From 91578e40704f2e4b57b8c2b51413bd917cd7baeb Mon Sep 17 00:00:00 2001 From: Arachnophobian <123899465+Arachnophobian@users.noreply.github.com> Date: Sat, 4 Feb 2023 12:33:19 -0500 Subject: [PATCH 1/5] Something happened; I messed up GitHub I can't click on anything and a huge portion of it won't display. I don't want to lose my answers thus far but have no idea how to save, so.... --- Project-1/README.md | 78 +++++++++++++++++++++++++++++++++++++++------ 1 file changed, 69 insertions(+), 9 deletions(-) diff --git a/Project-1/README.md b/Project-1/README.md index b67f3a56..31feda40 100644 --- a/Project-1/README.md +++ b/Project-1/README.md @@ -11,22 +11,82 @@ Tip #4: The work we do is challenging; that should be assumed. You are smart eno Note: The standard interpretation of the logical symbols - "∨", "∧", "→", "¬", "∀", "∃" - is assumed throughout. 1. Provide the truth tables for each of the following propositional logic formulas. State whether each is a tautology, a contradiction, or contingent: - ```(a) (¬A→B)∨((A∧¬C)→B) - (b) (A→B)∧(A→¬B) - (c) (A→(B∨C))∨(C→¬A) - (d) ((A→B)∧C)∨(A∧D) - + ```(a) (¬A→B)∨((A∧¬C)→B) <<< **This is a Tuatology** + +| A | B | C | ((¬A → B) ∨ ((A ∧ ¬C) → B)) | +|:-:|:-:|:-:|:---------------------------:| +| F | F | F | T | +| F | F | T | T | +| F | T | F | T | +| F | T | T | T | +| T | F | F | T | +| T | F | T | T | +| T | T | F | T | +| T | T | T | T | + + + (b) (A→B)∧(A→¬B) <<< **This is Contingent** + +| **A** | **B** | **((A → B) ∧ (A → ¬B))** | +|:-----:|:-----:|:------------------------:| +| F | F | T | +| F | T | T | +| T | F | F | +| T | T | F | + + + (c) (A→(B∨C))∨(C→¬A) <<< **This is a Tuatology** + +| A | B | C | ((A → (B ∨ C)) ∨ (C → ¬A)) | +|:-:|:-:|:-:|:--------------------------:| +| F | F | F | T | +| F | F | T | T | +| F | T | F | T | +| F | T | T | T | +| T | F | F | T | +| T | F | T | T | +| T | T | F | T | +| T | T | T | T | + + + (d) ((A→B)∧C)∨(A∧D) <<< **This is contingent** + +| A | B | C | D | (((A → B) ∧ C) ∨ (A ∧ D)) | +|:-:|:-:|:-:|:-:|:-------------------------:| +| F | F | F | F | F | +| F | F | F | T | F | +| F | F | T | F | T | +| F | F | T | T | T | +| F | T | F | F | F | +| F | T | F | T | F | +| F | T | T | F | T | +| F | T | T | T | T | +| T | F | F | F | F | +| T | F | F | T | T | +| T | F | T | F | F | +| T | F | T | T | T | +| T | T | F | F | F | +| T | T | F | T | T | +| T | T | T | F | T | +| T | T | T | T | T | + + 2. A _literal_ is an atomic formula or the negation of an atomic formula. We say a formula is in _conjunctive normal form_ (CNF) if it is the conjunction of the disjunction of literals. Find propositional logic formulas in CNF equivalent to each of the following: ```(a) (A→B)→C - + **(A∨B∨C) ∧ (A∨¬B∨C) ∧ (¬A∨¬B∨C)** + (b) (A→(B∨C))∨(C→¬A) - + **No CNF form is possible, as this is a Tuatology.** + (c) (¬A∧¬B∧C)∨(¬A∧¬C)∨(B∧C)∨A + **No CNF form is possible, as this is a Tuatology.** 3. Let V be the vocabulary of first-order logic consisting of a binary relation P and a unary relation F. Interpret P(x,y) as “x is a parent of y” and F(x) as “x is female.” Where possible define the following formulas in this vocabulary; where not possible, explain why: - (a) B(x,y) that says that x is a brother of y - (b) A(x,y) that says that x is an aunt of y + (a) B(x,y) that says that x is a brother of y + **This is not possible, unless "female" and "brother" are not restricted by sex/gender norms. If they are not, such that I may be a female *and* a brother, then: F(x)=B(x,y).