# Term Logic

## Abstract

**:**

## 1. Terminology

## 2. Language

#### 2.1. Grammar

_{1}, …, β

_{n}as arguments and forming an expression of category α, will be denoted as α⟨β

_{1}… β

_{n}⟩.

#### 2.2. Basic Vocabulary

#### 2.3. Basic Syntax

#### 2.4. Terms

#### 2.5. Sentences

## 3. Axioms

#### 3.1. Propositional Logic Background

#### 3.2. Term-Logical Axioms

#### 3.2.1. Intensional

for = ID a = a (Identity) LEIB a = b → (p[a] → p[b]) (Leibniz) |

#### Justification

for = and term conjunction IDEM aa = a (Idempotence) COMM ab = ba (Commutativity) ASSOC a(bc) = (ab)c (Associativity) |

#### Justification

for = and ’ DN a’’= a (Term Double Negation) |

#### Justification

for =, ’ and term conjunction DIST a(bc)’ = ((ab’)’(ac’)’)’ (Distribution) |

#### Justification

#### 3.2.2. Extensional

for N and Λ HEID NΛ (Heidegger’s Law) |

#### Justification

for N and term conjunction NWK Na → Nab (N-Weakening) |

#### Justification

for N, term conjunction and ’ TNC Naa’ (Term Non-Contradiction) |

#### Justification

NEXH Nab ∧ Nab’ → Na (N-Exhaustion) |

#### Justification

#### 3.3. Definitions

## 4. A Few Theorems

IDAEQ a = b → a ≡ b NC, LEIB, aeq AEQ (Identity entails Equivalence) |

EWK Eab → Ea NWK, contrap., EX (Existential Weakening) |

EEXH Ea → Eab ∨ Eab’ NWK, contrap., EX (Existential Exhaustion) |

TDS Ea ∧ Nab → Eab’ NEXH, contrap., EX (Term Disjunctive Syllogism) |

DIST1 a(b + c) = ab + ac (Distribution, First Form) |

**Proof.**

a(b + c) = a(b’c’)’ ADD = ((ab’’)’(ac’’)’)’ DIST = ((ab)’(ac)’)’ DN = ab + ac ADD |

DIST2 a + bc = (a + b)(a + c) (Distribution, Second Form) ☐ |

**Proof.**

a + bc = (a’(bc)’) ADD = (((a’b’)’(a’c’)’)’)’ DIST = ((a’b’)’(a’c’)’)’’ rewrite = (a’b’)’(a’c’)’ DN = (a + b)(a + c) ADD |

EXCL N(ab)’ ↔ Na’ ∧ Nb’ (Exclusion) ☐ |

**Proof.**

1. | N(ab)’ | A for CP (assumption for conditional proof) |

2. | N(ab)’ → Na’(ab)’ | nwk |

3. | Na’(ab) | nc, nwk, assoc, comm |

4. | Na’(ab)’ | 1, 2, MP |

5. | Na’ | 3, 4, NEXH |

6. | Nb’ | similiter |

7. | Na’ ∧ Nb’ | 5, 6 |

8. | N(ab)’ → Na’ ∧ Nb’ | 1–7 CP |

9. | Na’ ∧ Nb’ | A for CP |

10. | E(ab)’ | A for RAA (assumption for reductio ad absurdum) |

11. | Ea(ab)’ ∨ Ea’(ab)’ | 10, EEXH |

12. | Ea(ab)’ | 2nd disjunct incompatible with Na’ from 9 |

13. | Eab(ab)’ ∨ Eab’(ab)’ | 12, EEXH, term shuffling |

14. | Contradiction: first disjunct by TNC, second contradicts Nb’ from 9 | |

15. | N(ab)’ | 10, 14, reductio |

16. | Na’ ∧ Nb’ → N(ab)’ | 9–15 CP |

17. | N(ab)‘ ↔ Na’ ∧ Nb’ | 8, 16 ☐ |

**Corollary.**

BARBARA b ⊂ c, a ⊂ b ⊢ a ⊂ c |

**Proof.**

1. | b ⊂ c | A |

2. | a ⊂ b | A |

3. | Nbc’ | 1, ALL |

4. | Nab’ | 2, ALL |

5. | Nabc’ | 3, NWK |

6. | Nab’c’ | 4, nwk |

7. | Nac’ | 5, 6, NEXH |

8. | a ⊂ c | 7, all |

DARII b ⊂ c, a Δ b ⊢ a Δ c ☐ |

**Proof.**

1. | b ⊂ c | A |

2. | a Δ b | A |

3. | Nbc’ | 1, all |

4. | Eab | 2, SOM |

5. | Nabc’ | 3, NWK |

6. | Eabc | 4, 5, TDS, DN |

7. | Eac | 6, EWK |

8. | a Δ c | 7, som |

DARAPTI Eb, b ⊂ c, b ⊂ a ⊢ a Δ c ☐ |

**Proof.**

1. | Eb | A |

2. | b ⊂ c | A |

3. | b ⊂ a | A |

4. | Nbc’ | 2, ALL |

5. | Nba’ | 3, ALL |

6. | Eab | 1, 5, tds, dn, comm |

7. | Nabc’ | 4, NWK |

8. | Eabc | 6, 7, TDS, DN |

9. | Eac | 8, EWK |

10. | a Δ c | 9, SOM ☐ |

POSITIVE Eab, Nbc* ⊢ Eac (cf. DARII) |

NEGATIVE Nab*, Nbc ⊢ Nac (cf. CELARENT) |

IMPORT Ea, Nab*, Nbc* ⊢ Eac (cf. BARBARI) |

## 5. Intension and Extension

EXT a ≡ b → a = b |

## 6. Consistency

## 7. Decidability

^{n}individuals.

