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\title {Logic I \\ Logic (PH133)}

\maketitle

# Lecture 8

\def \ititle {Logic (PH133)}
\def \isubtitle {Lecture 8}
\begin{center}
{\Large
\textbf{\ititle}: \isubtitle
}

\iemail %
\end{center}
Readings refer to sections of the course textbook, \emph{Language, Proof and Logic}.

## ∀Intro

\section{∀Intro}

\section{∀Intro}
\begin{minipage}{\columnwidth}
Why is this proof incorrect?
\end{minipage}
12.4--12.5
*12.6--12.7
12.9--12.10

## There Is a Store for Everything

\section{There Is a Store for Everything}

\section{There Is a Store for Everything}
There is a store for everything:
\hspace{3mm} ∃y∀x StoreFor(y,x)
\hspace{3mm} ∀y∃x StoreFor(x,y)
Other sentences to translate:
\hspace{3mm} Wikipedia has an article about everything
\hspace{3mm} Everyone hurts someone they love
\hspace{3mm} Someone hurts everyone she loves
11.3
11.4, 11.8, 11.9
11.11, 11.13, *11.10
11.8, 11.9, *11.11

## Variables

\section{Variables}

\section{Variables}
Names : a, b, c, …
Variables : x, y, z, w, …
Variables are for saying several things about one thing even without specifying which thing it is
NB: Some x is a horse and x is dead' ain't English.

## Loving and Being Loved

\section{Loving and Being Loved}

\section{Loving and Being Loved}

#

## Quantifier Equivalences: ¬∀x Created(x) ⫤⊨ ∃x ¬Created(x)

\section{Quantifier Equivalences: ¬∀x Created(x) ⫤⊨ ∃x ¬Created(x)}

\section{Quantifier Equivalences: ¬∀x Created(x) ⫤⊨ ∃x ¬Created(x)}

\section{The End Is Near}

\section{The End Is Near}
‘The’ can be a quantifier, e.g. ‘the square is broken’. How to formalise it?
The square is broken \\ ⫤⊨ There is exactly one square and it is broken
Recall that we can translate There is exactly one square' as:
\hspace{5mm} ∃x ( Square(x) ∧ ∀y ( Square(y) → x=y ) )
So `There is exactly one square and it's broken':
\hspace{5mm} ∃x ( Sqr(x) ∧ ∀y ( Sqr(y) → x=y ) ∧ Broken(x) )

There is an end, and all ends are this end, and it is near.

∃x ( End(x) ∧ ∀y ( End(y) → x=y ) ∧ Near(x) )

14.26, 14.28
14.2
14.4, 14.5
14.10, 14.11
14.26, 14.28