histogram of gradesjonathanlivengood.net/2019 fall/phil 103 logic and... · review let ϕbe a...
TRANSCRIPT
![Page 1: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/1.jpg)
Histogram of grades
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First-Order Logic
Natural Deduction, Part 2
![Page 4: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/4.jpg)
Review
Let ϕ be a formula, let x be an arbitrary variable, and let c be an arbitrary constant.
ϕ[x/c] denotes the result of replacing all free occurrences of x in ϕ with c.
![Page 5: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/5.jpg)
Review
(Gzz → Hxz)
(∀x)(Sx → Px)
(Fy ᴧ (∃y)(By ᴧ Dx))
(Rxx ᴧ (∃x)Fx)
(~Dy → Kz)
![Page 6: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/6.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)
(Fy ᴧ (∃y)(By ᴧ Dx))
(Rxx ᴧ (∃x)Fx)
(~Dy → Kz)
![Page 7: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/7.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)
(Fy ᴧ (∃y)(By ᴧ Dx))
(Rxx ᴧ (∃x)Fx)
(~Dy → Kz)
(Gdd → Hxd)
![Page 8: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/8.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)[y/e]
(Fy ᴧ (∃y)(By ᴧ Dx))
(Rxx ᴧ (∃x)Fx)
(~Dy → Kz)
(Gdd → Hxd)
![Page 9: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/9.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)[y/e]
(Fy ᴧ (∃y)(By ᴧ Dx))
(Rxx ᴧ (∃x)Fx)
(~Dy → Kz)
(Gdd → Hxd)
(∀x)(Sx → Px)
![Page 10: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/10.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)[y/e]
(Fy ᴧ (∃y)(By ᴧ Dx))[y/c]
(Rxx ᴧ (∃x)Fx)
(~Dy → Kz)
(Gdd → Hxd)
(∀x)(Sx → Px)
![Page 11: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/11.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)[y/e]
(Fy ᴧ (∃y)(By ᴧ Dx))[y/c]
(Rxx ᴧ (∃x)Fx)
(~Dy → Kz)
(Gdd → Hxd)
(∀x)(Sx → Px)
(Fc ᴧ (∃y)(By ᴧ Dx))
![Page 12: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/12.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)[y/e]
(Fy ᴧ (∃y)(By ᴧ Dx))[y/c]
(Rxx ᴧ (∃x)Fx)[x/b]
(~Dy → Kz)
(Gdd → Hxd)
(∀x)(Sx → Px)
(Fc ᴧ (∃y)(By ᴧ Dx))
![Page 13: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/13.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)[y/e]
(Fy ᴧ (∃y)(By ᴧ Dx))[y/c]
(Rxx ᴧ (∃x)Fx)[x/b]
(~Dy → Kz)
(Gdd → Hxd)
(∀x)(Sx → Px)
(Fc ᴧ (∃y)(By ᴧ Dx))
(Rbb ᴧ (∃x)Fx)
![Page 14: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/14.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)[y/e]
(Fy ᴧ (∃y)(By ᴧ Dx))[y/c]
(Rxx ᴧ (∃x)Fx)[x/b]
(~Dy → Kz)[y/a]
(Gdd → Hxd)
(∀x)(Sx → Px)
(Fc ᴧ (∃y)(By ᴧ Dx))
(Rbb ᴧ (∃x)Fx)
![Page 15: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/15.jpg)
Review
(Gzz → Hxz)[z/d]
(∀x)(Sx → Px)[y/e]
(Fy ᴧ (∃y)(By ᴧ Dx))[y/c]
(Rxx ᴧ (∃x)Fx)[x/b]
(~Dy → Kz)[y/a]
(Gdd → Hxd)
(∀x)(Sx → Px)
(Fc ᴧ (∃y)(By ᴧ Dx))
(Rbb ᴧ (∃x)Fx)
(~Da → Kz)
![Page 16: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/16.jpg)
Review
Universal Elimination (∀E)
Suppose (∀x)ϕ is a well-formed formula for some variable x, and suppose c is an arbitrary constant.
