Natural deduction in the original forall x systems
This document gives a short description of how Carnap presents the systems of natural deduction from P.D. Magnus' forall x. At least some prior familiarity with Fitch-style proof systems is assumed.
There are several alternate versions (remixes) of forall x, which use slightly different syntax and/or rules. The versions supported by Carnap are:
- The original version of forall x by P.D. Magnus, described on this page.
- forall x: Calgary
- forall x: Mississippi State
- forall x: Pittsbugh
- forall x: UBC
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 system SL of P.D. Magnus forall
x, activated in Carnap by
Or, with a Fitch-style guides overlay (activated with
There is also a playground mode:
forall x System SL
The minimal system SL for P.D. Magnus' forall x (the system used in
a proofchecker constructed with
.ForallxSL in Carnap's Pandoc
Markup) has the following set of rules for direct
||φ ∧ ψ||φ/ψ|
||φ, ψ||φ ∧ ψ|
||¬ψ, φ ∨ ψ||φ|
|¬φ, φ ∨ ψ||ψ|
||φ||φ ∨ ψ|
|ψ||φ ∨ ψ|
||φ, φ → ψ||ψ|
||φ, φ ↔︎ ψ||ψ|
|ψ, φ ↔︎ ψ||φ|
We also have four rules for indirect inferences:
→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;
↔︎I, 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;
¬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.
¬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.
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
A/NOTETEXTHERE rather than
AS for an assumption.
Assumptions are only allowed on the first line of a subproof.
forall x System SL Plus
The extended system SL Plus for P.D. Magnus' forall x (the system used in a
proofchecker constructed with
.ForallxSLPlus in Carnap's Pandoc
Markup) also adds the
||φ ∨ ψ, φ → χ, ψ → χ||χ|
||φ → ψ, ψ → χ||φ → χ|
||φ → ψ, ¬ψ||¬φ|
As well as the following exchange rules, which can be used within a propositional context Φ:
||Φ(φ ∧ ψ)||Φ(ψ ∧ φ)|
|Φ(φ ∨ ψ)||Φ(ψ ∨ φ)|
|Φ(φ ↔︎ ψ)||Φ(ψ ↔︎ φ)|
||Φ(φ → ψ)||Φ(¬φ ∨ ψ)|
|Φ(¬φ ∨ ψ)||Φ(φ → ψ)|
|Φ(φ ∨ ψ)||Φ(¬φ → ψ)|
|Φ(¬φ → ψ)||Φ(φ ∨ ψ)|
||Φ(φ ↔︎ ψ)||Φ(φ → ψ ∧ ψ → φ)|
|Φ(φ → ψ ∧ ψ → φ)||Φ(φ ↔︎ ψ)|
||Φ(¬(φ ∧ ψ))||Φ(¬φ ∨ ¬ψ)|
|Φ(¬(φ ∨ ψ))||Φ(¬φ ∧ ¬ψ)|
|Φ(¬φ ∨ ¬ψ)||Φ(¬(φ ∧ ψ))|
|Φ(¬φ ∧ ¬ψ)||Φ(¬(φ ∨ ψ))|
The proof system for Magnus's forall x, QL, is activated using
The different admissible keyboard abbreviations for quantifiers and equality is as follows:
The forall x first-order systems do not contain sentence letters.
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 `F1, F2. The arity of a relation symbol is determined from context.
The available constants are a through w, with the infinitely many
subscripted letters c1, c2, … written
The available variables are x through z, with the infinitely many
subscripted letters x1, x2, … written
The first-order forall x systems QL (the systems used in
proofcheckers constructed with
.ForallxQL) extend the rules of the
system SL with the following set of new basic rules:
||σ = σ|
||σ = τ, φ(σ)/φ(τ)||φ(τ)/φ(σ)|
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
It also adds 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.
forall x QL Plus
The system QL Plus, activated with
.ForallxQLPlus, includes all the
rules of SL Plus, as well as the following exchange rules, which can
be used within a context Φ:
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.↩︎