I was shorter. I don t think Leibniz is gonna like this.
18 Identity Identity across worlds If I had a million dollars, I d be rich.
19 Identity Identity across worlds If I had a million dollars, I d be rich.
20 Identity Identity across worlds If I had a million dollars, I d be rich.
21 Identity Identity across worlds If I had a million dollars, I d be rich.
22 Identity Identity across worlds If I had a million dollars, I d be rich.
23 Identity Identity across worlds If I had a million dollars, I d be rich.
24 Identity We can use identity to translate sentences involving superlatives or numerical claims. Jim is the shortest man in the room. ( x)((mx ᴧ Rx) (Sjx v (j = x))) There is exactly one fish. ( x)(fx ᴧ ( y)(fy (y = x)))
25 First-Order Natural Deduction We now want to start thinking about how to prove things in first-order logic. To that end, we will add six new inference rules. We add introduction and elimination rules for the universal quantifier, the existential quantifier, and the identity relation.
26 First-Order Natural Deduction But before we add new rules of inference, we need to understand an operation: substitution. We use the substitution operation to describe the rules of inference for the quantifiers.
27 Substitution Let be a formula, let x be an arbitrary variable, and let c be an arbitrary constant. [x/c] denotes the result of replacing all free occurrences of x in with c.
28 Substitution (Fxy Myz) ( x)(gx Hy) (( y)(hy ᴧ Dx) ᴠ Fy) (Mxx ᴧ ( x)gx) ( Db Ky)
29 Substitution (Fxy Myz)[z/d] ( x)(gx Hy)[y/e] (( y)(hy ᴧ Dx) ᴠ Fy)[y/c] (Mxx ᴧ ( x)gx)[x/b] ( Db Ky)[y/a]
30 Substitution (Fxy Myz)[z/d] ( x)(gx Hy)[y/e] (( y)(hy ᴧ Dx) ᴠ Fy)[y/c] (Mxx ᴧ ( x)gx)[x/b] ( Db Ky)[y/a] (Fxy Myd) ( x)(gx He) (( y)(hy ᴧ Dx) ᴠ Fc) (Mbb ᴧ ( x)gx) ( Db Ka)
31 Universal Elimination ( E) Suppose ( x) is a well-formed formula for some variable x, and suppose c is an arbitrary constant. ( x) [x/c]
32 Let s try an example: { ( x)(gx ᴧ Hx), (Ga ( y)ky) } (Ha ᴧ Kb)
33 Assumptions Line Formula Justification 1 (1) ( x)(gx ᴧ Hx) A 2 (2) (Ga ( y)ky) A
34 Assumptions Line Formula Justification 1 (1) ( x)(gx ᴧ Hx) A 2 (2) (Ga ( y)ky) A 1 (3) (Ga ᴧ Ha) 1 E
35 Assumptions Line Formula Justification 1 (1) ( x)(gx ᴧ Hx) A 2 (2) (Ga ( y)ky) A 1 (3) (Ga ᴧ Ha) 1 E 1 (4) Ga 3 ᴧE
36 Assumptions Line Formula Justification 1 (1) ( x)(gx ᴧ Hx) A 2 (2) (Ga ( y)ky) A 1 (3) (Ga ᴧ Ha) 1 E 1 (4) Ga 3 ᴧE 1 (5) Ha 3 ᴧE
37 Assumptions Line Formula Justification 1 (1) ( x)(gx ᴧ Hx) A 2 (2) (Ga ( y)ky) A 1 (3) (Ga ᴧ Ha) 1 E 1 (4) Ga 3 ᴧE 1 (5) Ha 3 ᴧE 1,2 (6) ( y)ky 2,4 E
38 Assumptions Line Formula Justification 1 (1) ( x)(gx ᴧ Hx) A 2 (2) (Ga ( y)ky) A 1 (3) (Ga ᴧ Ha) 1 E 1 (4) Ga 3 ᴧE 1 (5) Ha 3 ᴧE 1,2 (6) ( y)ky 2,4 E 1,2 (7) Kb 6 E
39 Assumptions Line Formula Justification 1 (1) ( x)(gx ᴧ Hx) A 2 (2) (Ga ( y)ky) A 1 (3) (Ga ᴧ Ha) 1 E 1 (4) Ga 3 ᴧE 1 (5) Ha 3 ᴧE 1,2 (6) ( y)ky 2,4 E 1,2 (7) Kb 6 E 1,2 (8) (Ha ᴧ Kb) 5,7 ᴧI
40 Let s try another: { (Ga Hc) } (( y)gy Hc)