ϕ[x/c]
(∀x)ϕ
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Review
{ (∀x)(∀y)Rxy } Raa
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Review
Assumptions Line Formula Justification
1 (1) (∀x)(∀y)Rxy A
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Review
Assumptions Line Formula Justification
1 (1) (∀x)(∀y)Rxy A
1 (2) (∀y)Ray 1 ∀E
![Page 20: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/20.jpg)
Review
Assumptions Line Formula Justification
1 (1) (∀x)(∀y)Rxy A
1 (2) (∀y)Ray 1 ∀E
1 (3) Raa 2 ∀E
![Page 21: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/21.jpg)
New Rules
Universal Introduction (∀I)
Suppose ϕ is a well-formed formula, and c is a constant that does not appear in ϕ. Then
(∀x)ϕ
ϕ[x/c]
So long as ϕ[x/c] does not depend on any formula containing c.
![Page 22: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/22.jpg)
New Rules
The rule for universal introduction has two clauses that constrain how we can introduce universal quantifiers.
The formula ϕ[x/c] cannot depend on any formula containing c.
The constant c cannot appear in ϕ.
![Page 23: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/23.jpg)
New Rules
The easiest way to see how the constraints on universal introduction work is to see how they might be violated.
![Page 24: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/24.jpg)
New Rules
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 ϕ.
![Page 25: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/25.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)Fxx A
![Page 26: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/26.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)Fxx A
1 (2) Fbb 1 ∀E
![Page 27: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/27.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)Fxx A
1 (2) Fbb 1 ∀E
1 (3) (∀x)Fxb 2 ∀I
![Page 28: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/28.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)Fxx A
1 (2) Fbb 1 ∀E
1 (3) (∀x)Fxb 2 ∀I
NO!
![Page 29: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/29.jpg)
New Rules
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!
![Page 30: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/30.jpg)
New Rules
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!
Construct a small world showing invalidity.
![Page 31: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/31.jpg)
New Rules
And now a “proof” that violates the second constraint:
The formula ϕ[x/c] cannot depend on any formula containing c.
![Page 32: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/32.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fa A
![Page 33: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/33.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fa A
1 (2) (∀x)Fx 1 ∀I
![Page 34: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/34.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fa A
1 (2) (∀x)Fx 1 ∀I
NO!
![Page 35: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/35.jpg)
New Rules
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!
![Page 36: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/36.jpg)
New Rules
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!
Construct a small world showing invalidity.
![Page 37: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/37.jpg)
New Rules
Now, let’s see a correct example:
{ (∀x)(Fa → Gx) } (Fa → (∀x)Gx)
![Page 38: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/38.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(Fa → Gx) A
![Page 39: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/39.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(Fa → Gx) A
2 (2) Fa A (for CP)
![Page 40: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/40.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(Fa → Gx) A
2 (2) Fa A (for CP)
1 (3) (Fa → Gb) 1 ∀E
![Page 41: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/41.jpg)
New Rules
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
![Page 42: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/42.jpg)
New Rules
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
![Page 43: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/43.jpg)
New Rules
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
![Page 44: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/44.jpg)
New Rules
Existential Introduction (∃I)
Suppose (∃x)ϕ is a well-formed formula for some variable x, and suppose c is an arbitrary constant.
