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An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Computer Science & Applied Mathematics)

Peter B. Andrews

Chapter 3

Provability and Refutability - all with Video Answers

Educators

Section 1

Natural Deduction

01:15

Problem 1

Prove the following theorems in $\mathscr{N}$ :
$\sim \forall \mathbf{x} \mathbf{A} \equiv \exists \mathbf{x} \sim \mathbf{A}$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator

Problem 2

Prove the following theorems in $\mathscr{N}$ :
$\mathbf{.} \sim \exists \mathbf{x} \mathbf{A} \equiv \forall \mathbf{x} \sim \mathbf{A}$.

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Problem 3

Prove the following theorems in $\mathscr{N}$ :
$\forall u \forall v \forall w[P u v \vee P v w] \supset \exists x \forall y$ Pxy. (Hint: recall the advice given near the end of $\S 21$ about proving theorems of the form $\mathbf{A} \supset \mathbf{B}$.)

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06:48

Problem 4

Prove the following theorems in $\mathscr{N}$ :
Prove that the system $\mathscr{N}$ is sound in the sense of 2303

Brian Ketelobeter
Brian Ketelobeter
Numerade Educator
06:48

Problem 5

Prove the following theorems in $\mathscr{N}$ : .
Prove that the system $\mathscr{N}$ is complete in the sense of 2509.

Brian Ketelobeter
Brian Ketelobeter
Numerade Educator

Problem 6

Prove the following theorems in $\mathscr{N}$ :
Add rules of inference for $=$ to $\mathscr{N}$, and prove that the resulting system is sound and complete in the sense of 2609 and 2612.

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Problem 7

Prove the following theorems in $\mathscr{N}$ :
Are any of the rules of inference of $\mathscr{N}$ dependent (non-independent) in the sense of $\$ 13$ ?

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Problem 8

Prove the following theorems in $\mathscr{N}$ :
Find a system $\mathscr{N}^{\prime}$ which can be obtained from $\mathscr{N}$ by deleting certain rules of inference, and whose rules of inference are all independent. Prove the independence of these rules.

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