41 Assumptions Line Formula Justification 1 (1) (Ga Hc) A
42 Assumptions Line Formula Justification 1 (1) (Ga Hc) A 2 (2) ( y)gy A (for CP)
43 Assumptions Line Formula Justification 1 (1) (Ga Hc) A 2 (2) ( y)gy A (for CP) 2 (3) Ga 2 E
44 Assumptions Line Formula Justification 1 (1) (Ga Hc) A 2 (2) ( y)gy A (for CP) 2 (3) Ga 2 E 1,2 (4) Hc 1,3 E
45 Assumptions Line Formula Justification 1 (1) (Ga Hc) A 2 (2) ( y)gy A (for CP) 2 (3) Ga 2 E 1,2 (4) Hc 1,3 E 1 (5) (( y)gy Hc) 2,4 CP
46 Universal Introduction ( I) Suppose is a well-formed formula, and c is a constant that does not appear in. Then [x/c] ( x) So long as [x/c] does not depend on any formula containing c.
47 The rule for universal introduction has two clauses that constrain how we can introduce universal quantifiers. The constant c cannot appear in. The formula [x/c] cannot depend on any formula containing c.
48 The easiest way to see how the constraints on universal introduction work is to see how they might be violated.
49 The easiest way to see how the constraints on universal introduction work is to see how they might be violated. Let s see a proof that violates the first constraint: The constant c cannot appear in.
50 Assumptions Line Formula Justification 1 (1) ( x)fxx A
51 Assumptions Line Formula Justification 1 (1) ( x)fxx A 1 (2) Fbb 1 E
52 Assumptions Line Formula Justification 1 (1) ( x)fxx A 1 (2) Fbb 1 E 1 (3) ( x)fxb 2 I
53 Assumptions Line Formula Justification 1 (1) ( x)fxx A 1 (2) Fbb 1 E 1 (3) ( x)fxb 2 I NO!
54 Assumptions Line Formula Justification 1 (1) ( x)fxx A 1 (2) Fbb 1 E 1 (3) ( x)fxb 2 I NO! The formula Fxb contains the constant b!
55 And now a proof that violates the second constraint: The formula [x/c] cannot depend on any formula containing c.
56 Assumptions Line Formula Justification 1 (1) Fa A
57 Assumptions Line Formula Justification 1 (1) Fa A 1 (2) ( x)fx 1 I
58 Assumptions Line Formula Justification 1 (1) Fa A 1 (2) ( x)fx 1 I NO!
59 Assumptions Line Formula Justification 1 (1) Fa A 1 (2) ( x)fx 1 I NO! The formula Fa depends on a formula that contains the constant a, namely itself!
60 Now, let s see a correct example: { ( x)(fa Gx) } (Fa ( x)gx)
61 Assumptions Line Formula Justification 1 (1) ( x)(fa Gx) A
62 Assumptions Line Formula Justification 1 (1) ( x)(fa Gx) A 2 (2) Fa A (for CP)
63 Assumptions Line Formula Justification 1 (1) ( x)(fa Gx) A 2 (2) Fa A (for CP) 1 (3) (Fa Gb) 1 E
64 Assumptions Line Formula Justification 1 (1) ( x)(fa Gx) A 2 (2) Fa A (for CP) 1 (3) (Fa Gb) 1 E 1,2 (4) Gb 2,3 E
65 Assumptions Line Formula Justification 1 (1) ( x)(fa Gx) A 2 (2) Fa A (for CP) 1 (3) (Fa Gb) 1 E 1,2 (4) Gb 2,3 E 1,2 (5) ( x)gx 4 I
66 Assumptions Line Formula Justification 1 (1) ( x)(fa Gx) A 2 (2) Fa A (for CP) 1 (3) (Fa Gb) 1 E 1,2 (4) Gb 2,3 E 1,2 (5) ( x)gx 4 I 1 (6) (Fa ( x)gx) 2,5 CP
67 Next Time We will see rules for dealing with the existential quantifier.
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