** + (b) A(x,y) that says that x is an aunt of y + (c) C(x,y) that says that x and y are cousins (d) O(x) that says that x is an only child (e) T(x) that says that x has exactly two brothers From 6237b034b28131ba1948015ad083567c84525726 Mon Sep 17 00:00:00 2001 From: Arachnophobian <123899465+Arachnophobian@users.noreply.github.com> Date: Sun, 5 Feb 2023 15:10:56 -0500 Subject: [PATCH 2/5] Update README.md --- Project-1/README.md | 13 ++++++++----- 1 file changed, 8 insertions(+), 5 deletions(-) diff --git a/Project-1/README.md b/Project-1/README.md index 31feda40..7af5df06 100644 --- a/Project-1/README.md +++ b/Project-1/README.md @@ -11,7 +11,9 @@ Tip #4: The work we do is challenging; that should be assumed. You are smart eno Note: The standard interpretation of the logical symbols - "∨", "∧", "→", "¬", "∀", "∃" - is assumed throughout. 1. Provide the truth tables for each of the following propositional logic formulas. State whether each is a tautology, a contradiction, or contingent: - ```(a) (¬A→B)∨((A∧¬C)→B) <<< **This is a Tuatology** + ``` + +(a) (¬A→B)∨((A∧¬C)→B) <<< **This is a Tuatology** | A | B | C | ((¬A → B) ∨ ((A ∧ ¬C) → B)) | |:-:|:-:|:-:|:---------------------------:| @@ -84,9 +86,7 @@ Note: The standard interpretation of the logical symbols - "∨", "∧", "→", 3. Let V be the vocabulary of first-order logic consisting of a binary relation P and a unary relation F. Interpret P(x,y) as “x is a parent of y” and F(x) as “x is female.” Where possible define the following formulas in this vocabulary; where not possible, explain why: (a) B(x,y) that says that x is a brother of y - **This is not possible, unless "female" and "brother" are not restricted by sex/gender norms. If they are not, such that I may be a female *and* a brother, then: F(x)=B(x,y).** (b) A(x,y) that says that x is an aunt of y - (c) C(x,y) that says that x and y are cousins (d) O(x) that says that x is an only child (e) T(x) that says that x has exactly two brothers @@ -102,7 +102,8 @@ Note: The standard interpretation of the logical symbols - "∨", "∧", "→", 5. Select two formulas defined in ALC from question 4 to form the basis of a T-Box. Supplement this T-box with whatever other axioms you like, as well as an A-box, so that you ultimately construct a knowledge base K = (T,A). Provide a _model_ of K. This may be graphical or symbolic or both. 6. Explain the difference - using natural language - between the first-order prefixes: - ```(a) ∃x∀y and ∀x∃y + ``` + (a) ∃x∀y and ∀x∃y (b) ∃x∀y∃z and ∀x∃y∀z (c) ∀x∃y∀z∃w and ∃x∀y∃z∀w ``` @@ -110,11 +111,13 @@ Note: The standard interpretation of the logical symbols - "∨", "∧", "→", 7. Show that the following sentences are not equivalent by exhibiting a graph that models one but not both of these sentences: ``` ∀x∃y∀z(R(x,y) ∧ R(x,z) ∧ R(y,z)) + ∃x∀y∃z(R(x,y) ∧ R(x,z) ∧ R(y,z)) ``` 8. Using an online tableau proof generator - such as the one found here `https://www.umsu.de/trees/` - provide tree proofs of the following entailments, which are known as the De Morgan's laws: - ```(a) ∀x∀y(¬(Px ∧ Qx) → (¬Px ∨ ¬Qx)) + ``` + (a) ∀x∀y(¬(Px ∧ Qx) → (¬Px ∨ ¬Qx)) (b) ∀x∀y(¬(Px ∨ Qx) → (¬Px ∧ ¬Qx)) (c) ∀x∀y((¬Px ∨ ¬Qx) → ¬(Px ∧ Qx)) (d) ∀x∀y((¬Px ∨ ¬Qx) → ¬(Px ∧ Qx)) From a3a383a6ba64ca06d5b892d8ad7134f7d6389c17 