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:

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:

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

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

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:

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,….

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

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