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

Peter B. Andrews

Chapter 2

First-Order Logic - all with Video Answers

Educators

Section 1

Formation Rules for the System $\mathscr{F}$

Problem 1

Proceed as follows:
(a) Copy the wff, and underline the free occurrences of variables in it.
(b) List the free variables of the wff.
(c) List the bound variables of the wff.
(d) Decide whether $g x$ is free for $u$ in the wff.
(e) Decide whether $g x$ is free for $z$ in the wff.
(f) Decide whether $h x y$ is free for $u$ in the wff.
(g) Decide whether $u$ is free for $x$ in the wff.
$\sim[[\forall u P u z \wedge \sim \exists x=Q u x \vee \forall z \sim R x z] \supset \exists z Q u z]$

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Problem 2

Proceed as follows:
(a) Copy the wff, and underline the free occurrences of variables in it.
(b) List the free variables of the wff.
(c) List the bound variables of the wff.
(d) Decide whether $g x$ is free for $u$ in the wff.
(e) Decide whether $g x$ is free for $z$ in the wff.
(f) Decide whether $h x y$ is free for $u$ in the wff.
(g) Decide whether $u$ is free for $x$ in the wff.
$\forall x[P x u \vee \exists y \cdot Q x y \wedge \sim \forall u R u z] \supset \forall z Q z u$
In exercises $\mathrm{X} 2002-\mathrm{X} 2003$, interpret the predicate, function, and individual constants as indicated, and translate the given English expression [wff of $\mathscr{F}]$ into a wff of $\mathscr{F}$ [English expression].

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01:46

Problem 3

$H x$ means " $x$ is a horse", $t x$ means "the tail of $x$ ", $m x$ means "the mane of $x$ ", Bx means " $x$ is in the barn", $W y$ means " $y$ is white", $K y$ means " $y$ is black", $L x y$ means " $x$ likes $y$ ".
(a) "Every horse in the barn which has a white tail has a black mane."
(b) "White horses don't like horses with black manes."
(c) "No horse in the barn has a white tail."
(d) $(\exists x \in H)[B x \wedge(\forall y \in H) \cdot B y \wedge K t y \supset L x y]$
(e) $\sim(\exists x \in H)[B x \wedge \sim W t x]$

Christopher Stanley
Christopher Stanley
Numerade Educator
01:33

Problem 4

Bxy means " $x$ is a brother of $y$ ", $r$ means "Robert", $U x y$ means " $x$ is ancle of $y$ ", $f^1 x$ means "the father of $x$ ", $C x y$ means " $x$ is a cousin of $y$ ", Yxy means " $x$ is younger than $y$ ".
(a) "Any brother of Robert's father is an uncle of Robert."
(b) $\forall x \forall y\left[B f^{-1} x f^1 y>C x y\right]$
(c) "Robert has a cousin who is younger than one of Robert's brothers."

Hubert Agamasu
Hubert Agamasu
Numerade Educator
16:00

Problem 5

Formulate a definition of the set of terms of $\mathscr{F}$ in the style of the definition of the set of wffs of $\mathscr{P}$ in $\S 10$, and state and prove an analogue of Theorem 1000 .

Chris Trentman
Chris Trentman
Numerade Educator
16:00

Problem 6

Formulate a definition of the set of wffs of $\mathscr{F}$ in the style of the definition of the set of wffs of $\mathscr{P}$ in $\S 10$, and state and prove an analogue of Theorem 1000.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 7

Show that every wf part of a wff $\sim \mathbf{B}$ or $\forall \mathbf{x}$ B is either the entire wff or a wf part of $\mathbf{B}$.

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Problem 8

Show that every wf part of a wff $[\mathbf{B} \vee \mathbf{C}]$ is either the entire wff or a wf part of $\mathbf{B}$ or a wf part of $\mathbf{C}$. Would this still be true if brackets were omitted from the list of improper symbols and from formation rule (e)?

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01:31

Problem 9

Show that if $[\mathbf{B} \vee \mathbf{C}]$ and $[\mathbf{D} \vee \mathbf{E}]$ are the same wff, then $\mathbf{B}$ is $\mathbf{D}$ and $\mathbf{C}$ is $\mathbf{E}$.

AG
Ankit Gupta
Numerade Educator
01:51

Problem 10

Prove that if $\mathbf{X}$ and $\mathbf{y}$ are distinct individual variables such that $y$ is not free in $\mathbf{C}$ and $\mathbf{y}$ is free for $\mathbf{x}$ in $\mathbf{C}$, then $S_{\mathbf{x}}^y S_y^x \mathbf{C}=\mathbf{C}$.

Aman Gupta
Aman Gupta
Numerade Educator
06:27

Problem 11

Let $\mathbf{A}$ and $\mathbf{C}$ be wffs and let $\mathbf{m}$ be a propositional variable such that $\mathbf{C}$ is free for $\mathbf{m}$ in $\mathbf{A}$. Let $\mathbf{x}_1, \ldots, \mathbf{x}_n$ be the free individual variables of $\mathbf{C}$, and let $\mathbf{u}_1, \ldots, \mathbf{u}_n$ be distinct individual variables which do not occur in $\mathbf{C}$ or $\mathbf{A}$. examples to illustrate this metatheorem, and to show the necessity of various conditions in its hypotheses.

Rosina Dapaah
Rosina Dapaah
Numerade Educator