eolas/zk/Formal_proofs_in_propositional_logic.md

---
categories:
  - Logic
tags: [propositional-logic]
---

# Formal proofs in propositional logic

When we construct a formal proof in logic we are seeking to show that a certain
proposition is **derivable** from other propositions. We use the words
_derivation_ and _proof_ interchangeably.

> A sentence $P$ is derivable in a system of propositional logic from a finite
> set of sentences if and only if there is a derivation in this system in which
> all and only the members of the set are **primary assumptions** and $P$ is the
> sentence on the last line.

We express the above symbolically as $\Gamma \vdash P$ . (Be careful not to
confuse _derivable_ ($\vdash$) from _entails_ ($\vDash$).

Derivability is therefore a property that a proposition can possess relative to
a set.

For instance to demonstrate derivability for:

$$
\{ \lnot F \lor D, F, D \rightarrow (G \land H) \} \vdash G \land H
$$

We would establish $\lnot F \lor D, F, D \rightarrow (G \land H)$ as primary
assumptions and then, using the derivation rules of the system conclude with
$G\land H$. Every sentence in the derivation is either a **primary assumption**
or an **auxiliary** assumption or justified by the rules of the derivation.

An auxiliary assumption is an assumption belonging to a sub-derivation. Primary
assumptions belong to the main derivation. For any given derivation of the form
$\Gamma \vdash P$ there may be a number of ways of demonstrating the derivation
(more than one application of the rules governing the system) but one alone is
sufficient to establish derivability.

> We will tend to use the terms _derivation_ and _proof_ interchangeably but we
> should note that there is a technical distinction in that a **proof is a
> derivation in which all of the assumptions have been discharged**

## Constructing proofs

We place assumptions above derivations and mark them _A_ in the right-hand
margin. We use a shorthand for the derivation rules and also place these in the
right-hand margin.

We divide assumptions from derivations with a horizontal line. We number each
line and use this to refer to the line we are applying the derivation to.
Sub-proofs follow this structure recursively. This is known as _Fitch notation_

_Schematically_:

![](/img/proofs-drawio-Page-5.drawio.png)

_Applied example_:

![](/img/proofs-drawio-Page-6.drawio.png)

## Sub-proofs

When a sub-proof is terminated, the assumption with which it starts is said to
be _discharged_. It's conclusion can then be drawn upon and invoked within the
main proof by reference to its start and end line number. However statements
within the sub-proof cannot be referred to again from within the main proof,
only its result.

## Derivation rules

Derivation rules are
[syntactic](Syntax_of_sentential_logic.md) rather
than semantic. They are applied on the basis of their form rather than on the
basis of the truth conditions of the sentences they are applied to.

> Derivation rules can be applied without having an interpretation of the
> symbols in mind. A derivation rule tells us that: given a group of symbols
> with a certain structure, we can write down another group of symbols with a
> certain structure.

Each of the main logical connectives has an associated derivation rule. The
binary connectives have pairs of rules, one for the introduction of the
connective and one for its elimination.

The main derivation rules:

- [Negation Introduction](Negation_Introduction.md)
- [Negation Elimination](Negation_Elimination.md)
- [Conjunction Introduction](Conjunction_Introduction.md)
- [Conjunction Elimination](Conditional_Elimination.md)
- [Disjunction Introduction](Disjunction_Introduction.md)
- [Disjunction Elimination](Disjunction_Elimination.md)
- [Conditional Introduction](Conditional_Introduction.md)
- [Disjunction Elimination](Disjunction_Elimination.md)
- [Biconditional Introduction](Biconditional_Introduction.md)
- [Biconditional Elimination](Biconditional_Elimination.md)