Logic or Logics? - Universidad EAFIT

Logic or Logics?
Andrés Sicard-Ramírez
Formal methods seminar
EAFIT University
August 25, 2009
(Last modification: 17th January 2016)
Motivation
'
/ Non-classical
Classical p
logic g
logics
Universal
logic?
O
(New) problems/solutions
Non-classical logics
Is there a definition?
Sources
(Non-exhaustive!)
Reject of the classic logic principles
Reduction of the classic logical constants
Expansion of the classical logical constants
Reject of the classical properties of the consequence relation
Modifications to the mathematical structure of the classical
consequence relation
Non-classical logics (cont.)
Notation
For
𝛼, 𝛽, 𝛿, …
Δ, Γ, …
¬, ∧, ∨, →
⊢, ⊧, ⊩
Set of well-formed formulae
Formulas
Theories (⊆ For)
logical connectives
Consequence relations
Reject of the principle of bivalence
Principle of bivalence
Every proposition is either true or false
Many-valued logics
The number of truth values is not restricted to only two (Rescher 1969).
Truth values (Peña 1993, pp. 33-35).
designed
anti-designed
designed and anti-designed
neither designed nor anti-designed
no designed
no anti-designed
Nicholas Rescher (1969). Many-Valued Logic.
Lorenzo Peña (1993). Introducción A Las Lógicas No Clásicas.
Reject of the principle of bivalence (cont.)
Semantical universe
0: Minimal element, anti-designed and no designed
1: Maximal element, designed and no anti-designed
≤: partial order
∀𝛼 (0 ≤ /𝛼/ ≤ 1), where /𝛼/ is the truth-value of 𝛼
Reject of the principle of bivalence (cont.)
Example (Kleene’s 𝐾3 logic (Epstein 1990))
⎧1: True (designed)
{
Semantic universe ⎨ 12 : Undefined (anti-designed)
{0: False (anti-designed)
⎩
1
¬
0
∧
1
1
1
1
2
1
2
1
2
1
2
0
1
0
0
1
2
1
2
1
2
0
0
0
0
0
∨
1
1
2
0
1
1
1
1
1
2
0
1
→
1
1
2
1
2
1
2
1
2
1
0
0
A feature
There is not 𝛼 such that ⊧𝐾3 𝛼.
Richard L. Epstein (1990). The Semantic Foundations of Logic.
1
1
1
1
1
2
1
2
1
2
1
0
0
1
2
1
Reject of the principle of explosion
Principle of explosion (pseudo-Scotus, ex contradictione sequitur quod
libet)
∀Γ ∀𝛼 ∀𝛽 (Γ, 𝛼, ¬𝛼 ⊢𝐶𝐿 𝛽).
Paraconsistent logics (Bobenrieth 1996; Carnielli and Marcos 2002)
∃Γ ∃𝛼 ∃𝛽 (Γ, 𝛼, ¬𝛼 ⊮𝑃 𝛽).
M. Andrés Bobenrieth (1996). ¿Inconsistencias, Por Qué No?
Walter A. Carnielli and João Marcos (2002). A Taxonomy of C-systems.
Reject of the principle of explosion (cont.)
Example (da Costa’s 𝐶1 logic)
Bivalent semantic for 𝐶1 (Marcos 1999, p. 47):
A valuation for 𝐶1 is a function 𝜐 ∶ For(𝐶1 ) → {0, 1} such that:
𝜐(𝛼 ∗ 𝛽) has classical behavior (∗ ∈ {∧, ∨, →})
𝜐(𝛼) = 0 ⇒ 𝜐(¬𝛼) = 1,
𝜐(¬¬𝛼) = 1 ⇒ 𝜐(𝛼) = 1
A consequence
The semantic for 𝐶1 is not truth-functionality:
𝜐(𝛼) = 1 ⇏ 𝜐(¬𝛼) = 1,
𝜐(𝛼) = 1 ⇏ 𝜐(¬𝛼) = 0.
João Marcos (1999). Semânticas de Traduções Possíveis.
Reject of the principle of explosion (cont.)
Example (da Costa’s 𝐶1 logic (cont.))
