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

\maketitle

# Lecture 6

\def \ititle {Logic (PH133)}
\def \isubtitle {Lecture 6}
\begin{center}
{\Large
\textbf{\ititle}: \isubtitle
}

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

\section{DeMorgan: ¬(A ∧ B) ⫤⊨ ¬A ∨ ¬B}

\section{DeMorgan: ¬(A ∧ B) ⫤⊨ ¬A ∨ ¬B}
$\leftmodels\models$' means is logically equivalent to', so for now `has the same truth table as'.
A $\leftmodels\models$ ¬¬A
¬(A ∧ B) $\leftmodels\models$ (¬A ∨ ¬B)
¬(A ∨ B) $\leftmodels\models$ (¬A ∧ ¬B)
A → B $\leftmodels\models$ ¬A ∨ B
¬(A → B) $\leftmodels\models$ ¬(¬A ∨ B) $\leftmodels\models$ A ∧ ¬B
Here's a useful equivalence: double negations cancel out (at least in logic).

A ⫤⊨ ¬¬A

is logically equivalent to

i.e. has the same truth table as

 A ¬A ¬¬A ​T F T ​F T F
 ¬(A ∧ B) ⫤⊨ (¬A ∨ ¬B) ¬(A ∨ B) ⫤⊨ (¬A ∧ ¬B) A → B ⫤⊨ ¬A ∨ B ¬(A → B) ⫤⊨ ¬(¬A ∨ B) ⫤⊨ A ∧ ¬B
 A B A ∧ B ¬(A ∧ B) ¬A ¬B ¬A ∨ ¬B ​T T T F F F F ​T F F T F T T ​F T F T T F T ​F F F T T T T
 A B ¬(A ∨ B) ¬A ∧ ¬B ​T T F F ​T F F F ​F T F F ​F F T T
 A B A → B ¬A ∨ B ​T T T T ​T F F F ​F T T T ​F F T T
 A B A → B ¬(A → B) ¬(¬A ∨ B) A ∧ ¬B ​T T T F F F ​T F F T T T ​F T T F F F ​F F T F F F
3.19
4.15--18
7.1--7.2, *7.3--7.6
3.19
4.31

## Negation and the arrow: A → ¬B ⊭ ¬(A → B)

\section{Negation and the arrow: A → ¬B ⊭ ¬(A → B)}

\section{Negation and the arrow: A → ¬B ⊭ ¬(A → B)}

A → ¬B ¬(A → B)

is not a logical consequence of

i.e.
 ​ A → ¬B ​ ¬(A → B)
is not a logically valid argument
 A B A → ¬B A → B ¬(A → B) ​T T F T F ​T F T F T ​F T T T F ​F F T T F

A → ¬B ¬(A → B)

#

## ↔ : truth tables and rules

\section{↔ : truth tables and rules}

\section{↔ : truth tables and rules}
 A B A → B B → A (A → B) ∧ (B → A) A ↔ B ​T T T T T T ​T F F T F F ​F T T F F F ​F F T T T T

 ​ 1 A ↔ B ​ 2 A ​ 3 B ↔Elim: 1,2
 ​ 1 A ↔ B ​ 2 B ​ 3 A ↔Elim: 1,2

How do I intro ↔?

 A B A ↔ B ​T T T ​T F F ​F T F ​F F T
 A B A → B ​T T T ​T F F ​F T T ​F F T
 ​ 1 A ​ 2 ... ​ 3 B
 ​ 4 B ​ 5 ... ​ 6 A
7.A ↔ B
 ​ 1 A ​ 2 ... ​ 3 B
4.A → B
#

## All Cats Are Grey

\section{All Cats Are Grey}

\section{All Cats Are Grey}

Everything is broken

∀x Broken(x)

All squares are broken.

∀x ( Square(x)Broken(x) )

Everything of mine is broken.

∀x ( ThingofMine(x)Broken(x) )

Everybody loves chocolate or logic..

∀x ( Person(x)LovesChocolateOrLogic(x) )

## universal:arrow, existential:conjunction

\section{universal:arrow, existential:conjunction}

## Don't use ∃ with →

\section{Don't use ∃ with →}

\section{Don't use ∃ with →}
\begin{minipage}{\columnwidth}
Is true ∃x(Square(x) → Broken(x)) in this world?
\end{minipage}
∃x(Square(x) → Broken(x))
\hspace{3mm} ⫤⊨
∃x(¬Square(x) ∨ Broken(x))
\hspace{3mm} ⫤⊨
∃x(¬Square(x)) ∨ ∃x(Broken(x))

1.

∀x(Square(x) ∧ Broken(x))

2.

∀x(Square(x) → Broken(x))

⫤⊨
3.
∀x(¬Square(x) ∨ Broken(x))

4.
∃x(Square(x) ∧ Broken(x))

5.
∃x(Square(x) → Broken(x))

⫤⊨
6.
∃x(¬Square(x) ∨ Broken(x))

⫤⊨
7.
∃x(¬Square(x)) ∨ ∃x(Broken(x))

nb
∀x(¬Square(x) ∨ Broken(x)) ⊭ ∀x(¬Square(x)) ∨ ∀x(Broken(x))

1. is false

2. is true

4. is false

5. is ???

#
9.10
9.15--9.17
*9.18--9
9.10

## ¬Intro

\section{¬Intro}

\section{¬Intro}
6.7--6.10
*6.11--6.12
6.7--6.12
6.18--6.20
6.24--6.27
*6.40--6.42

Joint Action:

Which forms of shared agency underpin our social nature?

(Autumn Term)

Social Cognition:

What makes others’ minds and actions intelligible to us?

(Spring Term)

Origins of Mind: Philosophical Issues in Cognitive Development (PH357)

(http://origins-of-mind.butterfill.com)

## ¬Intro Proof Example

\section{¬Intro Proof Example}

\section{¬Intro Proof Example}
6.24--6.26

## Subproofs Are Tricky

\section{Subproofs Are Tricky}

\section{Subproofs Are Tricky}
What is wrong with the following apparent proof?

\section{∀Elim}