## 8. Tree Proof Techniques

- In a formula, terms may be replaced by terms identical to them according to the intensional axioms.
- Any sentence N(ab)’ may be replaced by the two sentences Na’, Nb’ by EXCL.
- These and DN may be used to drive term negations inwards so they only occur singly and modify term letters only.
- A branch containing ~Nab may be extended by ~Na, or ~Nb, or both.(In the next two rules, b is a term occurring in the premises but not occurring in a.)
- A branch containing Na may be extended by Nab and by Nab’
- A branch containing ~Na splits and continues with ~Nab in one branch and ~Nab’ in the other.
- Open branches are extended until all variables from the premises occur in any remaining branch, with term negations inmost, i.e., modifying a single term letter or constant term.Branches close under the following conditions:
- The branch contains two contradictory formulas, for example Eab’c and Nab’c
- The branch contains a formula ~NΛ
- The branch contains a formula ~Naa’.

## 9. Diagram Techniques

^{n}cells. Venn’s own curvilinear diagrams are inferior to the rectilinear ones proposed by Lewis Carroll, who ingeniously constructed diagrams for up to eight different simple terms, and indicated how to extend these further [9] (p. 245 ff.: “My Method of Diagrams”. Carroll was incidentally the first to use trees as an aid for solving logic problems: ibid., 279 ff. Since one of his problems (“Froggy’s Problem”, ibid., 338 ff.) is a sorites in 18 terms, which would require a diagram with 262,144 cells, taxing human capacity to solve, further aids were clearly needed.) The method for term logic as for syllogistic is to shade out those cells corresponding to N propositions, and indicate by crosses those cells corresponding to E propositions. The chief difficulty is that an E proposition whose term is not a maximal compound of simple terms and their negations must straddle several cells disjunctively, a problem compounded in any term-logical formula or inference employing disjunction or its equivalent. For this reason, diagrams are practicable only for relatively small and straightforward problems. Trees branch easily, but the only way to branch a diagram is to treat several diagrams disjunctively.

^{n}cells, represents the framework within which N and E propositions employing these variables are to be represented, and is neutral with respect to such propositions. The axioms for identity are then to be understood as indicating different but formally equivalent ways in which cells or groups of cells are indicated. This is why they play a different role in the logic from the N and E propositions.

## 10. Quantifiers

- All morphine is highly addictive
- Some pain medication is morphine:

_{≥2}a ↔ ∃x(Eax ∧ Eax’)

_{=1}a ↔ Ea ∧ ~E

_{≥2}a

## 11. Pegagogical Advantages of Term Logic

## Funding

## Conflicts of Interest

## References

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Category | Index | Expressions | Description |
---|---|---|---|

Monadic Connective | s⟨s⟩ | ~ | Sentential negation |

Dyadic Connectives | s⟨ss⟩ | ∧ ∨ → ↔ | [Standard] |

Term Variables | n | a, b, c, a_{1}, a_{2}, … | |

Term Constant | n | Λ | Empty term |

Monadic Term Functor | n⟨n⟩ | ’ | Term negation |

Dyadic Term Functor | n⟨nn⟩ | [juxtaposition] | Term conjunction |

Monadic Predicate | s⟨n⟩ | N | Non-existence predicate |

Dyadic Predicate | s⟨nn⟩ | = | Identity predicate |

Expression | Meaning | Example |
---|---|---|

Λ | Non-existing thing | |

a’ | non-a | non-animal |

ab | a which is a b | doctor who is a musician |

Na | there are no a | there are no unicorns |

a = b | to be a is (the same thing as) to be b | to be a widow is to be a woman whose husband has died |

Name | Definition | Description | Reading |
---|---|---|---|

UN | V = Λ’ | Universal Term | thing; object |

ADD | a + b = (a’b’)’ | Term Addition | a or b |

EX | Ea ↔ ~Na | Existence | There are a; a exist |

NO | a | b ↔ Nab | Universal Negative | No a are b |

ALL | a ⊂ b ↔ Nab’ | Universal Positive | All a are b |

SOM | a Δ b ↔ Eab | Particular Positive | Some a are b |

AEQ | a ≡ b ↔ Nab’ ∧ Nba’ | Term Equivalence | The a are the b |

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Simons, P.
Term Logic. *Axioms* **2020**, *9*, 18.
https://doi.org/10.3390/axioms9010018

**AMA Style**

Simons P.
Term Logic. *Axioms*. 2020; 9(1):18.
https://doi.org/10.3390/axioms9010018

**Chicago/Turabian Style**

Simons, Peter.
2020. "Term Logic" *Axioms* 9, no. 1: 18.
https://doi.org/10.3390/axioms9010018