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\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}.

\section{Subproofs Are Tricky: The Answer}

\section{∀Intro}

\emph{Reading:} §12.1, §12.3, §13.1

\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}

\emph{Reading:} §11.2, §11.3

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

\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.

\section{Loving and Being Loved}

\emph{Reading:} §11.2, §11.3

\section{Somebody Is Not Dead}

Some person is dead.

\hspace{5mm} ∃x(Person(x) ∧ Dead(x))

Some person is not dead.

\hspace{5mm} ∃x(Person(x) ∧ ¬Dead(x))

No person is dead.

\hspace{5mm} ¬∃x(Person(x) ∧ Dead(x))

Every person is dead.

\hspace{5mm} ∀x(Person(x) → Dead(x))

Every person is not dead.

\hspace{5mm} ∀x(Person(x) → ¬Dead(x))

Not every person is dead.

\hspace{5mm} ¬∀x(Person(x) → Dead(x))

Some person is dead.

∃x(Person(x) ∧ Dead(x))

Some person is not dead.

∃x(Person(x) ∧ ¬Dead(x))

No person is dead.

¬∃x(Person(x) ∧ Dead(x))

Every person is dead.

∀x(Person(x) → Dead(x))

Every person is not dead.

∀x(Person(x) → ¬Dead(x))

Not every person is dead.

¬∀x(Person(x) → Dead(x))

#

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

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

\emph{Reading:} §10.1, §10.3, §10.4

\section{The End Is Near}

\emph{Reading:} §14.3

‘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