# Positive Propositional Logic

## Notes

Hackstaff calls this system P+ (P sub plus), not to be confused with Church's P+

## Based on (Hilbert)

• Definitions (none)
• Rules
• Implication
• And
• Or
• PL6: p>(p+q)
• PL7: q>(p+q)
• PL8: (p>r)>((q>r)>((p+q)>r))
• Equivalence
• PL9: (p==q)>(p>q)
• PL10: (p==q)>(q>p)
• PL11: (p>q)>((q>p)>(p==q))

[Hilbert and Bernays, 1934, p68-71, plus table on p66]

## Based on (Hilbert II)

• Definitions (none)
• Rules
• Implication
• And
• Or
• PL6: p>(p+q)
• PL7: q>(p+q)
• PL8: (p>r)>((q>r)>((p+q)>r))
• Equivalence
• PL9: (p==q)>(p>q)
• PL10: (p==q)>(q>p)
• PL11: (p>q)>((q>p)>(p==q))

[Hilbert und Ackermann, 1959, p39]

## Based on (Hackstaff)

Note that this is (by inspection), the system ICI plus the Hilbert Axioms (above) for the individual operators other than implication. (Hackstaff specificly acknowledges the axioms from Hilbert) -JH

• Definitions (none)
• Rules
• Implication
• PL1: p>(q>p)
• PL2: (p>(q>r))>((p>q)>(p>r))
• And
• Or
• PL6: p>(p+q)
• PL7: q>(p+q)
• PL8: (p>r)>((q>r)>((p+q)>r))
• Equivalence
• PL9: (p==q)>(p>q)
• PL10: (p==q)>(q>p)
• PL11: (p>q)>((q>p)>(p==q))

[Hackstaff, 1966, p50]

## Basis for

PPL + appropriate axioms for NOT gives standard PC. [Hackstaff, 1966, p95]

PPL + appropriate axioms for NOT gives Intuitionist PC. [Hackstaff, 1966, p223]

PPL + appropriate axioms for NOT gives Fitch Calculus. [Hackstaff, 1966, p223]

PPL + appropriate axioms for NOT gives Johansson Minimal Calculus [Hackstaff, 1966, p229]