Hardegree, Set Theory ; Rules of Derivation 1 of 8
1. Sentential Logic
Henceforth, A, B, C, D are closed formulas.
1. Inference Rules
&I &O ∼∼∼&I/O
A A&B A&B ~(A&B) B –––––– –––––– ======–––––– A B A → ~B A&B
vI vO ∼∼∼vI/O
A B A∨B A∨B ~(A∨B) –––––– –––––– ∼A ∼B ======A∨B A∨B –––––– –––––– ∼A & ~B B A
→→→I →→→O ∼∼∼→→→I/O
∼A B A→B A→B ~(A→B) –––––– –––––– A ∼B ======A→B A→B –––––– –––––– A & ~B B ∼A
↔I ↔O ∼∼∼↔I/O
A→B A↔B A↔B ~(A↔B) B→A –––––– –––––– ======–––––– A→B B→A ∼A↔B A↔B
I O DN
A ~∼A ∼A ––– ===== –––– A A
Note: The ~O and ~I rules are combined, using a long equals sign ‘===’. Henceforth, any rule that is displayed with ‘===’ is a bi-directional rule, which can be used both as an in-rule and as an out-rule. Hardegree, Set Theory ; Rules of Derivation 2 of 8
2. Strategic Rules
Direct Derivation (DD) Indirect Derivation (ID) : A DD : A ID | ~A As : A
Conditional Derivation (CD) Tilde Indirect Derivation ( ∼∼∼D) : A→B CD : ∼A ∼D A As A As : B :
Ampersand Derivation (&D) Biconditional Derivation ( ↔D) : A&B &D : A↔B ↔D : A : A→B
: B : B→A
Wedge Indirect Derivation ( ∨∨∨ID) Separation of Cases (SC) D D D D D D : 1 ∨ 2 ∨ ... ∨ k ∨ID 1 ∨ 2 ∨ ... ∨ k ~D1 As : C SC ∼D2 As c1: D1 As … : C ∼D k : c2: D2 As : C
. . . D ck: k As : C
Hardegree, Set Theory ; Rules of Derivation 3 of 8
2. Quantifier Logic (Free Logic Version)
1. Introduction Classical first-order logic is based on the following two presuppositions: (C1) The domain (universe) of discourse is not empty; accordingly, the sentence ‘there is something’ is logically true, even though it is not necessarily true. (C2) Every singular term, no matter how silly, denotes an existing object (i.e., element of the domain). In contrast to classical logic, there is Free Logic , of which there are two variants. The more radical version (Universally-Free Logic) denies both (C1) and (C2). The less radical version of Free Logic denies (C2), but accepts (C1). In what follows, we pursue the more radical variant.
2. Constants In intro logic, the distinction between unquantified variables ("constants") (‘ a’, ‘ b’, ‘ c’, etc.) and proper nouns (‘Jay’, ‘Kay’, ‘the U.S.’, etc.) is not important. By contrast, in free logic, the distinction is very important . In particular, whereas constants always denote existing objects (in the domain of quantification), proper nouns need not denote anything.
In doing derivations in free logic, one treats constants as purely intra-derivational symbols. In particular, we have the following definition.
A constant is an atomic singular-term that is introduced by UD or ∃O.
Ordinarily, any atomic singular-term that occurs in the premises or conclusion is regarded as a proper noun , not a constant. However, in mathematics (including set theory), one often does derivations in which universal quantification is taken for granted, so constants are allowed in premises and show-lines. These are understood as having been introduced by UD.
A constant counts as old precisely when it occurs in a line that is neither boxed nor cancelled; otherwise, it counts as new . (as before) Hardegree, Set Theory ; Rules of Derivation 4 of 8
3. Quantifier Rules
In what follows, Φ is a formula, in which ν is the only variable (if any) that occurs free, and Φ[ε/ν] is the formula that results when ε replaces every occurrence of ν that is free in Φ. An expression is closed iff it contains no free occurrence of any variable. An occurrence of a variable ν is free in expression iff that occurrence does not lie within the scope of an operator binding ν – i.e., ∀ν, ∃ν, or ν.
Universal-Out ( ∀O) Existential-In ( ∃I) ∀νΦ Φ[o/ν] –––––– –––––––– Φ[o/ν] ∃νΦ o is any old constant.
Universal-Derivation (UD) Existential-Out (∃O) : ∀νΦ ∃νΦ |: Φ[n/ν] ––––––– || Φ[n/ν] n is any new constant.