Mon Sep 17 00:00:00 2001 From: Arachnophobian <123899465+Arachnophobian@users.noreply.github.com> Date: Sun, 5 Feb 2023 18:08:02 -0500 Subject: [PATCH 3/5] Only partly done I really meant it when I said logic was my weakness. --- Project-1/README.md | 67 ++++++++++++++++++++++++++++++++++++++------- 1 file changed, 57 insertions(+), 10 deletions(-) diff --git a/Project-1/README.md b/Project-1/README.md index 7af5df06..624bc0bf 100644 --- a/Project-1/README.md +++ b/Project-1/README.md @@ -85,45 +85,92 @@ Note: The standard interpretation of the logical symbols - "∨", "∧", "→", 3. Let V be the vocabulary of first-order logic consisting of a binary relation P and a unary relation F. Interpret P(x,y) as “x is a parent of y” and F(x) as “x is female.” Where possible define the following formulas in this vocabulary; where not possible, explain why: - (a) B(x,y) that says that x is a brother of y + (a) B(x,y) that says that x is a brother of y + **Not possible; unless "brother" is considered sex and/or gender neutral and the neutral, in which case (∃x(F(x)∧(B(x,y))))** + (b) A(x,y) that says that x is an aunt of y - (c) C(x,y) that says that x and y are cousins - (d) O(x) that says that x is an only child - (e) T(x) that says that x has exactly two brothers + ***Possible: (∃x(F(x)∧(A(x,y))))** + + (c) C(x,y) that says that x and y are cousins + **Possible: (∃x(C(x,y)))** + + (d) O(x) that says that x is an only child + **Not Possible: as (P(x,y)) is a parent relationship, and (O(x)) describes a *child_of* relationship, V does not have the relation necessary** + + (e) T(x) that says that x has exactly two brothers + **Possible: (∃x(P(x,y)∧T(x)))** 4. Let V be a vocabulary of the attribute (concept) language with complements (ALC) consisting of a role name "parent_of" and a concept name "Male". Interpret parent_of as "x is a parent of y" and M as "x is male". Where possible define the following formulas in this vocabulary; where not possible, explain why: - ```(a) B that says that x is a brother of y + ``` + (a) B that says that x is a brother of y + **Possible: MB ≡ M∩∃B** + (b) A that says that x is an aunt of y + **Not Possible, unless "aunt" is considered a gender neutral term, in which case; MA ≡ M∩∃A** + (c) C that says that x and y are cousins + **Possible: MC ≡ M∩∃C** + (d) O that says that x is an only child + **Not Possible, as nd O describes a *child_of* relationship, V does not have the relation necessary** + (e) T that says that x has exactly two brothers - + **Possible: I have no idea how.** 5. Select two formulas defined in ALC from question 4 to form the basis of a T-Box. Supplement this T-box with whatever other axioms you like, as well as an A-box, so that you ultimately construct a knowledge base K = (T,A). Provide a _model_ of K. This may be graphical or symbolic or both. + + **There's literally no way I can do this correctly.