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Pset 3
Make sure to read all instructions carefully! Any mistakes you make as a
Pset 3
Make sure to read all instructions carefully! Any mistakes you make as a result of missing the instructions are your responsibility.
When you type your proofs, Carnap will display them correctly, in the Fitch system, on the right. You can use your keyboard to type logical symbols as follows:
symbolhow to typeshortcut (proof-masters only!)
¬~ or –
∧/&
∨/|
→->>
↔︎<-><>
⊥_|_!?
∀A@
∃E3
Don’t use the shortcuts if you’re struggling with proofs! It’ll just make exams more confusing!
In order to enter your justifications, you must separate the sentence in the proof from the justification with a colon, followed immediately by the justification. Example: :/I 1,2.
In our textbook, you need to provide a justification on every line of a proof except for premises and temporary assumptions. (After all, they don’t require an inference rule!) On Carnap, though, you need to provide a justification even for premises and temporary assumptions: write :PR for premises and :AS for temporary assumptions.
On exercises where you get to check your work, each line will display a + if your justification is correct. If the justification is incorrect, it’ll display a ? or a ✗. If you hover your cursor over the ?/✗, Carnap will try to hint at what went wrong.
Tips:
Don’t include a space between the colon and the rule.
Don’t include a comma between the rule and the line numbers.
If you put a space in front of a line of your proof, Carnap will interpret you as introducing a subproof.
Justify any premises of a proof with :PR. If you put a space after :PR, your premises might not display correctly.
Justify the first line of a subproof — the temporary assumption — with :AS.
To open a subproof, indent by one space. (Tab won’t work.)
To close a subproof, use backspace to stop indenting.
To end a subproof and immediately start another, separate them with a separator line, –, one level of indentation back from the subproofs.
Remember: once a subproof ends, you can’t cite individual lines within it anymore; you can only cite the whole subproof (e.g. “3-5”, rather than “3,4,5”).
Subproofs in subproofs
Prove the following using only basic rules. For these exercises, you might need to nest a subproof within another:
3.A.1 (Q -> R)” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.A.1″ data-carnap-system=”thomasBolducAndZachFOL2019″ data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>R ⊢ P → (Q → R)
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3.A.2 R :|-: P -> (Q -> R)” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.A.2″ data-carnap-system=”thomasBolducAndZachFOL2019″ data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>(P ∧ Q) → R ⊢ P → (Q → R)
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3.A.3 ~J :|-: ~J” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.A.3″ data-carnap-system=”thomasBolducAndZachFOL2019″ data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>J → ¬J ⊢ ¬J
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3.A.4 L” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.A.4″ data-carnap-system=”thomasBolducAndZachFOL2019″ data-carnap-type=”proofchecker” class=”failure” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 151px;”>K ∧ L ⊢ K ↔ L⚠
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Submit
1.K ∧ L PR
2.K∧E 1
3.L∧E 1
3.A.5(C ∧ D) ∨ E ⊢ E ∨ D⚠
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Submit
1.(C ∧ D) ∨ E PR
2.C ∧ D AS
3.D∧E 2
4.E ∨ D ∨I 3
5.EAS
6.E ∨ D ∨I 5
3.A.6 D :|-: D” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.A.6″ data-carnap-system=”thomasBolducAndZachFOL2019″ data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;” rel=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;” rel=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>(Z ∧ K) ∨ (K ∧ M) , K → D ⊢ D
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3.A.7 Q) :|-: ~Q” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.A.7″ data-carnap-system=”thomasBolducAndZachFOL2019″ data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>¬(P → Q) ⊢ ¬Q
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3.A.8 ~R :|-: Q / E” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.A.8″ data-carnap-system=”thomasBolducAndZachFOL2019″ data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>P ∧ (Q ∨ R) , P → ¬R ⊢ Q ∨ E
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3.A.9 Q) :|-: P” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.A.9″ data-carnap-system=”thomasBolducAndZachFOL2019″ data-carnap-type=”proofchecker” class=”” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 151px;”>¬(P → Q) ⊢ P
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1.¬(P → Q)PR
2.P ∧ ¬Q ¬E 1
Theorems
When you’re asked to prove a theorem, Carnap puts a “⊤” before the turnstile “⊢”. As you know, our textbook doesn’t do this. The “⊤” is the symbol for tautology (the opposite of contradiction, “⊥”). You can ignore this.1 Note that “⊤” is not the same thing as the atomic sentence “T”! So don’t include “T” as a premise. Indeed, don’t use any premises! You only need temporary assumptions, within subproofs.