Quantifier-Negation (QN) ~∀νΦ ~∃νΦ ∃ν~Φ ∀ν~Φ
Tilde-Universal-Out ( ∼∀O) Tilde-Existential-Out (∼∃O) ∼∀O = QN+ ∃O ∼∃O = QN+ ∀O ~∀νΦ ~∃νΦ –––––––– –––––––– ∼Φ[n/ν] ∼Φ[o/ν]
n is any new constant. o is any old constant. Hardegree, Set Theory ; Rules of Derivation 5 of 8
3. Identity Logic
Reflexivity (R=) Symmetry (S=) Transitivity (T=) ρ = σ ∅ σ = τ σ = τ ––––– ––––– ––––– σ = σ τ = σ ρ = τ σ, τ, and ρ are any closed singular-terms.
LL
σ = τ σ = τ Φ[σ/ν] Φ[τ/ν] –––––– –––––– Φ[τ/ν] Φ[σ/ν]
Iota-Out ( O) Iota-In ( I)
c = νΦ ∀ν(Φ ↔ ν=c) ––––––––––––– ––––––––––––– ∀ν(Φ ↔ ν=c) c = νΦ
c must be a constant .
5. Short-Cut Rules
1. The Immediate Show-Cancel Rule
If a show-line follows from available lines ( earlier or later !) by a rule, then it can be cancelled by that rule. Annotation: cite the line number(s) and the rule.
2. The Conjunction Rule
Any available conjunctive line (with any number of conjuncts) can be treated as the appropriate number of separate lines, numbered (e.g.) 7a, 7b, 7c. And conversely, any number of available lines can be treated as the corresponding conjunction.
3. Rule-Multiplication
Any one-place rule can be multiplied, provided the particular rule also applies to the intermediate line. For example, ∀O+ ∀O = ∀O2; UD+UD = U2D; ~∃O+ ~∃O = ~∃O2. Hardegree, Set Theory ; Rules of Derivation 6 of 8
4. Further Rule Combinations
∀O, ∀O2, etc. can be combined with →O to produce ∀→O, ∀2→O, etc., and UD, UD2, etc., can be combined with CD to produce UCD, U2CD, etc. Similarly, UD, UD2, etc., can be combined with ↔D to produce UBD, U2BD, etc.
5. Contraposition Rule
For every genuine one-place rule , there is an associated contrapositive rule (–), which is obtained by reversing and negating and the premise and conclusion. NOTE CAREFULLY: ∃O is not a genuine inference rule , but is rather an assumption rule .
6. Alphabetic Variance (AV)
Φ[u] ––––– Φ[v]
Here, u, v are variables, Φ[u] is a formula in which v does not occur, Φ[v] is a formula in which u does not occur, and Φ[v] results when every occurrence of u in Φ[u] is replaced by an occurrence of v.
7. The Rule SL
If a line can be derived from available lines using only SL rules, then it may be written down by the rule SL. This rule may be used in place of any combination of SL rules, including inference rules and show rules.
8. The Rule QL
If a line can be derived from previous available lines using only quantifier rules, but it cannot be derived using just SL rules, then it may be written down by the rule QL. This rule may be used in place of any QL inference rule, as well as any combination of QL rules, including inference rules and show rules. NOTE CAREFULLY: The rule ∃O, and hence ~∀O, are not genuine inference rules, but assumption rules . No instance of ∃O is valid by QL! So when you cite ∃O or ~∀O, do not cite it as QL.
9. The Rule IL
If a line can be derived from previous available lines using only identity rules, but it cannot be derived using just QL rules, then it may be written down by the rule IL. This rule may be used in place of any IL inference rule, as well as any combination of IL rules, including inference rules and show rules.
Hardegree, Set Theory ; Rules of Derivation 7 of 8
6. Theories (including Set Theory)
1. Appealing to a Definition
Sentential definitions may be used like rules; each such definition serves as three different rules; out-rule, in-rule, show-rule. The annotation is the same for all three uses – “Def __”, where ‘__’ is filled by the particular defined symbol; see examples below. A sentential definition is one in which the definiens and definiendum are both sentences (formulas).
General form of rule:
Given an instance of a definition “ Φ1 Φ2”, where Φ1 and Φ2 are formulas ,
Out-Rule:
If one has a line Φ1, then one may infer Φ2.
In-Rule:
If one has a line Φ2, then one may infer Φ1.
