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\title {Logic I \\ Lecture 13}

Logic I

Lecture 13

\def \ititle {Logic I}
\def \isubtitle {Lecture 13}
\textbf{\ititle}: \isubtitle
\iemail %
Readings refer to sections of the course textbook, \emph{Language, Proof and Logic}.
10.24–7, *10.28–9
\section{There Is a Store for Everything}
\emph{Reading:} §11.2, §11.3
\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
11.4, 11.8, 11.9
11.11, 11.13, *11.10
11.8, 11.9, *11.11

How Big Is a Truth-Table?

\section{How Big Is a Truth-Table?}
\section{How Big Is a Truth-Table?}
How many truth-functions can be constructed using 2 sentence letters?

Truth-functional completeness

\section{Truth-functional completeness}
\emph{Reading:} §7.4
\section{Truth-functional completeness}
‘A set of truth-functors is said to be \emph{expressively adequate} (or sometimes \emph{functionally complete}) iff, for every truth-function whatever, there is a formula containing only those truth-functors which express that truth-function, i.e. which has as its truth-table the truth-table specifying that function.’ (Bostock, \emph{Intermediate Logic} p. 45)
Illustration of the proof that $\{$¬, ∧, ∨$\}$ is truth-functionally complete:
\emph{Exercise} assuming $\{$¬,∨,∧$\}$ is truth-functionally complete, show that $\{$¬,∨$\}$ is.
7.25, 7.26, *7.28, 7.29
7.25, 7.26, *7.28, 7.29