# Natural Deduction in Logic Book Systems

This document gives a short description of how Carnap presents the systems of natural deduction from Bergmann Moore and Nelson's Logic Book.

## Propositional Systems

### Notation

The different admissible keyboard abbreviations for the different connectives are as follows:

Connective Keyboard
->, =>,>
& /\, &, and
\/, |, or
↔︎ <->, <=>
~ -, ~, not

The available sentence letters are A through Z, together with the infinitely many subscripted letters P1, P2, … written P_1, P_2 and so on.

Proofs consist of a series of lines. A line is either an assertion line containing a formula followed by a : and then a justification for that formula, or a separator line containing two dashes, thus: --.1 A justification consists of zero or more numbers (citations of particular lines) and pairs of numbers separated by a dash (citations of a subproof contained within the given line range), followed by the name of a rule being applied.

A subproof is begun by increasing the indentation level. The first line of a subproof should be more indented than the containing proof, and the lines directly contained in this subproof should maintain this indentation level. (Lines indirectly contained, by being part of a sub-sub-proof, will need to be indented more deeply.) The subproof ends when the indentation level of the containing proof is resumed; hence, two contiguous sub-proofs of the same containing proof can be distinguished from one another by inserting a separator line between them at the same level of indentation as the containing proof. The final unindented line of a derivation will serve as the conclusion of the entire derivation.

Here's an example derivation, using system SD from the Logic Book

Playground
P :AS P :AS P :2 R P > P :2-3 CI P > (P > P) :1-4 CI

and here is one using system PD

Playground
(Ax)Fx :PR (Ax)(Fx > Gx) :PR Fa :1 AE Fa > Ga :2 AE Ga :3,4 >E (Ax)Gx :5 AI

### Basic Rules

#### Logic Book System SD

The minimal system SD from the Logic Book (the system used in a proofchecker constructed with .LogicBookSD in Carnap's Pandoc Markup) has the following set of rules for direct inferences:

Rule Abbreviation Premises Conclusion
And-Elim. &E φ&ψ φ/ψ
And-Intro. &I φ, ψ φ&ψ
Or-Intro \/I φ/ψ φ ∨ ψ
Conditional-Elim ->E.CE φ, φ ⊃ ψ ψ
Biconditional-Elim <->E,BE φ/ψ, φ ↔︎ ψ ψ/φ
Reiteration R φ φ

We also have five rules for indirect inferences:

1. ->I (also denoted CI), which justifies an assertion of the form φ ⊃ ψ by citing a subproof beginning with the assumption φ and ending with the conclusion ψ;
2. <->I, (also denoted BI) which justifies an assertion of the form φ↔︎ψ by citing two subproofs, beginning with assuptions φ, ψ, respectively, and ending with conclusions ψ, φ, respectively;
3. ~I, which justifies an assertion of the form  φ by citing a subproof beginning with the assumption φ and ending with a pair of lines ψ, ψ.
4. ~E, which justifies an assertion of the form φ by citing a subproof beginning with the assumption  φ and ending with a pair of lines ψ, ψ.
5. ∨E, which justifies an assertion of the form χ by citing a line φ∨ψ, and two subproofs beginning the assumptions φ, ψ respectively and both ending with a line χ.

Finally, PR or Assumption (to keep with the Logic Book terminology) can be used to justify a line asserting a premise, and AS can be used to justify a line making an assumption. A/SOMEADDITIONALTEXT (where SOMEADDITIONALTEXT indicates some additional text to be included alongside the assumption in the rendered proof) can also be used to justify an assumption. Assumptions are only allowed on the first line of a subproof.

#### Logic Book System SD Plus

The extended system SD Plus (the system used in a proofchecker constructed with .LogicBookSDPlus in Carnap's Pandoc Markup) also adds the following rules:

Rule Abbreviation Premises Conclusion
Modus Tollens MT φ ⊃ ψ,  ∼ ψ  ∼ φ
Hypothetical Syllogism HS φ ⊃ ψ, ψ ⊃ χ φ ⊃ χ
Disjunctive Syllogism DS  ∼ ψ, φ ∨ ψ φ
∼ φ, φ ∨ ψ ψ

As well as the following exchange rules, which can be used within a propositional context Φ:

