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\title {Logic I \\ Lecture 17}
 
\maketitle
 

Logic I

Lecture 17

\def \ititle {Logic I}
\def \isubtitle {Lecture 17}
\begin{center}
{\Large
\textbf{\ititle}: \isubtitle
}
 
\iemail %
\end{center}
Readings refer to sections of the course textbook, \emph{Language, Proof and Logic}.
 
\section{Revison: Definitions}
 
\section{Revison: Definitions}
\emph{Exercise} State the rules of proof for the following two connectives: ∧ →
What is a logically valid argument?
What is ... logical consequence, a tautology, a contradiction, a counterexample, a subproof, ...
What is a proof?
 

Revison: Truth tables

 
\section{Revison: Truth tables}
 
\section{Revison: Truth tables}
Use truth tables to establish whether the following arguments are valid. If any arguments are invalid, state counterexamples to them. If any arguments are valid, explain carefully using the truth tables why they are valid.
\begin{enumerate}
\item
\begin{equation*}
\begin{fitch}
\fh P \to Q \\
\fa \lnot P \lor Q \\
\end{fitch}
\end{equation*}
\item
\begin{equation*}
\begin{fitch}
\fh P ↔ (Q \to Q) \\
\fa P \lor Q \\
\end{fitch}
\end{equation*}
\item
\begin{equation*}
\begin{fitch}
\fh P \lor \lnot(Q \land R) \\
\fa P \lor (\lnot Q \land R) \\
\end{fitch}
\end{equation*}
\end{enumerate}

substituion of equivalents

No lecture this Thursday

 

Revison: Proofs (propositional)

 
\section{Revison: Proofs (propositional)}
 
\section{Revison: Proofs (propositional)}
\begin{enumerate}
\item
\begin{equation*}
\begin{fitch}
\fh \lnot P \land R \\
\fa \lnot P \\
\end{fitch}
\end{equation*}
\item
\begin{equation*}
\begin{fitch}
\fh \lnot P \lor R \\
\fa P \to R \\
\end{fitch}
\end{equation*}
\end{enumerate}
 

Revison: Proofs (with quantifiers)

 
\section{Revison: Proofs (with quantifiers)}
 
\section{Revison: Proofs (with quantifiers)}
\begin{enumerate}
\item
\begin{equation*}
\begin{fitch}
\fh ∀x S(x) \\
\fh ∀x \lnot S(x) \\
\fa ⊥ \\
\end{fitch}
\end{equation*}
\item
\begin{equation*}
\begin{fitch}
\fh ∀x ( F(x) → x=a ) \\
\fa ¬∃x ( F(x) ∧ ¬x=a ) \\
\end{fitch}
\end{equation*}
\item
\begin{equation*}
\begin{fitch}
\fh ∃x ∀y ( F(y) → ¬G(x,y) ) \\
\fa ∀y ∃x ( F(y) → ¬G(x,y) ) \\
\end{fitch}
\end{equation*}
\end{enumerate}

No lecture this Thursday

 

Revison: Translation from English to awFOL

 
\section{Revison: Translation from English to awFOL}
 
\section{Revison: Translation from English to awFOL}
\emph{Exercise} Translate the following sentences of English into awFOL using the interpretation below:
\hspace{5mm} L(x,y) : x is a logical consequence of y
\hspace{5mm} N(x,y) : x is the negation of y
\hspace{5mm} S(x) : x is a sentence
\hspace{5mm} a : ‘Fire melts ice’
i. ‘Fire melts ice’ is a sentence
ii. There is a sentence
iii. There is a sentence which is the negation of ‘Fire melts ice’
iv. Some sentences are contradictions and all contradictions are logically equivalent.

i. ‘Fire melts ice’ is a sentence

S(a)

ii. There is a sentence

∃y S(y)

iii. There is a sentence which is the negation of ‘Fire melts ice’

∃y ( S(y) ∧ N(y,a) )

iv. Some sentences are contradictions and all contradictions are logically equivalent.

This kind of thing is just here to make you cry like a baby; you’re not supposed to be able to get 100% without being exceptionally good at alogic.
But let’s look at how you could get something approximately this complex ...

i. Some sentences are logically equivalent.

∃x ∃y ( S(y) ∧ S(x) ∧ LgcllyEqvlnt(x,y) )

LgcllyEqvlnt(x,y) = L(x,y) ∧ L(y,x)

ii. Some sentences are contradictions.

∃y ( S(y) ∧ Contradiction(y) )

Contradiction(y) = ∀x L(y,x)

iii. All contradictions are logically equivalent.

∀x ∀y ( Contradiction(x) ∧ Contradiction(y) → LgcllyEqvlnt(y) )

 

Proof Example: ∃x Dead(x) ⊢ ¬∀x¬ Dead(x).

 
\section{Proof Example: ∃x Dead(x) ⊢ ¬∀x¬ Dead(x).}
 
\section{Proof Example: ∃x Dead(x) ⊢ ¬∀x¬ Dead(x).}
13.43--13.45
 

Proof Example: ¬∀x Dead(x) ⊢ ∃x¬ Dead(x).

 
\section{Proof Example: ¬∀x Dead(x) ⊢ ∃x¬ Dead(x).}
 
\section{Proof Example: ¬∀x Dead(x) ⊢ ∃x¬ Dead(x).}
13.49--13.50
 
\section{The End Is Near}
\emph{Reading:} §14.3
 
\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