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
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}, … 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:
Sentential logic
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.
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)) |
Quantificational logic
The proof system for Magnus's forall x, QL, is activated using .ForallxQL
.
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}, … 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 c_{1}, c_{2}, … written c_1, c_2,…
.
The available variables are x through z, with the infinitely many
subscripted letters x_{1}, x_{2}, … written x_1, x_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.
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.↩︎