** + + 1) B that says that x is a brother of y + T1 = {Brother ⊆ Entity + X ⊆ exists.Entity + Y ⊆ exists.Entity + No.... this can't be right. There's an existential quantifier on page 18... the existence is... presumed? But the problem with it is that they're using "Teacher" in the example, but they don't mean it as a concept, but as a "thing"... I have no idea how to go about this with an X and a Y, which are not "things which exist". + + + 2) O that says that x is an only child. + See above. Obviously I have no idea what I'm doing. + 6. Explain the difference - using natural language - between the first-order prefixes: ``` (a) ∃x∀y and ∀x∃y + ** These read "There exists some X such that for all y..." and "For all x there exists some y such that..." There is no way to explain the difference beyond this because there is no meaning associated with these prefixes. They are analogous to "sentence fragments" from natural language where the Subject was missing and the only verb is "is"; "Is large and blue." ** + (b) ∃x∀y∃z and ∀x∃y∀z + ** These read "There exists some X such that for all Y and some z..." and "For all X there exists some Y such that all Z. .. " There is no way to explain the difference beyond this because there is no meaning associated with these prefixes. ** + (c) ∀x∃y∀z∃w and ∃x∀y∃z∀w + ** These read "For all X there exists some Y such that all Z and some W. . ." and "There exists some X for all Y such that some some Z and all W . . . " There is no way to explain the difference beyond this because there is no meaning associated with these prefixes. ** + ``` 7. Show that the following sentences are not equivalent by exhibiting a graph that models one but not both of these sentences: ``` ∀x∃y∀z(R(x,y) ∧ R(x,z) ∧ R(y,z)) - ∃x∀y∃z(R(x,y) ∧ R(x,z) ∧ R(y,z)) +** I have no idea where to even begin. Do I just *make up* verticies and edges? How would I have any idea how many I needed?** + ``` 8. Using an online tableau proof generator - such as the one found here `https://www.umsu.de/trees/` - provide tree proofs of the following entailments, which are known as the De Morgan's laws: ``` (a) ∀x∀y(¬(Px ∧ Qx) → (¬Px ∨ ¬Qx)) + ![proof(1)](https://user-images.githubusercontent.com/123899465/216850416-7fb691da-f18f-4164-877c-bc1064085130.png) + (b) ∀x∀y(¬(Px ∨ Qx) → (¬Px ∧ ¬Qx)) + ![proof(2)](https://user-images.githubusercontent.com/123899465/216850442-db0062d8-5ffb-4e46-bec6-1135c47a53a8.png) + (c) ∀x∀y((¬Px ∨ ¬Qx) → ¬(Px ∧ Qx)) - (d) ∀x∀y((¬Px ∨ ¬Qx) → ¬(Px ∧ Qx)) + ![proof(3)](https://user-images.githubusercontent.com/123899465/216850473-a00e0b27-ee52-43a3-a2b8-e7060cd0abac.png) + + (d) ∀x∀y((¬Px ∨ ¬Qx) → ¬(Px ∧ Qx)) + ![proof(4)](https://user-images.githubusercontent.com/123899465/216850506-16aea313-2ca2-4297-8615-668f8d5a3701.png) + ``` -9. Using a natural deduction proof generator - such as the one found here `https://proofs.openlogicproject.org/` - provide natural deduction proofs for each of De Morgan's laws. +9. Using a natural deduction proof generator - such as the one found here `https://proofs.openlogicproject.org/` - provide natural deduction proofs for each of De Morgan's laws. +I honestly couldn't figure out how to use the Natural Deduction Proof checker. There's nothing to see here but my utter lack of knowledge. 10. Compare and contrast the proofs provided for (a) in your answers to questions 8 and 9. Explain the different assumptions, strategies, etc. exhibited in tree proofs vs natural deduction proofs. - +I couldn't begin to and this was due an hour ago. From cb035d81b1ce38244181dbc220b74e9dbe7b9737 Mon Sep 17 00:00:00 2001 From: Arachnophobian <123899465+Arachnophobian@users.noreply.github.com> Date: Sat, 11 Feb 2023 10:12:06 -0500 Subject: [PATCH 4/5] Update README.md --- Project-1/README.md | 189 +++++++++++++++++++++++++++++++++++--------- 1 file changed, 150 insertions(+), 39 deletions(-) diff --git a/Project-1/README.md b/Project-1/README.md index 624bc0bf..f7037ecf 100644 --- a/Project-1/README.md +++ b/Project-1/README.md @@ -85,63 +85,106 @@ Note: The standard interpretation of the logical symbols - "∨", "∧", "→", 3. Let V be the vocabulary of first-order logic consisting of a binary relation P and a unary relation F. Interpret P(x,y) as “x is a parent of y” and F(x) as “x is female.” Where possible define the following formulas in this vocabulary; where not possible, explain why: - (a) B(x,y) that says that x is a brother of y - **Not possible; unless "brother" is considered sex and/or gender neutral and the neutral, in which case (∃x(F(x)∧(B(x,y))))** - - (b) A(x,y) that says that x is an aunt of y - ***Possible: (∃x(F(x)∧(A(x,y))))** - - (c) C(x,y) that says that x and y are cousins - **Possible: (∃x(C(x,y)))** - - (d) O(x) that says that x is an only child - **Not Possible: as (P(x,y)) is a parent relationship, and (O(x)) describes a *child_of* relationship, V does not have the relation necessary** - - (e) T(x) that says that x has exactly two brothers - **Possible: (∃x(P(x,y)∧T(x)))** + ``` + (a) B(x,y) that says that x is a brother of y + B(x,y) := ∃z.(P(zx) ∧ P(zy) ∧ ¬(z=y)) ∧ ¬(z=x) ¬(x=y) ¬F(x) + (b) A(x,y) that says that x is an aunt of y + A(x,y) := ∃z∃w.(P(zy) ∧ P(wz) ∧ P(wx) ∧ ¬(z=x)) F(x) + (c) C(x,y) that says that x and y are cousins + C(x,y) := ∃u∃w∃z(P(w,z)∧P(w,u)∧P(z,x)∧P(u,y)∧¬(z=u))∧¬(x=y) + (d) O(x) that says that x is an only child + O(x) := ∃y.(P(yx) ∧ ∀z.(P(yz) → (w=y))) + (e) T(x) that says that x has exactly two brothers + T(x) := ∃u∃y∃z(¬(y=z)∧¬(x=y)∧¬(x=z)∧P(u,x)∧P(u,y)∧P(u,z)∧¬F(y)∧¬F(z))∧∀w(P(u,w)∧¬F(w)→ w=x ∨ w=y ∨ w=z) + + ``` 4. Let V be a vocabulary of the attribute (concept) language with complements (ALC) consisting of a role name "parent_of" and a concept name "Male". Interpret parent_of as "x is a parent of y" and M as "x is male". Where possible define the following formulas in this vocabulary; where not possible, explain why: ``` (a) B that says that x is a brother of y - **Possible: MB ≡ M∩∃B** - + + Per (Person) ≡ M ⊔ ¬M + p2 (Parent of 2) ≡ ∃parentof. ≥ 2 ⊓ ¬(∃parentof. = 1) (this condition is meant to avoid that the two persons are the same, but I am not entirely sure it works or whether there are better ways to do so) + + B ≡ M ⊓ ∃p2¯.Per + (b) A that says that x is an aunt of y - **Not Possible, unless "aunt" is considered a gender neutral term, in which case; MA ≡ M∩∃A** - + + ∃p2¯.parentof.parentof.Per + + A ≡ ¬M ⊓ (∃p2¯. (parentof. Per ≥ 2).(parentof.Per)) ⊔ ¬parentof.Per + + U(uncle) ≡ M ⊓ (∃p2¯. (parentof. Per ≥ 2).(parentof.Per)) ⊔ ¬parentof.Per + (c) C that says that x and y are cousins - **Possible: MC ≡ M∩∃C** - + + gp2(Grandparent with at least two children which each have children) ≡ ∃parentof ≥ 2.parentof.Per + + C ≡ gp2¯.per + + But this would only describe one of the two cousins, i.e. "being a cousin of someone" or "having a cousin". + We cannot define "being cousins" in description logic. + (d) O that says that x is an only child - **Not Possible, as nd O describes a *child_of* relationship, V does not have the relation necessary** - + + O ≡ ¬ ∃p2¯.Per + (e) T that says that x has exactly two brothers - **Possible: I have no idea how.