A feature
def
The logic 𝐶1 admits a strong negation ∼ 𝛼 = ¬𝛼 ∧ 𝛼∘ , where ∘ is the
well-behavior operator. The negation ∼ is a classical negation.
Reject of the principle of the excluded third
Principle of the excluded third
∀𝛼 (⊢𝐶𝐿 𝛼 ∨ ¬𝛼).
Intuitionistic logics (Sørensen and Urzyczyn 2006; van Dalen 2004)
Computational meaning of the logical constants
Proofs are constructions (programs)
Dirk van Dalen (2004). Logic and Structure.
Morten-Heine Sørensen and Paul Urzyczyn (2006). Lectures on the Curry-Howard
Isomorphism.
Reject of the principle of the excluded third (cont.)
The Brouwer-Heyting-Kolmogorov (BHK) interpretation
A construction of
𝛼1 ∧ 𝛼2
𝛼1 ∨ 𝛼2
𝛼1 → 𝛼2
⊥
¬𝛼
∃𝑥 ∈ 𝑈 .𝜙(𝑥)
∀𝑥 ∈ 𝑈 .𝜙(𝑥)
Consists of
A construction of 𝛼1 and a construction of 𝛼2
An indicator 𝑖 ∈ {1, 2} and a construction of 𝛼𝑖
A method (function) which takes any construction
of 𝛼1 to a construction of 𝛼2
There is not construction
A method (function) which takes any construction
of 𝛼 into a nonexistent object
An element 𝑎 ∈ 𝑈 and a construction of 𝜙(𝑎)
A method (function) which takes any element 𝑥 ∈ 𝑈
to a construction of 𝜙(𝑥)
Reject of the principle of the excluded third (cont.)
A feature
The proofs by contradiction are invalids
Prove 𝛼
¬𝛼
⋮
⊥
¬𝛼 → ⊥ ≡ ¬¬𝛼
𝛼
But, disprove 𝛼 is valid
𝛼
⋮
⊥
𝛼 → ⊥ ≡ ¬𝛼
Invalid!
Expansion of the logical constants
Modal logics (Hughes and Cressivell 1998)
(□ ∶ necessity,♢ ∶ possibility)
Temporal logics
Epistemic logics
Deontic logics
G. E. Hughes and M. J. Cressivell (1998). A New Introduction to Modal Logic.
Reduction of the logical constants
Possible reductions (Rasiowa 1974)
Positive logics
Implicative logics
…
General question
What is a logical constant? (for example {¬, ∧, ∨, →})
Helena Rasiowa (1974). An Algebraic Approach to Non-Classical Logics.
A general definition of logic?
Definition
A logic 𝔏 is a structure 𝔏 =< For, ⊩> where the consequence relation
⊩⊆ 𝑃 (For) × For satisfies (Béziau 2000; Carnielli and Marcos 2002):
Reflexivity: If 𝛼 ∈ Γ, then Γ ⊩ 𝛼
Monotony: If Γ ⊩ 𝛼 and Γ ⊆ Δ, then Δ ⊩ 𝛼
Transitivity: If Γ ⊩ 𝛼 and Δ, 𝛼 ⊩ 𝛽, then Γ, Δ ⊩ 𝛽
Jean-Yves Béziau (2000). What is Paraconsistent Logic?
Walter A. Carnielli and João Marcos (2002). A Taxonomy of C-systems.
Walter Carnielli and Mariana Matulovic (2015). The Method of Polynomial Ring
Calculus and its Potentialities.
A general definition of logic?
Definition
A logic 𝔏 is a structure 𝔏 =< For, ⊩> where the consequence relation
⊩⊆ 𝑃 (For) × For satisfies (Béziau 2000; Carnielli and Marcos 2002):
Reflexivity: If 𝛼 ∈ Γ, then Γ ⊩ 𝛼
Monotony: If Γ ⊩ 𝛼 and Γ ⊆ Δ, then Δ ⊩ 𝛼
Transitivity: If Γ ⊩ 𝛼 and Δ, 𝛼 ⊩ 𝛽, then Γ, Δ ⊩ 𝛽
Remark: A ‘Tarksian logic’ is a logic whose consequence relation satisfies
the above three properties (Carnielli and Matulovic 2015).
