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\title {Logic I \\ Lecture 12}

\maketitle

# Lecture 12

\def \ititle {Logic I}
\def \isubtitle {Lecture 12}
\begin{center}
{\Large
\textbf{\ititle}: \isubtitle
}

\iemail %
\end{center}
Readings refer to sections of the course textbook, \emph{Language, Proof and Logic}.
9.20
13.1--13.4, 13.10--13.15, *13.16

\section{Relations: Reflexive, Symmetric}

\section{Relations: Reflexive, Symmetric}
A \emph{reflexive} relation is one that everything bears to itself. (E.g. everything is the SameShape as itself. E.g. of non-reflexive: not everything is LeftOf itself).
A \emph{symmetric} relation is one such that if x bears it to y, then y bears it to x. (E.g. Adjacent(x,y) is symmetric, LeftOf(x,y) is not symmetric.)

\section{Relations: Transitivity}

\section{Relations: Transitivity}
A \emph{transitive} relation is one such that if x bears it to y and y bears it to z then x bears it to z. (E.g. LeftOf is transitive; NotAdjacent is not transitive.)
\begin{minipage}{\columnwidth}
If NotAdjacent were transitive, the following argument would be logically valid:
\end{minipage}
\begin{minipage}{\columnwidth}
A counterexample to this argument:
\end{minipage}

express the counterexample formally

Domain

{0, 1, 2}

Names

a : 0

b : 1

c : 2

Predicates

<0,1>, <1,2>,

<1,0>, <2,1>

}

Domain

{0, 1, 2}

Names

a : 0

b : 1

c : 2

Predicates

<0,1>, <1,2>,

<1,0>, <2,1>

}

Domain

{0, 1, 2}

Names

a : 0

b : 1

c : 2

Predicates

<0,1>, <1,2>,

<1,0>, <2,1>

}

## Relations: Some Examples

\section{Relations: Some Examples}

\section{Relations: Some Examples}
Artificial relations ...
EqualToOrLeftOf(x, y) iff
\hspace{3mm} x = y or LeftOf(x, y)
JohnOrAyesha(x, y) iff
\hspace{3mm} x = John and y = Ayesha
\hspace{3mm} or x = Ayesha and y = John
JohnToAyesha(x, y) iff
\hspace{3mm} x = John and y = Ayesha

## Defining Relations Using Dot-Arrow Diagrams

\section{Defining Relations Using Dot-Arrow Diagrams}

## Expressing Relations with Quantifiers

\section{Expressing Relations with Quantifiers}

\section{Expressing Relations with Quantifiers}
\begin{minipage}{\columnwidth}
A \emph{reflexive} relation is one that everything bears to itself. (E.g. SameShape)
reflexive: ∀x R(x,x)
\end{minipage}
\begin{minipage}{\columnwidth}
A \emph{symmetric} relation is one such that if x bears it to y, then y bears it to x. (E.g. Adjacent(x,y))
symmetric: ∀x∀y ( R(x,y) → R(y,x) )
\end{minipage}
\begin{minipage}{\columnwidth}
A \emph{transitive} relation is one such that if x bears it to y and y bears it to z then x bears it to z. (E.g. LeftOf is transitive; DifferentShape is not transitive)
transitive: ∀x∀y∀z ( ( R(x,y) ∧ R(y,z) ) → R(x,z) )
\end{minipage}
15.33--15.40 (second edition)
15.33, 15.37--15.39 (second edition)

feedback

‘You could recommend an introduction to logic or an introduction to thinking about logic’

2. Mark Sainsbury,Logical Forms

## Negating Identity

\section{Negating Identity}

\section{Negating Identity}
 ¬a=b ¬(a=b) } these mean the same thing

Ayesha is not Beatrice.

It’s not true that: Ayesha is Beatrice

Ayesha is this: not Beatrice (?)

a=¬b <== don’t do this!

(¬a)=b <== don’t do this either!

## There Does Not Exist

\section{There Does Not Exist}

\section{There Does Not Exist}
Everything is not broken:
\hspace{3mm} ∀x ¬Broken(x)
Not everything is broken:
\hspace{3mm} ¬∀x Broken(x)

 ​ 1 ​ 2 a=a =Intro ​ 3 ∃x (x=x) ∃Intro: 2

Everything is broken.

∀x Broken(x)

Everything is not broken.

∀x ¬Broken(x)

Not everything is broken.

¬∀x Broken(x)

1.

 ​ 2 Dead(a) ​ 3 ∃x Dead(x) ∃Intro: 2 ​ 4 ⊥ ⊥Intro: 1,3

Counterexample:

Domain: {Ayesha, Beatrice}

a : Ayesha ; b : Beatrice

... and ¬Dead(a) is true; so the premise, ∃x ¬Dead(x), is true

9.12
9.18--9.19

## Expressing Counterexamples Formally

\section{Expressing Counterexamples Formally}

\section{Expressing Counterexamples Formally}
Give a counterexample to this argument:
Informally:
Formally:
\hspace{3mm} Domain: \{a, b\}
\hspace{3mm} R: \{<a,a>, <a,b>, <b,b>\}
Here’s a quick look at how to enter the counterexample in zoxiy. If you want more detail, please see the user guide (which is linked to on the home page of zoxiy).

## Summary of Relations

\section{Summary of Relations}

## Proof Example: A∧B therefore ¬(¬A∨¬B).

\section{Proof Example: A∧B therefore ¬(¬A∨¬B).}

\section{Proof Example: A∧B therefore ¬(¬A∨¬B).}

## Proof Example: P therefore ¬¬P.

\section{Proof Example: P therefore ¬¬P.}

\section{Proof Example: P therefore ¬¬P.}

## Proof Example: S∨(Q∧R) therefore S∨Q.

\section{Proof Example: S∨(Q∧R) therefore S∨Q.}

\section{Proof Example: S∨(Q∧R) therefore S∨Q.}