eolas/zk/Validity_and_entailment.md

---
tags: [propositional-logic]
---

## Validity

### Informal definition

In order to say whether an argument is 'good' or 'bad' we must have criteria of
evaluation. in logic there are different criteria of evaluation:

- **Deductive validity**

  An **argument is deductively valid if and only if it is not possible for the
  premises to be true and the conclusion false**. Linking to consistency: it is
  not possible to consistently assert all of the premises but deny the
  conclusion.

- **Inductive strength**

  We do not say that inductive arguments have 'validity' because despite
  inductive premises being true, the conclusion may be falsifiable. Therefore we
  say inductive 'strength' rather than 'validity'. An argument is inductively
  strong if and only if the conclusion is probably true given the premises.

#### Demonstration

The Socrates demonstration above is an example of deductive validity.

The following is an example of an argument that is inductively strong:

```
99% of deaf persons have no musical talent.
Beethoven was deaf.
___________________________________________
Beethoven had no musical talent.
```

The test for a strong inductive argument is not whether the conclusion is true,
rather it concerns the evidence the premises provide in support of the
conclusion.

> In propositional logic we are concerned solely with deductive validity or
> invalidity.

### Formal definition

> An argument is truth-functionally valid if and only if there is no
> truth-assignment on which all the premises are true and the conclusion is
> false.

Linking this to derivation, we say:

> In a system of derivation in propositional logic, an argument is valid if the
> conclusion of the argument is derivable within the system of derivation from
> the set consisting of the premises, and invalid otherwise.

### Demonstration

The inference from the set ${P, P \rightarrow Q}$ to $Q$ is valid

### Truth-table

| $P$ | $Q$ | $P \rightarrow Q$ | $P$ | $Q$ | Assessment |
| --- | --- | ----------------- | --- | --- | ---------- |
| T   | T   | T                 | T   | T   | Valid      |
| T   | F   | F                 | T   | F   |            |
| F   | T   | T                 | F   | T   |            |

## Entailment

### Informal definition

Entailment as a concept is almost identical to validity. We say that a
proposition is entailed by a set of propositions if it is not possible for every
member of this set to be true and the proposition to be false.

The difference with validity resides in the fact that the propositions are
distinguished in terms of whether they are premises or a conclusion. So,
technically, validity is a subclass of entailment. A case of entailment where we
distinguish propositions in terms of whether they are premises or conclusions. A
proposition may be entailed by a given set without that proposition being the
_conclusion_ of the set and where the set is a syllogism.

### Formal definition

> A finite set of sentences $\Gamma$ $\vdash$ $P$ if and only if there is no
> truth-assignment in which every member of $\Gamma$ is true and $P$ is false.

#### Informal demonstration

```
It is raining.
If it is raining then the pavement will be wet.
The pavement is wet.
```

#### Formal demonstration

$$
  \{ P, P\rightarrow Q   \} \vdash Q
$$

#### Truth-table

| $P$ | $Q$ | $P \rightarrow Q$ | $P$ | $Q$ | Assessment |
| --- | --- | ----------------- | --- | --- | ---------- |
| T   | T   | T                 | T   | T   | Valid      |
| T   | F   | F                 | T   | F   |            |
| F   | T   | T                 | F   | T   |            |