Rule Abbreviation Premises Conclusion
Commutation Comm Φ(φ&ψ) Φ(ψ&φ)
Φ(φ ∨ ψ) Φ(ψ ∨ φ)
Association Assoc Φ(φ&(ψ&χ)) Φ((φ&ψ)&χ)
Φ((φ&ψ)&χ) Φ(φ&(ψ&χ))
Φ(φ ∨ (ψ ∨ χ)) Φ((φ ∨ ψ) ∨ χ)
Φ((φ ∨ ψ) ∨ χ) Φ(φ ∨ (ψ ∨ χ))
Implication Impl Φ(φ ⊃ ψ)/Φ( ∼ φ ∨ ψ) Φ(φ ⊃ ψ)/Φ( ∼ φ ∨ ψ)
Double Negation DN Φ(φ)/Φ( ∼  ∼ φ) Φ( ∼  ∼ φ)/Φ(φ)
Idempotence Idem Φ(φ) Φ(φ&φ)
Φ(φ&φ) Φ(φ)
Φ(φ) Φ(φ ∨ φ)
Φ(φ ∨ φ) Φ(φ)
DeMorgan's Laws DeM Φ( ∼ (φ&ψ)) Φ( ∼ φ ∨  ∼ ψ)
Φ( ∼ (φ ∨ ψ)) Φ( ∼ φ& ∼ ψ)
Φ( ∼ φ ∨  ∼ ψ) Φ( ∼ (φ&ψ))
Φ( ∼ φ& ∼ ψ) Φ( ∼ (φ ∨ ψ))
Transposition Trans Φ(φ ⊃ ψ) Φ( ∼ ψ ⊃  ∼ φ))
Φ( ∼ ψ ⊃  ∼ φ)) Φ(φ ⊃ ψ)
Exportation Exp Φ(φ ⊃ (ψ ⊃ χ)) Φ(φ&ψ ⊃ χ)
Φ(φ&ψ ⊃ χ) Φ(φ ⊃ (ψ ⊃ χ))
Distribution Dist Φ(φ&(ψ ∨ χ)) Φ((φ&ψ) ∨ (φ&χ))
Φ((φ&ψ) ∨ (φ&χ)) Φ(φ&(ψ ∨ χ))
Φ(φ ∨ (ψ&χ)) Φ((φ ∨ ψ)&(φ ∨ χ))
Φ((φ ∨ ψ)&(φ ∨ χ)) Φ(φ ∨ (ψ&χ))
Equivalence Equiv Φ(φ ↔︎ ψ) Φ(φ ⊃ ψ&ψ ⊃ φ)
Φ(φ ⊃ ψ&ψ ⊃ φ) Φ(φ ↔︎ ψ)
Φ(φ ↔︎ ψ) Φ((φ&ψ) ∨ ( ∼ ψ& ∼ φ))
Φ((φ&ψ) ∨ ( ∼ ψ& ∼ φ)) Φ(φ ↔︎ ψ)

## First-Order Systems

### Notation

The different admissible keyboard abbreviations for quantifiers are as follows:

Connective Keyboard
A
E

The Logic Book first order systems contain sentence letters A through Z, together with the infinitely many subscripted letters P1, P2, … written P_1, P_2 and so on.

Application of a relation symbol is indicated by directly appending the arguments to the symbol.

The available relation symbols are A through Z, together with the infinitely many subscripted letters F1, F2, … written F_1, F_2,…. The arity of a relation symbol is determined from context.

The available constants are a through v, with the infinitely many subscripted letters c1, c2, … written c_1, c_2,….

The available variables are w through z, with the infinitely many subscripted letters x1, x2, … written x_1, x_2,….

Quantificational phrases are formed by appending a variable to a quantifier, and wrapping the result in parentheses.

### Basic Rules

The first-order Logic Book systems PD and PD+ (the systems used in proofcheckers constructed with .LogicBookPD, and LogicBookPDPlus respectively) extend the rules of the system SD and SD+ respectively with the following set of new basic rules:

Rule Abbreviation Premises Conclusion
Existential Introduction EI φ(σ) (∃x)φ(x)
Universal Elimination AE (∀x)φ(x) φ(σ)
Universal Introduction AI φ(σ) (∀x)φ(x)

Where Universal Introduction is subject to the restriction that σ must not appear in φ(x), or any undischarged assumption or in any premise of the proof.2

It also adds one new rule for indirect derivations: EE, which justifies an assertion ψ by citing an assertion of the form (∃x)φ(x) and a subproof beginning with the assumption φ(σ) and ending with the conclusion ψ, where σ does not appear in ψ, (∃x)φ(x), or in any of the undischarged assumptions or premises of the proof.

To these rules, PD+ also adds the exchange rule:

Rule Abbreviation Premises Conclusion
Quantifier Negation QN Φ( ∼ (∀xφ)(x)) Φ((∃x) ∼ φ(x))
Φ((∃x) ∼ φ(x)) Φ( ∼ (∀x)φ(x))
Φ( ∼ (∃xφ)(x)) Φ((∀x) ∼ φ(x))
Φ((∀x) ∼ φ(x)) Φ( ∼ (∃x)φ(x))

1. These are intended for use when you have two contiguous subproofs and need to separate them - since they're at the same level of indentation, you need some extra indication to show that they're distinct subproofs.↩︎

2. Technically, Carnap checks only the assumptions and premises that are used in the derivation of φ(σ). This has the same effect in terms of what's derivable, but is a little more lenient.↩︎