Jean-Yves Béziau (2000). What is Paraconsistent Logic?
Walter A. Carnielli and João Marcos (2002). A Taxonomy of C-systems.
Walter Carnielli and Mariana Matulovic (2015). The Method of Polynomial Ring
Calculus and its Potentialities.
Reject of the properties of the consequence relation
Non-reflexivity logics
Alfabar logics (Krause and Béziau 1997)
Example
Let 𝔏 =< For, ⊩> be a logic such that Γ ⊩ 𝛼 iff exists Γ′ such that
Γ′ ⊆ Γ and Γ′ is consistent, and Γ′ ⊢𝐶𝐿 𝛼. Therefore, 𝑝 ∧ ¬𝑝 ⊮ 𝑝 ∧ ¬𝑝.
Non-monotonic logics
“family of formal frameworks…in which reasoners draw conclusions
tentatively, reserving the right to retract them in the light of further
information.” (Strasser and Antonelli 2014)
Non-transitive logics?
Décio Krause and Jean-Yves Béziau (1997). Relativizations of the Principle of
Identity.
Christian Strasser and G. Aldo Antonelli (2014). Non-monotonic Logic.
Modifications to the mathematical structure of the
consequence relation
Multiple consequence
⊩⊆ 𝑃 (For) × 𝑃 (For)
Sub-structural logics (Restall 2004)
Multi-set ≠ set: ({𝐴, 𝐴, 𝐵} ≠ {𝐴, 𝐵}), therefore 𝛼, 𝛼, 𝛽 ⊩ 𝛾 does not
imply 𝛼, 𝛽 ⊩ 𝛾
𝛼, 𝛽 ⊩ 𝛾 does not imply 𝛽, 𝛼 ⊩ 𝛾
In general, a theory Γ has not to be a set
Greg Restall (2004). An Introduction to Substructural Logics.
Towards an universal logic?
Béziau’s approach
What is…? (Béziau 2002, 2000; Béziau, de Freitas and Viana 2001)
Some questions
Others approach to the consequence relations (e.g. visual inference)
Equivalence criteria between semantics, syntax and algebra for a logic
Jean-Yves Béziau, Renata Pereira de Freitas and Jorge Petrucio Viana (2001). What
is Classical Propositional Logic? (A Study in Universal Logic).
Jean-Yves Béziau (2000). What is Paraconsistent Logic?
Jean-Yves Béziau (2002). Are paraconsistents negations negations?
Towards an universal logic? (cont.)
Some questions (cont.)
Equivalence criteria between logics (e.g. Possible-translation
semantics)
Minimal properties of the logical connectives. (e.g. What is a
negation?)
Compatibility between the logical connectives
High-order logic extensions
Possible applications
Mathematical theories construction (Mortensen 1995)
Hypercomputation (Agudelo and Sicard 2004; Sylvan and Copeland 1998)
“Or maybe paraconsistent logic will save us from the
tricephalous CGC-monster (CGC for Cantor-Gödel-Church)
by providing foundations for finite decidable complete
mathematics.” (Béziau 1999, p. 16)
Type : Type (Martin-Löf 1971, 1984)
Chris Mortensen (1995). Inconsistent Mathematics.
Richard Sylvan and Jack Copeland (1998). Computability is Logic-Relative.
Juan C. Agudelo and Andrés Sicard (2004). Máquinas de Turing Paraconsistentes:
Una Posible Definición.
Jean-Yves Béziau (1999). The Future of Paraconsistent Logic.
Per Martin-Löf (1971). A Theory of Types.
Per Martin-Löf (1984). Intuitionistic Type Theory.
Conclusions
Tolerance principle in Mathematics (Newton da Costa, 1958):
“Desde el punto de vista sintáctico-semántico, toda teoría es
admisible, desde que no sea trivial. En sentido amplio,
existe, en matemática, lo que no sea trivial.” (Bobenrieth
1996, p. 180)
Logical pluralism (Bueno 2002)
A new crisis? New opportunities?
M. Andrés Bobenrieth (1996). ¿Inconsistencias, Por Qué No?
Otávio Bueno (2002). Can a Paraconsistent Theorist be a Logical Monist?