(∃x)ϕ
ϕ[x/c]
![Page 45: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/45.jpg)
New Rules
Let’s try an example:
{ Fa } (∃x)Fx
![Page 46: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/46.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fa A
![Page 47: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/47.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fa A
1 (2) (∃x)Fx 1 ∃I
![Page 48: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/48.jpg)
New Rules
Let’s try an example:
{ } (Fbb → ((∃x)Fxb ᴧ (∃x)Fbx)
![Page 49: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/49.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fbb A (for CP)
![Page 50: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/50.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fbb A (for CP)
1 (2) (∃x)Fxb 1 ∃I
![Page 51: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/51.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fbb A (for CP)
1 (2) (∃x)Fxb 1 ∃I
1 (3) (∃x)Fbx 1 ∃I
![Page 52: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/52.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fbb A (for CP)
1 (2) (∃x)Fxb 1 ∃I
1 (3) (∃x)Fbx 1 ∃I
1 (4) ((∃x)Fxb ᴧ (∃x)Fbx) 2,3 ᴧI
![Page 53: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/53.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Fbb A (for CP)
1 (2) (∃x)Fxb 1 ∃I
1 (3) (∃x)Fbx 1 ∃I
1 (4) ((∃x)Fxb ᴧ (∃x)Fbx) 2,3 ᴧI
(5) (1) → (4) 1,4 CP
![Page 54: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/54.jpg)
New Rules
And one more:
{ (∀x)(∀y)Rxy } (∃x)(∃y)Rxy)
![Page 55: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/55.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(∀y)Rxy A
![Page 56: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/56.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(∀y)Rxy A
1 (2) (∀y)Ray 1 ∀E
![Page 57: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/57.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(∀y)Rxy A
1 (2) (∀y)Ray 1 ∀E
1 (3) Rab 2 ∀E
![Page 58: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/58.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(∀y)Rxy A
1 (2) (∀y)Ray 1 ∀E
1 (3) Rab 2 ∀E
1 (4) (∃y)Ray 3 ∃I
![Page 59: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/59.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(∀y)Rxy A
1 (2) (∀y)Ray 1 ∀E
1 (3) Rab 2 ∀E
1 (4) (∃y)Ray 3 ∃I
1 (5) (∃x)(∃y)Rxy 4 ∃I
![Page 60: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/60.jpg)
Putting It Together
Give a proof of the conclusion of this argument from its premisses:
(∀x)(∃y)Rxy
(∃x)(∃y)Rxy → (∀x)(Fx → Gx)
((∀x)Fx → (∀x)Gx)
![Page 61: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/61.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
![Page 62: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/62.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
3 (3) (∀x)Fx ACP
![Page 63: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/63.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
3 (3) (∀x)Fx ACP
1 (4) (∃y)Ray 1 ∀E
![Page 64: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/64.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
3 (3) (∀x)Fx ACP
1 (4) (∃y)Ray 1 ∀E
1 (5) (∃x)(∃y)Rxy 4 ∃I
![Page 65: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/65.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
3 (3) (∀x)Fx ACP
1 (4) (∃y)Ray 1 ∀E
1 (5) (∃x)(∃y)Rxy 4 ∃I
1,2 (6) (∀x)(Fx → Gx) 2,5 →E
![Page 66: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/66.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
3 (3) (∀x)Fx ACP
1 (4) (∃y)Ray 1 ∀E
1 (5) (∃x)(∃y)Rxy 4 ∃I
1,2 (6) (∀x)(Fx → Gx) 2,5 →E
3 (7) Fb 3 ∀E
![Page 67: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/67.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
3 (3) (∀x)Fx ACP
1 (4) (∃y)Ray 1 ∀E
1 (5) (∃x)(∃y)Rxy 4 ∃I
1,2 (6) (∀x)(Fx → Gx) 2,5 →E
3 (7) Fb 3 ∀E
1,2 (8) (Fb → Gb) 6 ∀E
![Page 68: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/68.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
3 (3) (∀x)Fx ACP
1 (4) (∃y)Ray 1 ∀E
1 (5) (∃x)(∃y)Rxy 4 ∃I
1,2 (6) (∀x)(Fx → Gx) 2,5 →E
3 (7) Fb 3 ∀E
1,2 (8) (Fb → Gb) 6 ∀E
1,2,3 (9) Gb 7,8 →E
![Page 69: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/69.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
3 (3) (∀x)Fx ACP
1 (4) (∃y)Ray 1 ∀E
1 (5) (∃x)(∃y)Rxy 4 ∃I
1,2 (6) (∀x)(Fx → Gx) 2,5 →E
3 (7) Fb 3 ∀E
1,2 (8) (Fb → Gb) 6 ∀E
1,2,3 (9) Gb 7,8 →E
1,2,3 (10) (∀x)Gx 9 ∀I
![Page 70: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/70.jpg)
Putting It TogetherAssumptions Line Formula Justification
1 (1) (∀x)(∃y)Rxy A
2 (2) (∃x)(∃y)Rxy → (∀x)(Fx → Gx) A
3 (3) (∀x)Fx ACP
1 (4) (∃y)Ray 1 ∀E
1 (5) (∃x)(∃y)Rxy 4 ∃I
1,2 (6) (∀x)(Fx → Gx) 2,5 →E
3 (7) Fb 3 ∀E
1,2 (8) (Fb → Gb) 6 ∀E
1,2,3 (9) Gb 7,8 →E
1,2,3 (10) (∀x)Gx 9 ∀I
1,2 (11) ((∀x)Fx → (∀x)Gx) 3,10 CP
![Page 71: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/71.jpg)
Next Time
We’ll look at one more rule for the existential quantifier and two rules for the identity relation.
