19 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
19.1 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 \(P_1, P_2,\ldots\) 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 .ForallxSL:
Or, with a Fitch-style guides overlay (activated with guides="fitch"):
There is also a playground mode:
19.2 Sentential logic
19.2.1 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 inferences:
| Rule | Abbreviation | Premises | Conclusion |
|---|---|---|---|
| And-Elim | ∧E |
\(φ∧ψ\) | \(φ/ψ\) |
| And-Intro | ∧I |
\(φ,ψ\) | \(φ∧ψ\) |
| Or-Elim | ∨E |
\(¬ψ, φ∨ψ\) | \(φ\) |
| \(¬φ, φ∨ψ\) | \(ψ\) | ||
| Or-Intro | ∨I |
\(φ\) | \(φ∨ψ\) |
| \(ψ\) | \(φ∨ψ\) | ||
| Conditional-Elim | →E |
\(φ,φ→ψ\) | \(ψ\) |
| Biconditional-Elim | ↔︎E |
\(φ, φ↔ψ\) | \(ψ\) |
| \(ψ, φ↔ψ\) | \(φ\) | ||
| Reiteration | R |
\(φ\) | \(φ\) |
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 \(ψ\),\(¬ψ\).
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.
19.2.2 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 following rules:
| Rule | Abbreviation | Premises | Conclusion |
|---|---|---|---|
| Dilemma | DIL |
\(φ∨ψ,φ→χ,ψ→χ\) | \(χ\) |
| Hypothetical Syllogism | HS |
\(φ→ψ, ψ→χ\) | \(φ→χ\) |
| Modus Tollens | MT |
\(φ→ψ,¬ψ\) | \(¬φ\) |
As well as the following exchange rules, which can be used within a propositional context Φ:
| Rule | Abbreviation | Premises | Conclusion |
|---|---|---|---|
| Commutativity | Comm |
\(Φ(φ∧ψ)\) | \(Φ(ψ∧φ)\) |
| \(Φ(φ∨ψ)\) | \(Φ(ψ∨φ)\) | ||
| \(Φ(φ↔ψ)\) | \(Φ(ψ↔φ)\) | ||
| Double Negation | DN |
\(Φ(φ)/Φ(¬¬φ)\) | \(Φ(¬¬φ)/Φ(φ)\) |
| Material Conditional | MC |
\(Φ(φ→ψ)\) | \(Φ(¬φ∨ψ)\) |
| \(Φ(¬φ∨ψ)\) | \(Φ(φ→ψ)\) | ||
| \(Φ(φ∨ψ)\) | \(Φ(¬φ→ψ)\) | ||
| \(Φ(¬φ→ψ)\) | \(Φ(φ∨ψ)\) | ||
| BiConditional Exchange | ↔︎ex |
\(Φ(φ↔ψ)\) | \(Φ(φ→ψ∧ψ→φ)\) |
| \(Φ(φ→ψ∧ψ→φ)\) | \(Φ(φ↔ψ)\) | ||
| DeMorgan’s Laws | DeM |
\(Φ(¬(φ∧ψ))\) | \(Φ(¬φ∨¬ψ)\) |
| \(Φ(¬(φ∨ψ))\) | \(Φ(¬φ∧¬ψ)\) | ||
| \(Φ(¬φ∨¬ψ)\) | \(Φ(¬(φ∧ψ))\) | ||
| \(Φ(¬φ∧¬ψ)\) | \(Φ(¬(φ∨ψ))\) |
19.3 Quantificational logic
The proof system for Magnus’s forall x, QL, is activated using .ForallxQL.
19.3.1 Notation
The different admissible keyboard abbreviations for quantifiers and equality is as follows:
| Connective | Keyboard |
|---|---|
| ∀ | A |
| ∃ | E |
| = | = |
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 \(F_1, F_2, \ldots\) written `F_1, F_2. The arity of a relation symbol is determined from context.
The available constants are a through w, with the infinitely many subscripted letters \(c_1, c_2, \ldots\) written c_1, c_2,….
The available variables are \(x\) through \(z\), with the infinitely many subscripted letters \(x_1, x_2,\dots\) written x_1, x_2,….
19.3.2 Basic Rules
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:
| Rule | Abbreviation | Premises | Conclusion |
|---|---|---|---|
| Existential Introduction | ∃I |
\(φ(σ)\) | \(∃xφ(x)\) |
| Universal Elimination | ∀E |
\(∀xφ(x)\) | \(φ(σ)\) |
| Universal Introduction | ∀E |
\(φ(σ)\) | \(∀xφ(x)\) |
| Equality Introduction | =I |
\(σ=σ\) | |
| Equality Elimination | =E |
\(σ=τ,φ(σ)/φ(τ)\) | \(φ(τ)/φ(σ)\) |
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.
19.3.3 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:
| Rule | Abbreviation | Premises | Conclusion |
|---|---|---|---|
| Quantifier Negation | QN |
\(Φ(∀x¬φ(x)))\) | \(Φ(¬∃xφ(x)))\) |
| \(Φ(¬∀xφ(x)))\) | \(Φ(∃x¬φ(x)))\) | ||
| \(Φ(∃x¬φ(x)))\) | \(Φ(¬∀xφ(x)))\) | ||
| \(Φ(¬∃xφ(x)))\) | \(Φ(∀x¬φ(x)))\) |
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.↩︎