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"
|align="center" bgcolor="#FFEBAD" colspan="4"|<math>P \to Q, \neg Q \vdash \neg P</math>
|+Suppes–Lemmon style proof (first example)
|-
!Assumption set
!Line number
!Formula (''wff'')
!Annotation
|-
|bgcolor="#bbffbb"|1
|bgcolor="#bbffbb"|1
|bgcolor="#bbffbb"|<math>P \to Q</math>
|bgcolor="#bbffbb"|A
|-
|bgcolor="#bbffbb"|2
|bgcolor="#bbffbb"|2
|bgcolor="#bbffbb"|<math>\neg Q</math>
|bgcolor="#bbffbb"|A
|-
|bgcolor="#bbffbb"|3
|bgcolor="#bbffbb"|3
|bgcolor="#bbffbb"|<math>P</math>
|bgcolor="#bbffbb"|A
|-
|bgcolor="#bbffbb"|1, 3
|bgcolor="#bbffbb"|4
|bgcolor="#bbffbb"|<math>Q</math>
|bgcolor="#bbffbb"|1, 3 →E
|-
|bgcolor="#bbffbb"|1, 2
|bgcolor="#bbffbb"|5
|bgcolor="#bbffbb"|<math>\neg P</math>
|bgcolor="#bbffbb"|2, 4 RAA
|-
|align="center" bgcolor="#BBBBFF" colspan="4"|Q.E.D
|-
|}
This proof will become clearer when the inference rules and their appropriate annotations are specified – see {{section link||Propositional inference rules (Suppes–Lemmon style)}}.