![Page 72: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/72.jpg)
New Rules
Existential Elimination (∃E)
Suppose φ and ψ are well-formed formulas in which the constant c does not appear. Then
ψ
(φ[x/c] → ψ)
So long as (φ[x/c] → ψ) does not depend onany formula containing c.
(∃x)φ
![Page 73: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/73.jpg)
New Rules
As with Universal Introduction, Existential Elimination has two constraints on its application:
The formula (φ[x/c] → ψ) cannot rest on any formula containing c.
The constant c cannot appear in φ or ψ.
![Page 74: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/74.jpg)
New Rules
Again, it will be easier to see how the constraints work by seeing how they might be violated.
![Page 75: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/75.jpg)
New Rules
Consider a case where the first constraint is violated:
The constant c cannot appear in φ or ψ.
![Page 76: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/76.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Fx A
![Page 77: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/77.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Fx A
2 (2) Fa A (for CP)
![Page 78: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/78.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Fx A
2 (2) Fa A (for CP)
(3) (Fa → Fa) 2 CP
![Page 79: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/79.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Fx A
2 (2) Fa A (for CP)
(3) (Fa → Fa) 2 CP
1 (4) Fa 1,3 ∃E
![Page 80: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/80.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Fx A
2 (2) Fa A (for CP)
(3) (Fa → Fa) 2 CP
1 (4) Fa 1,3 ∃E
NO!
![Page 81: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/81.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Fx A
2 (2) Fa A (for CP)
(3) (Fa → Fa) 2 CP
1 (4) Fa 1,3 ∃E
NO!
Here, the constant a in Fx[x/a] on line (2) also appears in ψ = Fa.
![Page 82: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/82.jpg)
New Rules
Now, consider a case where the second constraint is violated:
The formula (φ[x/c] → ψ) cannot rest on any formula containing c.
![Page 83: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/83.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Gx A
2 (2) ~Ga A
![Page 84: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/84.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Gx A
2 (2) ~Ga A
2 (3) (Ga → Gb) 2 Lemma
![Page 85: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/85.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Gx A
2 (2) ~Ga A
2 (3) (Ga → Gb) 2 Lemma
1,2 (4) Gb 1,4 ∃E
![Page 86: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/86.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Gx A
2 (2) ~Ga A
2 (3) (Ga → Gb) 2 Lemma
1,2 (4) Gb 1,4 ∃E
NO!
![Page 87: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/87.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)Gx A
2 (2) ~Ga A
2 (3) (Ga → Gb) 2 Lemma
1,2 (4) Gb 1,4 ∃E
NO!
The formula (Gx[x/a] → ψ) on line (3) depends on a formula that contains the constant a.
![Page 88: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/88.jpg)
New Rules
Now, let’s see a correct example:
{ (∀x)(Fx → Gb) } ((∃x)Fx → Gb)
![Page 89: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/89.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(Fx → Gb) A
![Page 90: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/90.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(Fx → Gb) A
2 (2) (∃x)Fx A (for CP)
![Page 91: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/91.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(Fx → Gb) A
2 (2) (∃x)Fx A (for CP)
1 (3) (Fa → Gb) 1 ∀E
![Page 92: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/92.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(Fx → Gb) A
2 (2) (∃x)Fx A (for CP)
1 (3) (Fa → Gb) 1 ∀E
1,2 (4) Gb 2,3 ∃E
![Page 93: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/93.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∀x)(Fx → Gb) A