** + + This cannot be done without a role name for brother, we only have a concept name for it. + + ``` 5. Select two formulas defined in ALC from question 4 to form the basis of a T-Box. Supplement this T-box with whatever other axioms you like, as well as an A-box, so that you ultimately construct a knowledge base K = (T,A). Provide a _model_ of K. This may be graphical or symbolic or both. - **There's literally no way I can do this correctly.** - - 1) B that says that x is a brother of y - T1 = {Brother ⊆ Entity - X ⊆ exists.Entity - Y ⊆ exists.Entity - No.... this can't be right. There's an existential quantifier on page 18... the existence is... presumed? But the problem with it is that they're using "Teacher" in the example, but they don't mean it as a concept, but as a "thing"... I have no idea how to go about this with an X and a Y, which are not "things which exist". - + TBox = { +B ⊑ M +B ⊑ Per +M ⊑ Per +A ⊑ ¬M +A ⊑ Per +} + +ABox = { John : M +John : parentof. Chris +John: B +Chris: M +Mary : ¬M +Mary: A +Alexander : parentof. Mary +Alexander : parentof. John +} + +ΔI = {John, Mary, Alexander, Chris} + +Named Individuals: +JohnI = J, +MaryI = M, +AlexanderI = A, +Chris = C + +Concept Assignments: +PersonI = {J, M, A, C}, +MaleI = {J, A, C}, +FemaleI = {M}, +BrotherI = {J}, +AuntI = {M} + +Role Assignments: +parentofI = {(J, C), (A, J), (A, M)} - 2) O that says that x is an only child. - See above. Obviously I have no idea what I'm doing. 6. Explain the difference - using natural language - between the first-order prefixes: ``` (a) ∃x∀y and ∀x∃y - ** These read "There exists some X such that for all y..." and "For all x there exists some y such that..." There is no way to explain the difference beyond this because there is no meaning associated with these prefixes. They are analogous to "sentence fragments" from natural language where the Subject was missing and the only verb is "is"; "Is large and blue." ** + ** These read "There exists some X such that for all y..." and "For all x there exists some y such that..." There is no way to explain the difference beyond this because these are incomplete sentences. They are analogous to "sentence fragments" from natural language where the Subject was missing and the only verb is "is"; "Is large and blue." ** (b) ∃x∀y∃z and ∀x∃y∀z - ** These read "There exists some X such that for all Y and some z..." and "For all X there exists some Y such that all Z. .. " There is no way to explain the difference beyond this because there is no meaning associated with these prefixes. ** + ** These read "There exists some X such that for all Y and some z..." and "For all X there exists some Y such that all Z. .. " There is no way to explain the difference beyond this because these are incomplete sentences ** (c) ∀x∃y∀z∃w and ∃x∀y∃z∀w - ** These read "For all X there exists some Y such that all Z and some W. . ." and "There exists some X for all Y such that some some Z and all W . . . " There is no way to explain the difference beyond this because there is no meaning associated with these prefixes. ** + ** These read "For all X there exists some Y such that all Z and some W. . ." and "There exists some X for all Y such that some some Z and all W . . . " There is no way to explain the difference beyond this because these are incomplete sentences. ** ``` @@ -149,7 +192,12 @@ Note: The standard interpretation of the logical symbols - "∨", "∧", "→", ``` ∀x∃y∀z(R(x,y) ∧ R(x,z) ∧ R(y,z)) ∃x∀y∃z(R(x,y) ∧ R(x,z) ∧ R(y,z)) -** I have no idea where to even begin. Do I just *make up* verticies and edges? How would I have any idea how many I needed?