Modal Logic

Truth-Functionality

The connectives of propositional logic are truth-functional:

This means that the truth of $\phi\wedge\psi$ is fully determined by the truth of $\phi$ and $\psi$.

If a proposition is not truth-functional we can describe it like so:

\[\phi = \begin{cases} p & \text{That happens to be true}\\ \top & \text{That is always true} \end{cases}\]

We are not sure whether $\phi$ will continue to be true.

In modal logic we have an extra unary connective:

\[\phi::=p\mid\neg\phi\mid\phi\wedge\phi\mid\square\phi\]

$\lozenge\phi$ is an abbreviation for $\neg\square\neg\phi$.

$\square \phi$ has many meanings depending on the circumstance:

Circumstance	Meaning
Alethic	$\phi$ is necessarily true
Epistemic	I know that $\phi$ is true
Doxastic	I believe that $\phi$ is true
Temporal	At every time in the future, $\phi$ will be true
Denotic	$\phi$ should be true
Legal	$\phi$ is legally required to be true

$\lozenge \phi$ has many meanings depending on the circumstance:

Circumstance	Meaning
Alethic	$\phi$ is possibly true
Epistemic	As far as I know, $\phi$ might be true
Doxastic	I believe that $\phi$ might be true
Temporal	At some time in the future, $\phi$ will be true
Denotic	$\phi$ is allowed to be true
Legal	it is legal for $\phi$ to be true

Temporal: $\square\lozenge p$:
- At every point in the future, $p$ will be true some late time.
Deontic: $\square p\implies\lozenge p$:
- If $p$ is mandatory then it is also permitted.
Legal: $\neg\lozenge\square\neg p$:
- It is not permitted to forbid $p$.
Epistemic: $\square p\implies\square\square p$:
- If I know $p$, then I know that I know $p$.

“I know that Today is Thursday”:
- $p=\text{today is Thursday}$
- $\square p$
“If it it legal to walk here then is it legal to stand here”:
- $p=\text{walk here}$
- $q=\text{stand here}$
- $\lozenge p\implies\lozenge q$

There are also quotes which can be parsed multiple ways:

When translating an ambiguous sentence to logic, choose a disambiguation. All disambiguations will be acceptable in the exam.