2 (2) (∃x)Fx A (for CP)
1 (3) (Fa → Gb) 1 ∀E
1,2 (4) Gb 2,3 ∃E
1 (5) ((∃x)Fx → Gb) 2,4 CP
![Page 94: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/94.jpg)
New Rules
Let’s try a more complicated example:
{ (∃x)~Fx } ~(∀x)Fx
![Page 95: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/95.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)~Fx A
![Page 96: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/96.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)~Fx A
2 (2) (∀x)Fx A*
![Page 97: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/97.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)~Fx A
2 (2) (∀x)Fx A*
3 (3) ~Fa A (for CP)
![Page 98: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/98.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)~Fx A
2 (2) (∀x)Fx A*
3 (3) ~Fa A (for CP)
2 (4) Fa 2 ∀E
![Page 99: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/99.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)~Fx A
2 (2) (∀x)Fx A*
3 (3) ~Fa A (for CP)
2 (4) Fa 2 ∀E
(5) ((∀x)Fx → Fa) 2,4 CP
![Page 100: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/100.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)~Fx A
2 (2) (∀x)Fx A*
3 (3) ~Fa A (for CP)
2 (4) Fa 2 ∀E
(5) ((∀x)Fx → Fa) 2,4 CP
3 (6) ((∀x)Fx → ~Fa) 3 →I
![Page 101: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/101.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)~Fx A
2 (2) (∀x)Fx A*
3 (3) ~Fa A (for CP)
2 (4) Fa 2 ∀E
(5) ((∀x)Fx → Fa) 2,4 CP
3 (6) ((∀x)Fx → ~Fa) 3 →I
3 (7) ~(∀x)Fx 5,6 ~I
![Page 102: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/102.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)~Fx A
2 (2) (∀x)Fx A*
3 (3) ~Fa A (for CP)
2 (4) Fa 2 ∀E
(5) ((∀x)Fx → Fa) 2,4 CP
3 (6) ((∀x)Fx → ~Fa) 3 →I
3 (7) ~(∀x)Fx 5,6 ~I
(8) (~Fa → ~(∀x)Fx) 3,7 CP
![Page 103: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/103.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (∃x)~Fx A
2 (2) (∀x)Fx A*
3 (3) ~Fa A (for CP)
2 (4) Fa 2 ∀E
(5) ((∀x)Fx → Fa) 2,4 CP
3 (6) ((∀x)Fx → ~Fa) 3 →I
3 (7) ~(∀x)Fx 5,6 ~I
(8) (~Fa → ~(∀x)Fx) 3,7 CP
1 (9) ~(∀x)Fx 1,8 ∃E
![Page 104: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/104.jpg)
New Rules
Identity is a very special relation. And we give it special treatment.
Unlike the relations “… is taller than ---,” “… is made of ---,” and so on, the identity relation gets its own introduction and elimination rules.
![Page 105: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/105.jpg)
New Rules
Identity Introduction (=I)
On any line in a proof, you may write the formula (∀x)(x = x) without writing anything in the assumption column.
(∀x)(x = x)
![Page 106: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/106.jpg)
New Rules
Let’s see an example:
{ } ((a=a) ᴧ (b=b))
![Page 107: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/107.jpg)
New Rules
Assumptions Line Formula Justification
(1) (∀x)(x = x) =I
![Page 108: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/108.jpg)
New Rules
Assumptions Line Formula Justification
(1) (∀x)(x = x) =I
(2) (a = a) 1 ∀E
![Page 109: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/109.jpg)
New Rules
Assumptions Line Formula Justification
(1) (∀x)(x = x) =I
(2) (a = a) 1 ∀E
(3) (b = b) 1 ∀E
![Page 110: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/110.jpg)
New Rules
Assumptions Line Formula Justification
(1) (∀x)(x = x) =I
(2) (a = a) 1 ∀E
(3) (b = b) 1 ∀E
(4) ((a = a) ᴧ (b = b)) 2,3 ᴧI
![Page 111: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/111.jpg)
New Rules
The identity introduction rule is simple. In order to state the identity elimination rule, we need a new kind of operation: partial substitution.
![Page 112: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/112.jpg)
New Rules
Partial Substitution:
Let b and c be arbitrary constants, and let ϕbe an arbitrary formula. Then ϕ[[b/c]] represents any formula obtained from ϕ by replacing some occurrences of the constant b with the constant c.
