Editing Natural deduction (section)

=== Suppes–Lemmon notation ===
Many textbooks use [[Suppes–Lemmon notation]],{{sfn|Pelletier|Hazen|2024}} so this article will also give that – although as of now, this is only included for [[Propositional calculus|propositional logic]], and the rest of the coverage is given only in Gentzen style. A '''proof''', laid out in accordance with the [[Suppes–Lemmon notation]] style, is a sequence of lines containing sentences,{{sfn|Allen|Hand|2022}} where each sentence is either an assumption, or the result of applying a rule of proof to earlier sentences in the sequence.{{sfn|Allen|Hand|2022}} Each '''line of proof''' is made up of a '''sentence of proof''', together with its '''annotation''', its '''assumption set''', and the current '''line number'''.{{sfn|Allen|Hand|2022}} The assumption set lists the assumptions on which the given sentence of proof depends, which are referenced by the line numbers.{{sfn|Allen|Hand|2022}} The annotation specifies which rule of proof was applied, and to which earlier lines, to yield the current sentence.{{sfn|Allen|Hand|2022}} Here's an example proof:
{| class="wikitable" style="margin:auto;"
|+Suppes–Lemmon style proof (first example)
!Assumption set
!Line number
!Sentence of proof
!Annotation
|-
|{{EquationRef|1}}
|{{EquationRef|1}}
|<math>P \to Q</math>
|A
|-
|{{EquationRef|2}}
|{{EquationRef|2}}
|<math>-Q</math>
|A
|-
|{{EquationRef|3}}
|{{EquationRef|3}}
|<math>P</math>
|A
|-
|{{EquationRef|1}}, {{EquationRef|3}}
|{{EquationRef|4}}
|<math>Q</math>
|{{EquationRef|1}}, {{EquationRef|3}} →E
|-
|{{EquationRef|1}}, {{EquationRef|2}}
|{{EquationRef|5}}
|<math>-P</math>
|{{EquationRef|2}}, {{EquationRef|4}} RAA
|}
This proof will become clearer when the inference rules and their appropriate annotations are specified – see {{section link||Propositional inference rules (Suppes–Lemmon style)}}.