eolas/zk/Logical_truth_and_falsity.md

---
tags:
  - propositional-logic
  - logic
---

# Logical truth and falsity

We say of certain propositions that they are logically true or logically false.

## Logical falsity

### Informal definition

A proposition is logically false if and only if **it is not possible for the
proposition to be true**. The proposition itself cannot be consistently
asserted.

**Demonstration**

```
There is a country that is not a country.
Apples are fruits and apples are not fruits
```

Neither proposition can be true because the truth of the first clause is
contradicted by the second. By the principle of
[consistency](Logical_consistency.md), it is not
possible for both clauses to be true at once therefore the proposition, overall
has the truth value of false.

```
It is raining and it is not raining.
```

### Formal definition

> A proposition P is truth-functionally false if and only if P is false on every
> truth-value assignment

### Formal expression

$$ P \land \lnot P $$

### Truth-table

| $P$ | $P \land \lnot P$ |
| --- | ----------------- |
| T   | F                 |
| T   | F                 |

## Logical truth

### Informal definition

A proposition is logically true if and only if it is not possible for the
proposition to be false. The proposition itself cannot be consistently denied.

**Demonstration**

```
A rose is a rose.
Today is Tuesday unless today is not Tuesday.
```

Regardless of any facts obtaining in the world, these propositions cannot be
false.should be avoided in arguments, they 'prove' everything whi

As with logically false propositions, logical truth can also apply to compound
propositions:

```
A rose is a rose and a shoe is a shoe
```

### Formal definition

> A proposition P is truth-functionally true if and only if P is true on every
> truth-value assignment

$$ P \lor \lnot P$$

### Truth-table

| $P$ | $P \lor \lnot P$ |
| --- | ---------------- |
| T   | T                |
| F   | T                |

### Consequences

The existence of logically false and logically true propositions affects the
validity and soundness of arguments in which they are used. These are
technicalities that have philosophically interesting consequences.

- If an argument contains premises which are logically false than this argument
  will perforce be valid. This is because one cannot consistently assert the
  premises and deny the conclusion which is the definition of
  [validity](Validity_and_entailment.md). However the
  _reason_ why one cannot consistently assert the premises and deny the
  conclusions is because one cannot consistently assert the premises - they
  conflict with each other. Furthermore as the argument contains false premises,
  it cannot be sound.

  ```
  (P1) Russia is a country.
  (P2) Russia is not a country
  (P3) All countries have languages.
  ____________________________________________
  (C) Russian is a language.
  ```

- Any argument with a logically true conclusion is valid. Because the conclusion
  cannot be consistently denied it follows that we cannot consistently assert
  the premises _and_ deny the conclusion. Whether or not the argument is sound
  remains an open question however. If the premises happen to be true then the
  argument will be sound on the strength of the conclusion being logically true
  but if the premises are false it will be unsound regardless of the truth of
  the conclusion.

  ```
  (P1) Horses have legs.
  (P2) Animals with legs can move.
  ____________________________________________
  (C) A horse is a horse
  ```