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\title {Logic I \\ Lecture 16}
 
\maketitle
 

Logic I

Lecture 16

\def \ititle {Logic I}
\def \isubtitle {Lecture 16}
\begin{center}
{\Large
\textbf{\ititle}: \isubtitle
}
 
\iemail %
\end{center}
Readings refer to sections of the course textbook, \emph{Language, Proof and Logic}.
 
\section{There Is Exactly One}
 
\section{There Is Exactly One}
There is one creator (at least one, maybe more).
\hspace{3mm} ∃x Creator(x)
Ahura Mazda is the one and only creator.
\hspace{3mm} Creator(a) ∧ ∀x( Creator(x) → x=a )
All squares are broken.
\hspace{3mm} ∀x( Sqr(x) → Brkn(x) )
There is one and only one creator.
\hspace{3mm} ∃y( Creator(y) ∧ ∀x( Creator(x) → x=y ) )
\hspace{3mm} or:
\hspace{3mm} ∃y ∀x( Creator(x) ↔ x=y )

There is one creator (at least one, maybe more).

∃x Creator(x)

Ahura Mazda is the one and only creator.

∃y( Creator(a) ∧ ∀x( Creator(x) → x=a )

All squares are broken.

∀x( Sqr(x) → Brkn(x) )

There is one and only one creator.

∃y( Creator(y) ∧ ∀x( Creator(x) → x=y ) )

Look, two quantifiers. But not just two quantifiers, one is existential and the other is universal. Mixed quantifiers!
Here's another, shorter way of expressing the same proposition (these are logically equivalent)

or:

∃y( ∀x( Creator(x) ↔ x=y ) )

Ex: There is one and only one female creator.

Ex: There is one and only one creator and she is female.

14.10--14.12, *14.13
11.10, 11.13
14.2

No Lecture Next Thursday (17th March 2016)

 

Every Time I Go to the Dentist Someone Dies

 
\section{Every Time I Go to the Dentist Someone Dies}
\emph{Reading:} §11.2
 
\section{Every Time I Go to the Dentist Someone Dies}
∀t (
\hspace{5mm} ( Time(t) ∧ ToDentist(a,t) )
\hspace{5mm} →
\hspace{5mm} ∃x ( Person(x) ∧ TimeOfDeath(x,t) )
)
 

Could There Be Nothing?

 
\section{Could There Be Nothing?}
\emph{Reading:} §13.2
 
\section{Could There Be Nothing?}
 

Proofs about Proofs

 
\section{Proofs about Proofs}
 
\section{Proofs about Proofs}

let’s get meta

\begin{minipage}{\columnwidth}
\textbf{If A ⊢ B then ⊢ A→B}
Proof Given a proof for A ⊢ B …
… we can turn it into a proof for ⊢ A→B:
\end{minipage}
\textbf{If ⊢ A→B then A ⊢ B}
\begin{minipage}{\columnwidth}
\textbf{If A ⊢ B then A ⊢ ¬¬B}
Proof:
\end{minipage}
\textbf{If A ⊢ C then A ⊢ B→C}
\textbf{If A ⊢ B and A ⊢ ¬C then A ⊢ ¬(B→C)}
 

Does ‘if’ mean what ‘→’ means?

 
\section{Does ‘if’ mean what ‘→’ means?}
\emph{Reading:} §7.3
 
\section{Does ‘if’ mean what ‘→’ means?}
\begin{minipage}{\columnwidth}
These two arguments are valid: does that mean that `if' means what `→' means?
\end{minipage}
\begin{minipage}{\columnwidth}
The English argument isn't valid; the awFOL argument is valid; therefore `if' can't mean what `→' means?
\end{minipage}

No Lecture Next Thursday (17th March 2016)