** +∀x∃y∀z(R(x,y) ∧ R(x,z) ∧ R(y,z)) +∃x∀y∃z(R(x,y) ∧ R(x,z) ∧ R(y,z)) +``` + {(a,b)(b,a)(a,a)} + The second statement is true in this model. There is an x that relates every y (a, related to a and b) There is an x that relates to a z (where a is x and b is z), all ys (a and b) are related to some z (a is related to a and b, b is related to a) + The first statement is not true. It is not true that all xs (a and b) are related to all zs (b is not related to a) ``` @@ -170,7 +218,70 @@ Note: The standard interpretation of the logical symbols - "∨", "∧", "→", ``` 9. Using a natural deduction proof generator - such as the one found here `https://proofs.openlogicproject.org/` - provide natural deduction proofs for each of De Morgan's laws. -I honestly couldn't figure out how to use the Natural Deduction Proof checker. There's nothing to see here but my utter lack of knowledge. +a) +assume -(Px & Qx). + assume -(-Px + -Qx). + assume -Px. + -Px + -Qx. + F. + therefore --Px. + assume Qx. + Px. + assume -(-Px + -Qx). + Px & Qx. + F. + therefore --(-Px + -Qx). + F. + therefore -Qx. + -Px + -Qx. + F. + therefore --(-Px + -Qx). + -Px + -Qx. +therefore -(Px & Qx) => -Px + -Qx. + +b) +assume -(Px + Qx). + assume Px. + Px + Qx. + F. + therefore -Px. + assume Qx. + Px + Qx. + F. + therefore -Qx. + -Px & -Qx. +therefore -(Px + Qx) => -Px & -Qx. + +c) +assume -Px + -Qx. + assume Px & Qx. + Px. + Qx. + assume -Px. + F. + therefore -Px => F. + assume -Qx. + F. + therefore -Qx => F. + F. + therefore -(Px & Qx). +therefore -Px + -Qx => -(Px & Qx). + +d) +assume -Px & -Qx. + assume Px + Qx. + assume Px. + -Px. + F. + therefore Px => F. + assume Qx. + -Qx. + F. + therefore Qx => F. + F. + therefore -(Px + Qx). +therefore -Px & -Qx => -(Px + Qx). 10. Compare and contrast the proofs provided for (a) in your answers to questions 8 and 9. Explain the different assumptions, strategies, etc. exhibited in tree proofs vs natural deduction proofs. -I couldn't begin to and this was due an hour ago. + +The tableau proofs are more systematizable because they take the negation of each statement; this is a fixed starting point which does not require any choice between options. The natural proof method is more flexible and depends on dividing the original statement into an assumption/consequent pair; this is less easy to systematize, because the division needs to be chosen. From a9debadf87c4aa9b9e6cf7a655a255dd68cf7c9f Mon Sep 17 00:00:00 2001 From: Arachnophobian <123899465+Arachnophobian@users.noreply.github.com> Date: Sun, 12 Feb 2023 20:52:01 -0500 Subject: [PATCH 5/5] Update README.md --- Project-1/README.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/Project-1/README.md b/Project-1/README.md index f7037ecf..0b153d19 100644 --- a/Project-1/README.md +++ b/Project-1/README.md @@ -192,8 +192,7 @@ parentofI = {(J, C), (A, J), (A, M)} ``` ∀x∃y∀z(R(x,y) ∧ R(x,z) ∧ R(y,z)) ∃x∀y∃z(R(x,y) ∧ R(x,z) ∧ R(y,z)) -∀x∃y∀z(R(x,y) ∧ R(x,z) ∧ R(y,z)) -∃x∀y∃z(R(x,y) ∧ R(x,z) ∧ R(y,z)) + ``` {(a,b)(b,a)(a,a)} The second statement is true in this model. There is an x that relates every y (a, related to a and b) There is an x that relates to a z (where a is x and b is z), all ys (a and b) are related to some z (a is related to a and b, b is related to a)