Models of Propositional Logic

“We have to live today by what truth we can get today and be ready tomorrow to call it falsehood.”
—William James

Is the WFF

(P\lor \lnot Q)

true? It depends on what we mean by $P$ and $Q$ , which depends on the situation!

The technical name for a situation in logic is a model (though sometimes it is called an assignment or an interpretation).

It turns out that in Classical Propositional Logic, if we want to know whether a WFF like $(P\lor \lnot Q)$ is true, we don’t need to know exactly which propositions our propositional variables $P$ and $Q$ stand for, e.g.,

2+2=4.
3+4=99.
The moon is made of green cheese.
It’s raining outside

All that matters is whether $P$ and $Q$ represent true propositions or false propositions.

Definition

A model in Classical Propositional Logic is an assignment of truth values (T or F) to all relevant propositional variables.

Example

If our WFFs contain only the propositional variables $P$ and $Q$ , then there are four possible models:

$P$ is true, $Q$ is false
$P$ is false, $Q$ is true
$P$ is false, $Q$ is false
$P$ is true, $Q$ is true

If the WFFs we care about instead contained variables $R_1$ , $R_2$ , and $R_3$ , then there would be eight ( $2^3$ ) possible models.

Truth of a WFF in a Model

We cannot say that a WFF like $(P\land Q)$ is true or false until we specify the model.

In Classical Propositional Logic, the truth of a formula is determined by the standard truth table operations.

Conjunction (And)

Propositions of the form $({\cal A}\land {\cal B})$ (pronounced ” $\cal A$ and $\cal B$ ”) are called conjunctions.

Definition

A WFF of the form $({\cal A}\land {\cal B})$ is true in a model when both $\cal A$ and $\cal B$ are true in the model:

$A\cal$	$\cal B$	$({\cal A}\land{\cal B})$
T	T	T
T	F	F
F	T	F
F	F	F

Example

In the model

$P$ is true, $Q$ is false, $R$ is true

the WFF $(P\land R)$ is true, since both parts of the conjunction are true in this model;
the WFF $(P\land (Q\land R))$ is false, since $(Q\land R)$ is false in this model.

Disjunction (Or)

Propositions of the form $({\cal A}\lor {\cal B})$ (pronounced ” ${\cal A}$ or ${\cal B}$ ”) are called disjunctions.

Definition

A WFF of the form $({\cal A}\lor {\cal B})$ is true in a model when either ${\cal A}$ and ${\cal B}$ or both are true.

${\cal A}$	${\cal B}$	$({\cal A}\lor {\cal B})$
T	T	T
T	F	T
F	T	T
F	F	T

Example

In the model where $P$ is true, $Q$ is false, $R$ is true:

the WFF $(P\lor R)$ is true, since both parts of the disjunction are true;
the WFF $(P\lor Q)$ is true, since the first part of the disjunction is true;
the WFF $(Q\lor (R\land Q))$ is false, since both $Q$ and $(R\land Q)$ are false in this model.

Negation (Not)

Propositions of the form $\lnot {\cal A}$ (pronounced “not ${\cal A}$ ”) are called negations.

Definition

A WFF of the form $\lnot {\cal A}$ is true in a model when ${\cal A}$ is false in that model:

${\cal A}$	$\lnot {\cal A}$
T	F
F	T

Example

In the model where $P$ is true, $Q$ is false, $R$ is true:

the WFF $\lnot P$ is false, since $P$ is true in the model;
the WFF $\lnot\lnot P$ is true, since $\lnot P$ is false in the model;
the WFF $(\lnot P\lor \lnot Q)$ is true, since $\lnot Q$ is true.

Implication (If … then …)

Propositions of the form $({\cal A}\to {\cal B})$ (pronounced ” ${\cal A}$ implies ${\cal B}$ ”) are called implications.

In an implication $({\cal A}\to {\cal B})$ , we call ${\cal A}$ the premise and ${\cal B}$ the conclusion. (Others call ${\cal A}$ the antecedent and ${\cal B}$ the consequent.)

Definition

An implication of the form $({\cal A}\to {\cal B})$ is true in a model if ${\cal A}$ is false in the model or ${\cal B}$ is true in the model.

Equivalently, the implication $({\cal A}\to {\cal B})$ is only false when ${\cal A}$ is true in the model and ${\cal B}$ is false in the model.

${\cal A}$	${\cal B}$	$({\cal A}\to {\cal B})$
T	T	T
T	F	F
F	T	T
F	F	T

Example

In the model where $P$ is true, $Q$ is false, $R$ is true:

the WFF $(P\to R)$ is true, since $P$ and $R$ are true;
the WFF $(P\to Q)$ is false, since $P$ is true but $Q$ is false;
the WFF $((P\to Q)\to R)$ is true in the model, since $(P\to Q)$ is false and $R$ is true;

Implication defined as in this truth table is officially called material implication, and is the if-then relationship used in math and logic. Unlike the use of “if…then…” in everyday speech, there is no need for a cause-effect relationship between the premise and conclusion.

Example

The following are considered logically true statements in the model corresponding to the real world:

“If $2+2=4$ , then the moon is not made of green cheese.”
(true implies true)
“If the moon is made of green cheese, then $2+2=4$ .”
(false implies true)
“If the moon is made of green cheese, then the moon is made of spam.”
(false implies false)
“If you pick a guinea pig up by the tail, then its eyes will fall out.”
(guinea pigs don’t have tails, and false implies anything)
“If you scare a pregnant guinea pig, then her babies will be born without tails.”
(guinea pigs don’t have tails, and anything implies true)

$\top$ (“top”) and $\bot$ (“bottom”)

Definition

$\top$ (pronounced “top”) is the WFF that is true in every model.

$\bot$ (pronounced “bottom”) is the WFF that is false in every model.

Example

In the model where $P$ is true, $Q$ is false, $R$ is true:

the WFF $(P\to \bot)$ is false, since $P$ is true and $\bot$ is false;
the WFF $(R\to(Q\lor\top))$ is true, since $R$ is true and $(Q\lor\top)$ is true.

We use $\top$ and $\bot$ when writing logical formulas (WFFs) as strings of symbols, but write T and F when speaking philosophically of meanings, models, and truth or falsity.

Models of Propositional Logic

Definition

Example

Truth of a WFF in a Model

Conjunction (And)

Definition

Example

Disjunction (Or)

Definition

Example

Negation (Not)

Definition

Example

Implication (If … then …)

Definition

Example

Example

⊤\top⊤ (“top”) and ⊥\bot⊥ (“bottom”)

Definition

Example

$\top$ (“top”) and $\bot$ (“bottom”)