Prove the following theorems using only basic rules
3.B.1⊤ ⊢ ¬(A ∧ ¬A)✓
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1.A ∧ ¬A AS
2.A∧E 1
3.¬A∧E 1
4.⊥¬E 2, 3
5.¬(A ∧ ¬A)¬I 1-4
3.C.2 D” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.C.2″ data-carnap-system=”thomasBolducAndZachTFL2019″ data-carnap-type=”proofchecker” class=”success” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 151px;” rel=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 151px;” rel=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 151px;”>⊤ ⊢ (C ∧ D) → D✓
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1.C ∧ D AS
2.D∧E 1
3.(C ∧ D) → D →I 1-2
3.B.3 C) / (B -> C)) -> ((A / B) -> C)” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.B.3″ data-carnap-system=”thomasBolducAndZachFOL2019″ data-carnap-type=”proofchecker” class=”” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;” rel=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;” rel=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>⊤ ⊢ ((A → C) ∧ (B → C)) → ((A ∨ B) → C)
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Proofs where additional/derived rules are allowed.
NOTE: for some reason, for this set of rules, the Fitch bar under argument premises isn’t rendering in the proof display on the right. It’s a bug; hopefully it’ll be fixed soon. In the meantime, remember that you do need a line under your premises in handwritten proofs, e.g. on exams.
Fill in justifications for the proof below:
3.C.1 B) / (~A -> B)” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.C.1″ data-carnap-system=”thomasBolducAndZachTFL” data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 284px;”>⊤ ⊢ ((A → B) ∨ (¬A → B))
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Below is a partial proof. Complete it! (Note: you’ll need to add extra lines after ‘¬¬A’, so don’t worry about whether the second subproof starts on line 9.)
3.D.2(A ∨ B), (¬A ∨ C) ⊢ (B ∨ C)
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See whether you can prove the next argument using DeM and without using subproofs. (Hint: you can prove it in 5 lines, including premises!)
3.C.3(A ∨ (B ∧ C)), ¬B ⊢ A
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See whether you can prove the next argument using LEM.
3.C.4 ~A, A -> B :|-: ~A” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.C.4″ data-carnap-system=”thomasBolducAndZachTFL” data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>(B → ¬A), (A → B) ⊢ ¬A
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3.C.5 C” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.C.5″ data-carnap-system=”thomasBolducAndZachTFL” data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>⊤ ⊢ ((A ∧ ¬A) → C)
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3.C.6 C” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.C.6″ data-carnap-system=”thomasBolducAndZachTFL” data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>⊤ ⊢ ((¬(A ∨ B) ∧ B) → C)
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Using two the proofs, show that the sentences ‘¬(¬A ∨ ¬B)’ is provably equivalent to ‘(A ∧ B)’:
3.C.7¬(¬A ∨ ¬B) ⊢ (A ∧ B)
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3.C.8(A ∧ B) ⊢ ¬(¬A ∨ ¬B)
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Show that ‘(P ∨ Q)’ and ‘(¬P → Q)’ are provably equivalent:
3.C.9 Q)” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.C.9″ data-carnap-system=”thomasBolducAndZachTFL” data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>(P ∨ Q) ⊢ (¬P → Q)
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3.C.10 Q) :|-: (P / Q)” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.C.10″ data-carnap-system=”thomasBolducAndZachTFL” data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>(¬P → Q) ⊢ (P ∨ Q)
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Show that the following set of sentences are provably inconsistent:
3.C.11(A ∧ B), (¬A ∧ C) ⊢ ⊥
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3.C.12 (B / C), ~A -> C :|-: _|_” data-carnap-guides=”fitch” data-carnap-options=”indent resize render fonts” data-carnap-submission=”saveAs:3.C.12″ data-carnap-system=”thomasBolducAndZachTFL” data-carnap-type=”proofchecker” style=”display: inline-block; vertical-align: top; position: relative; width: 400px; min-height: 110pt; margin-bottom: 50pt; line-height: 1.2em; font-size: 10pt; height: 168px;”>¬C, (A → (B ∧ C)), (¬A → C) ⊢ ⊥
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Reminder: always press the submit button when you’ve completed a problem. –>
The idea is this: tautologies are always true, no matter what. So they don’t tell you anything new about the world. They’re true at all possible worlds; they don’t rule any possible worlds out. So we don’t need to assume any specific premises, which may be true only at some possible worlds. Instead, we can substitute in any tautology as a premise, no matter what sentence letters it uses — because when we prove a theorem, we won’t need to make any use of the premise.
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