The category of the logics (bonus slides)
Definition
A category 𝐶 is given the following data:
A class of objects 𝑂𝑏𝑗(𝐶)
A class of arrows or morphisms 𝑀 𝑜𝑟(𝐶)
The functions 𝑑𝑜𝑚, 𝑐𝑜𝑑 ∶ 𝑀 𝑜𝑟(𝐶) → 𝑂𝑏𝑗(𝐶)
Notation: 𝑓 ∶ 𝐴 → 𝐵 ≡ 𝑓 ∈ 𝑀 𝑜𝑟(𝐶), 𝑑𝑜𝑚 𝑓 = 𝐴, 𝑐𝑜𝑑 𝑓 = 𝐵
For 𝐴 ∈ 𝑂𝑏𝑗(𝐶), the identity arrow 1𝐴 ∶ 𝐴 → 𝐴
A composition operator ∘ ∶ 𝑀 𝑜𝑟(𝐶) × 𝑀 𝑜𝑟(𝐶) → 𝑀 𝑜𝑟(𝐶)
The category of the logics (cont.)
These data are subject to the conditions:
𝑔 ∘ 𝑓 is defined iff 𝑐𝑜𝑑 𝑔 = 𝑑𝑜𝑚 𝑓
If 𝑔 ∘ 𝑓 is defined, then
𝑑𝑜𝑚(𝑔 ∘ 𝑓) = 𝑑𝑜𝑚 𝑓 and 𝑐𝑜𝑑(𝑔 ∘ 𝑓) = 𝑐𝑜𝑑 𝑔
For any 𝑓 ∶ 𝐴 → 𝐵,
1𝐵 ∘ 𝑓 = 𝑓 and 𝑓 ∘ 1𝐴 = 𝑓
For any 𝑓 ∶ 𝐴 → 𝐵, 𝑔 ∶ 𝐵 → 𝐶, ℎ ∶ 𝐶 → 𝐷,
ℎ ∘ (𝑔 ∘ 𝑓) = (ℎ ∘ 𝑔) ∘ 𝑓
The category of the logics (cont.)
Example (The category Set)
𝑂𝑏𝑗(Set): Sets
𝑀 𝑜𝑟(Set): functions
The identity arrow 1𝐴 : The identity function
The composition operator ∘: The composition of functions
Technical remark
The usual definition of a function 𝑓 ∶ 𝐴 → 𝐵 as a set 𝑓 ⊆ 𝐴 × 𝐵 which is
single-valued and totally defined is not sufficient to uniquely determine
𝑐𝑜𝑑 𝑓. Therefore it is necessary to define 𝑓 as a triple (𝐴, 𝑔𝑟𝑎𝑝ℎ(𝑓), 𝐵).
The category of the logics (cont.)
Example (The category 3)
1𝐵
?𝐵@
 @@@ 𝑔

@@

@@


/𝐶
2𝐴
Q
ℎ
𝑓
1𝐴
1𝐶
The category of the logics (cont.)
Example
Almost every known example of a mathematical structure with the
appropriate structure-preserving map yields a category.
Category
Set
Pfn
Vect
Top
Poset
CPO
Lat
Objects
Sets
Sets
Vector spaces
Topological spaces
Posets
Complete posets
Lattices
Morphisms
Functions
Partial functions
Linear transforms
Continuous functions
Monotone functions
Continuous functions
Structure preserving homomorphisms
The category of the logics (cont.)
Example
A deductive system ⊩𝐷 can be turned on a category D
𝑂𝑏𝑗(D): Formulas
𝑀 𝑜𝑟(D): Proofs
The identity arrow 1𝐴 ∶ 𝐴 → 𝐴: A proof of 𝐴 ⊩𝐷 𝐴
The composition operator ∘: Transitivity of the ⊩𝐷
𝑓 ∶𝐴→𝐵
𝑔∶𝐵→𝐶
𝑔∘𝑓 ∶𝐴→𝐶
References
Juan C. Agudelo and Andrés Sicard. Máquinas de Turing Paraconsistentes:
Una Posible Definición. Matemáticas: Enseñanza Universitaria XII.2
(2004), pp. 37–51. eprint: http://revistaerm.univalle.edu.co.