![Page 113: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/113.jpg)
New Rules
Examples of Partial Substitution:
Ga
Haa
Hab
(a=b)
(∀x)Hxb
![Page 114: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/114.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa
Hab
(a=b)
(∀x)Hxb
![Page 115: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/115.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa
Hab
(a=b)
(∀x)Hxb
Gb
![Page 116: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/116.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa[[a/b]]
Hab
(a=b)
(∀x)Hxb
Gb
![Page 117: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/117.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa[[a/b]]
Hab
(a=b)
(∀x)Hxb
Gb
Hab, Hba, Hbb
![Page 118: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/118.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa[[a/b]]
Hab[[b/c]]
(a=b)
(∀x)Hxb
Gb
Hab, Hba, Hbb
![Page 119: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/119.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa[[a/b]]
Hab[[b/c]]
(a=b)
(∀x)Hxb
Gb
Hab, Hba, Hbb
Hac
![Page 120: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/120.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa[[a/b]]
Hab[[b/c]]
(a=b)[[b/c]]
(∀x)Hxb
Gb
Hab, Hba, Hbb
Hac
![Page 121: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/121.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa[[a/b]]
Hab[[b/c]]
(a=b)[[b/c]]
(∀x)Hxb
Gb
Hab, Hba, Hbb
Hac
(a=c)
![Page 122: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/122.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa[[a/b]]
Hab[[b/c]]
(a=b)[[b/c]]
(∀x)Hxb[[b/a]]
Gb
Hab, Hba, Hbb
Hac
(a=c)
![Page 123: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/123.jpg)
New Rules
Examples of Partial Substitution:
Ga[[a/b]]
Haa[[a/b]]
Hab[[b/c]]
(a=b)[[b/c]]
(∀x)Hxb[[b/a]]
Gb
Hab, Hba, Hbb
Hac
(a=c)
(∀x)Hxa
![Page 124: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/124.jpg)
New Rules
Identity Elimination (=E)
Let b and c be arbitrary constants, and let ϕbe an arbitrary formula. Then
ϕ[[b/c]]
(b = c)
ϕ
ϕ[[b/c]]
(c = b)
ϕ
![Page 125: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/125.jpg)
New Rules
Let’s see an example:
{ Gab, (b = c) } Gac
![Page 126: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/126.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Gab A
2 (2) (b = c) A
![Page 127: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/127.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Gab A
2 (2) (b = c) A
1,2 (3) Gac 1,2 =E
![Page 128: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/128.jpg)
New Rules
Let’s see another example:
{ (a = b) } (b = a)
![Page 129: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/129.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (a = b) A
![Page 130: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/130.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (a = b) A
(2) (∀x)(x = x) =I
![Page 131: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/131.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (a = b) A
(2) (∀x)(x = x) =I
(3) (a = a) 2 ∀E
![Page 132: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/132.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) (a = b) A
(2) (∀x)(x = x) =I
(3) (a = a) 2 ∀E
1 (4) (b = a) 1,3 =E
![Page 133: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/133.jpg)
New Rules
Okay, last example:
{ Ha, ~Hb } ~(a = b)
![Page 134: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/134.jpg)
New Rules
Okay, last example:
{ Ha, ~Hb } (a ≠ b)
We’ll write (a ≠ b) instead of ~(a = b).
![Page 135: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/135.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Ha A
2 (2) ~Hb A
![Page 136: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/136.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Ha A
2 (2) ~Hb A
3 (3) (a = b) A*
![Page 137: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/137.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Ha A
2 (2) ~Hb A
3 (3) (a = b) A*
2,3 (4) ~Ha 2,3 =E
![Page 138: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/138.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Ha A
2 (2) ~Hb A
3 (3) (a = b) A*
2,3 (4) ~Ha 2,3 =E
2 (5) ((a = b) → ~Ha) 3,4 CP
![Page 139: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/139.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Ha A
2 (2) ~Hb A
3 (3) (a = b) A*
2,3 (4) ~Ha 2,3 =E
2 (5) ((a = b) → ~Ha) 3,4 CP
1 (6) ((a = b) → Ha) 1 →I
![Page 140: Histogram of gradesjonathanlivengood.net/2019 Fall/PHIL 103 Logic and... · Review Let ϕbe a formula, let x be an arbitrary variable, and let c be an arbitrary constant. ϕ[x/c]](https://reader036.vdocuments.site/reader036/viewer/2022063012/5fc8faa2bac9456057776ccf/html5/thumbnails/140.jpg)
New Rules
Assumptions Line Formula Justification
1 (1) Ha A
2 (2) ~Hb A
3 (3) (a = b) A*
2,3 (4) ~Ha 2,3 =E
2 (5) ((a = b) → ~Ha) 3,4 CP
1 (6) ((a = b) → Ha) 1 →I
1,2 (7) (a ≠ b) 5,6 ~I
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Next Time
We’ll start thinking about set theory.