# Natural Deduction in the forall x: Pittsburgh systems

This document gives a short description of how Carnap presents the systems of natural deduction from forall x: Pittsburgh, the remix by Dimitri Gallow of Aaron Thomas-Bolduc and Richard Zach's Calgary version of P.D. Magnus's forall x.

The syntax of formulas accepted is described in the Systems Reference.

## Truth-functional logic

### 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: `--`. A justification consists of a rule abbreviation followed by 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).

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 the system `.GallowSL`:

Ex
A \/ B:AS A:AS B \/ A:\/I 2 -- B:AS B \/ A:\/I 5 B \/ A:\/E 1,2-3,5-6 (A \/ B) -> (B \/ A):->I 1-7

Or, `.GallowSLPlus` with a Fitch-style guides overlay (activated with `guides="fitch"`):

Playground
A \/ B:AS A:AS B \/ A:\/I 2 -- B:AS B \/ A:\/I 5 B \/ A:\/E 1,2-3,5-6 (A \/ B) -> (B \/ A):->I 1-7

Simple indent guides overlay (activated with `guides="indent"`) with system `.GallowPL`:

Playground
A \/ B:AS A:AS B \/ A:\/I 2 -- B:AS B \/ A:\/I 5 B \/ A:\/E 1,2-3,5-6 (A \/ B) -> (B \/ A):->I 1-7

### The SL systems

The system `.GallowSLPlus` allows all rules below. `.GallowSL` is like `.GallowSLPlus` except it disallows all derived rules, i.e., the only allowed rules are `R`, and the I and E rules for the connectives and for ⊥.

It has the following set of rules for direct inferences:

Basic rules:

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
Contradiction-Intro `⊥I` φ, ¬φ
Contradiction-Elim `⊥E` ψ
Biconditional-Elim `↔︎E` φ/ψ, φ ↔︎ ψ ψ/φ
Reiteration `R` φ φ

Derived rules:

Rule Abbreviation Premises Conclusion
Disjunctive Syllogism `DS` ¬ψφ, φ ∨ ψ φ/ψ
Modus Tollens `MT` φ → ψ, ¬ψ ¬φ
Double Negation Elim. `DNE` ¬¬φ φ
DeMorgan's Laws `DeM` ¬(φψ) ¬φ ∨ ¬ψ
¬(φψ) ¬φ ∧ ¬ψ
¬φ ∨ ¬ψ ¬(φψ)
¬φ ∧ ¬ψ ¬(φψ)

We also have five rules for indirect inferences:

1. `→I`, 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`, which justifies an assertion of the form φ ↔︎ ψ by citing two subproofs, beginning with assumptions 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 conclusion .
4. `∨E`, which justifies an assertion of the form φ by citing a disjunction ψ ∨ χ and two subproofs beginning with assumptions ψ, χ respectively and each ending with the conclusion φ.
5. `¬E` (indirect proof), which justifies an assertion of the form φ by citing a subproof beginning with the assumption ¬φ and ending with a conclusion .
6. `LEM` (Law of the Excluded Middle), which justifies an assertion of the form ψ by citing two subproofs beginning with assumptions φ, ¬φ respectively and each ending with the conclusion ψ. LEM is a derived rule.

Finally, `PR` can be used to justify a line asserting a premise, and `AS` can be used to justify a line making an assumption. A note about the reason for an assumption can be included in the rendered proof by writing `A/NOTETEXTHERE` rather than `AS` for an assumption. Assumptions are only allowed on the first line of a subproof.

## Predicate logic

There are two proof systems corresponding to the Pittsburgh remix of forall x. All of them allow sentence letters in first-order formulas. The available relation symbols are the same as for SL: A through Z, together with the infinitely many subscripted letters F1, F2, … written `F_1, F_2` etc. However, the available constants and function symbols are only a through v, together 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,…`.

Connective Keyboard
`A`, `@`
`E`, `3`

The predicate logic system `.GallowPL` of forall x: Pittsburgh extend the rules of the system `.GallowSL` with the following set of new basic rules:

Rule Abbreviation Premises Conclusion
Existential Introduction `∃I` φ(σ) xφ(x)
Universal Elimination `∀E` xφ(x) φ(σ)
Universal Introduction `∀E` φ(σ) xφ(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.1

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

The proof system `.GallowPLPlus` for the Pittsburgh version of forall x adds, in addition to the (basic and derived) rules of `.GallowPL`, the rules

Rule Abbreviation Premises Conclusion
Conversion of Quantifiers `CQ` ¬∀xφ(x) x¬φ(x)
x¬φ(x) ¬∀xφ(x)
¬∃xφ(x) x¬φ(x)
x¬φ(x) ¬∃xφ(x)

1. 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.↩︎