Show-Rule:
If one has a show-line “show: Φ1”, then one can resolve this show-line to the show-line “show: Φ2”. Note: there is a single associated box for both show-lines.
Examples from Set Theory:
Def : AB ∀x(x∈A → x∈B) [note: ‘ A’ and ‘ B’ are schematic.] Instance: A∪B B ∩C ∀x(x ∈ A ∪B → x ∈ B ∩C) [note: ‘A’ and ‘B’ are constants.] out: 1. A∪B B ∩C given 2. ∀x(x ∈ A ∪B → x ∈ B ∩C) 1, Def in: 1. ∀x(x ∈ A ∪B → x ∈ B ∩C) given 2. A∪B B ∩C 1, Def show: 1. A∪B B ∩C Def 2. ∀x(x ∈ A ∪B → x ∈ B ∩C) | |
2. Appealing to an Axiom or Previously Proved Theorem
Axioms and previously proved theorems can be treated as (unstated) premises. The annotation cites the axiom/theorem and line numbers (optional when line is directly below).
Example from Set Theory: show: 1. A∪∅ = A AE (axiom of extentionality) 2. ∀x(x ∈ A ∪∅ ↔ x ∈ A) | | Hardegree, Set Theory ; Rules of Derivation 8 of 8
7. Axioms a1. ∀x∀y[∀z(z ∈ x ↔ z ∈ y) → x=y] [Extensionality] a2. ∃x∼∃y[y ∈ x] [Empty Set] a3. ∀x∃y∀z[z ∈ y ↔ ( z ∈ x & )] [[ y nor free in ] [Separation] a4. ∀x∀y∃z∀w(w ∈ z ↔ [ w ∈ x ∨ w ∈ y]) [Simple Unions] a5. ∀x∃y∀z(z ∈ y ↔ z=x) [Singletons] a6. ∀x∃y∀z[z ∈ y ↔ ∃w(w ∈ x & z ∈ w)] [General Unions] a7. ∀x∃y∀z[z ∈ y ↔ ∀w(w ∈ z → w ∈ x)] [Power Sets] 8. Definitions d0.1 a ≠ b ∼[a = b] [negation] d0.2 a ∉ b ∼[a ∈ b] [negation] d1. ∃!ν ∃x∀ν( ↔ ν=x) [unique existence] d2. Σν ∃!S∀ν(ν ∈ S ↔ ) [legitimacy] d3. A ⊆ B ∀x(x ∈ A → x ∈ B) [inclusion] d4. A ⊂ B A ⊆ B & A≠B [proper inclusion] d5. A ⊇ B B ⊆ A [converse inclusion] d6. A ⊃ B B ⊂ A [converse proper inclusion] d7. A ⊥ B ∼∃x(x ∈ A & x ∈ B) [exclusion] d8. {ν:} S∀ν(ν ∈ S ↔ ) [set-abstract] d9.1 {a} { x : x=a} [singleton] d9.2 {a,b} { x : x=a ∨ x=b} [doubleton] d9.3 {a,b,c} { x : x=a ∨ x=b ∨ x=c} [tripleton] etc. d10. { x : x=x} [universal set] d11. ∅ { x : x≠x} [empty set] d12. A∩B { x : x ∈ A & x ∈ B} [simple intersection] d13. A−B { x : x ∈ A & x ∉ B} [set-difference] d14. A∪B { x : x ∈ A ∨ x ∈ B} [simple union] d15. A+B ( A−B)∪(B−A) [Boolean sum] d16. C { x : ∃y(y ∈ C & x ∈ y)} [general union] d17. C { x : ∀y(y ∈ C → x ∈ y)} [gen intersection] d18. (A) { X : X ⊆ A} [power set] 9. Definitions (Sentential Versions) cd8. a ∈ { ν:} [a/ν] cd9.1 a ∈ { b} a=b cd9.2 a ∈ { b,c} a=b ∨ a=c cd9.3 a ∈ { b,c,d} a=b ∨ a=c ∨ a=d etc. cd11. a ∈ ∅ a≠a cd12. a ∈ A∩B a ∈ A & a ∈ B cd13. a ∈ A−B a ∈ A & a ∉ B cd14. a ∈ A∪B a ∈ A ∨ a ∈ B cd16. a ∈ C ∃X(X ∈ C & a ∈ X) cd17. a ∈ C ∀X(X ∈ C → a ∈ X) [provided C≠∅] cd18. A ∈ (B) A ⊆ B