2. Logical connectives

Logical Connectives

Logical connective

A logical connective is an expression used to modify a statement or combine two or more statements.

Negation

Negation

Let $p$ be a proposition. The negation of $p$, which is denoted by $\neg p$, is the proposition that "it is not the case that $p$".

$$v(\neg A)=\{\begin{array}{ll}0,& \text{if }v(A)=1\\ 1,& \text{if }v(A)=0\end{array}$$

$p$	$\neg p$
T	F
F	T

For example:

• Let $p$ be that “Today is Tuesday”.
• $\neg p$ means “It is not the case that today is Tuesday”.
  • Or otherwise “Today is not Tuesday” or “It is not Tuesday today”.

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Conjuction

Conjunction

Let $p$ and $q$ be propositions. The conjunction of $p$ and $q$, denoted by $p\wedge q$, is the proposition that "$p$ and $q$".

$$v(A\wedge B)=\{\begin{array}{ll}1,& \text{if }v(A)=1\text{ and }v(B)=1\\ 0,& \text{if }v(A)=0\text{ or }v(B)=0\end{array}$$

$p$	$q$	$p\wedge q$
T	T	T
T	F	F
F	T	F
F	F	F

For example:

• Let $p$ be that “Today is Tuesday”.
• Let $q$ be that “It is raining today”.
• $p\wedge q$ is that “Today is Tuesday and it is raining today”.

But in English, $p \wedge q$ and $q \wedge p$ do not always mean the same thing.
- "Bob became sick and he visited the doctor."
- "Bob visited the doctor and he became sick."
 
This can be handled by dynamic conjunctions.

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Disjunction

Disjunction

Let $p$ and $q$ be propositions. The disjunction of $p$ and $q$, denoted by $p\vee q$, is the proposition that "$p$ (inclusive) or $q$".

$$v(A\vee B)=\{\begin{array}{ll}1,& \text{if }v(A)=1\text{ or }v(B)=1\\ 0,& \text{if }v(A)=0\text{ and }v(B)=0\end{array}$$

$p$	$q$	$p\vee q$
T	T	T
T	F	T
F	T	T
F	F	F

For example: “Students who have taken either maths of physics A-levels are admitted.”

Exclusive OR

$p$	$q$	$p\oplus q$
T	T	F
T	F	T
F	T	T
F	F	F

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Conditional Statements

Conditional statement

Let $p$ and $q$ be propositions. The conditional statement $p\to q$, is the proposition that "if $p$ then $q$". This is also called an implication.

$$v(A\to B)=\{\begin{array}{ll}1,& \text{if }v(A)=0\text{ or }v(B)=0\\ 0,& \text{if }v(A)=1\text{ and }v(B)=0\end{array}$$

$p$	$q$	$p\to q$
T	T	T
T	F	F
F	T	T
F	F	T

For example:

• If it is sunny today, we will go to the beach.
• If I am elected, I’ll lower taxes.

You may encounter some weird statements which are still true.
- "If the Moon is made of green cheese then $2\times2=5$." $(F \rightarrow F)$.
- "If the Moon is made of green cheese then $2\times2=4$." $(F \rightarrow T)$.
- "If London is in the UK then $2\times2=4$." $(T \rightarrow T)$.
  This is also known as irrelevant logic.

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Biconditional Statements

Biconditional statement

Let $p$ and $q$ be propositions. The biconditional statement $p⟺q$, is the proposition that "$p$ if and only if $q$". This is also called an bi-implication or an equivalence.

$$v(A⟺B)=\{\begin{array}{ll}1,& \text{if }v(A)=v(B)\\ 0,& \text{if }v(A)\ne v(B)\end{array}$$

$p$	$q$	$p⟺q$
T	T	T
T	F	F
F	T	F
F	F	T

There are some implicit uses of biconditional statements in English, for example:
- If you finish your meal then you can have desert, $p \rightarrow q$.
 
But we actually mean $p \iff q$:
- You can have desert if and only if you finish your meal.

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