Jean-Yves Béziau. Are paraconsistents negations negations? In:
Paraconsistency. The Logical Way to the Inconsistent. Ed. by
Walter A. Carnielli, Marcello E. Coniglio and Ítala M. L. D’Ottaviano.
Marcel Dekker, 2002, pp. 465–486.
Jean-Yves Béziau. The Future of Paraconsistent Logic. Logical Studies 2
(1999).
Jean-Yves Béziau. What is Paraconsistent Logic? In: Frontiers of
Paraconsistent Logic. Ed. by Diderik Batens et al. Hertfordshire, England:
Research Studies Press LTD., 2000, pp. 95–111.
Jean-Yves Béziau, Renata Pereira de Freitas and Jorge Petrucio Viana.
What is Classical Propositional Logic? (A Study in Universal Logic).
Logical Studies 7 (2001).
References
M. Andrés Bobenrieth. ¿Inconsistencias, Por Qué No? Santafé de Bogotá:
Tercer Mundo Editores, División Gráfica, 1996.
Otávio Bueno. Can a Paraconsistent Theorist be a Logical Monist? In:
Paraconsistency. The Logical Way to the Inconsistent. Ed. by
Walter A. Carnielli, Marcello E. Coniglio and Ítala M. L. D’Ottaviano.
Marcel Dekker, 2002, pp. 535–552.
Walter A. Carnielli, Marcello E. Coniglio and Ítala M. L. D’Ottaviano, eds.
Paraconsistency. The Logical Way to the Inconsistent. Marcel Dekker,
2002.
Walter A. Carnielli and João Marcos. A Taxonomy of C-systems. In:
Paraconsistency. The Logical Way to the Inconsistent. Ed. by
Walter A. Carnielli, Marcello E. Coniglio and Ítala M. L. D’Ottaviano.
Marcel Dekker, 2002, pp. 1–94.
Walter Carnielli and Mariana Matulovic. The Method of Polynomial Ring
Calculus and its Potentialities. Theoretical Computer Science 606.42–56
(2015).
References
Richard L. Epstein. The Semantic Foundations of Logic. Vol. 1.
Propositional Logics. Kluwer Academic Publishers, 1990.
G. E. Hughes and M. J. Cressivell. A New Introduction to Modal Logic.
Routledge, 1998.
Décio Krause and Jean-Yves Béziau. Relativizations of the Principle of
Identity. Logic Journal of the IGPL 5.3 (1997), pp. 327–338.
João Marcos. Semânticas de Traduções Possíveis. MA thesis. U. Estadual
de Campinas, 1999.
Per Martin-Löf. A Theory of Types. Tech. rep. University of Stockholm,
1971.
Per Martin-Löf. Intuitionistic Type Theory. Bibliopolis, 1984.
Chris Mortensen. Inconsistent Mathematics. Kluwer Academic Publishers,
1995.
References
Lorenzo Peña. Introducción A Las Lógicas No Clásicas. Instituto de
Investigaciones Filosóficas. Colección: Cuadernos. México D.F.: Universidad
Autónoma de México, 1993.
Helena Rasiowa. An Algebraic Approach to Non-Classical Logics. Vol. 78.
Studies in Logic and the Foundations of Mathematics. North-Holland
Publishing Company, 1974.
Nicholas Rescher. Many-Valued Logic. McGraw-Hill, 1969.
Greg Restall. An Introduction to Substructural Logics. Taylor & Francis,
2004.
Morten-Heine Sørensen and Paul Urzyczyn. Lectures on the Curry-Howard
Isomorphism. Vol. 149. Studies in Logic and the Foundations of
Mathematics. Elsevier, 2006.
Christian Strasser and G. Aldo Antonelli. Non-monotonic Logic. The
Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.) 2014.
Richard Sylvan and Jack Copeland. Computability is Logic-Relative.
Philosophy Research Papers, University of Canterbury 5 (1998), pp. 1–16.
References
Dirk van Dalen. Logic and Structure. 4th ed. Springer, 2004.