THE PROBLEM OF IMPLICATION

"When our minds are fixed on inference, it seems natural to take 'implication' as the fundamental primitive relationship" (Bertand Russell).

"A logical consequence is a necessary and formal consequence" (Alfred Tarski).

"All theorems in mathematics are implications between statements" (Peano).

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Implication vs. Conditional

In logic, implication is essential, since every logical system makes use of this mechanism. Its simplest form is p→q (read "p implies q"), where p and q are propositions: the antecedent and consequent of the implication, respectively. The natural meaning of implication is that if p is true then q, which is the result of the implication, is inferred. If p is false, nothing is inferred.

• In implication, p and q are related at the logical level; hence, implication is also called "logical implication".
  
  
• In implication, propositions p and q can be specific or generic. An example of specific implicature is: "It is raining" → "I do not leave the house". An example of generic implicature is: "x is male" → "x is mortal".
  
  
• In logic programming languages (such as Prolog), implication is used in the rules that define the logic of inference or deduction.

The conditional is different from the implication. The conditional has the form "if p then q", which we will also symbolize as p→q. The conditional realizes the logic of decision, while the implication realizes the logic of inference.

In procedural or operational programming languages (such as Basic, C, Pascal, etc.), the conditional performs the "decision logic", selecting the block of statements q if the condition p is true. This is the traditional control structure "If.... Then..." (in its simple form) and "If.... Then... Else..." (in its complete form).

In propositional logic, p→q is considered to have a truth value that depends on the truth values of p and q. Its truth table (which comes from Boole) is:

p	q	p→q
F	F	T
F	T	T
T	F	F
T	T	T

That is, the conditional is always true except when p is true and q is false. Therefore, the propositional logic expression p→q is equivalent to ¬p∨q (non-p or q).

This interpretation of the conditional is called "material interpretation" and the conditional is called "material implication" (or weak implication) because it refers to a basic, material, superficial level, as opposed to "logical implication" (or strong implication), which is of a higher type. The material implication used in propositional logic is called "propositional logical implication".

Russell and Whitehead's "Principia Mathematica" is based on material implication.

Some properties of the conditional are:

1. Idempotence.
   p → p (idempotence).
   
   
2. Commutative.
   p → (q → r) ≡ q → (p → r)
   
   
3. Transitive.
   (p → q) → (q → r) → (p → r)
   
   
4. Distributive.
   (p → (q → r)) → ((p → q) → (p → r))

In the conditional, p and q need not be conceptually related. This circumstance makes it possible to construct "shocking" true statements, with strange meanings, without any relation of meaning between antecedent and consequent. The following sentences are formally true, in spite of not having any meaning.

1. "If the world is square, then 2+2=4" is a true sentence because p ("the world is square") is false and "2+2=4" is true.
   
   
2. "If the world is square, 2+2=5" is also true because p ("the world is square") is false and q ("2+2 = 5") is also false.

With material implication paradoxes occur, for example, we have the following tautologies:

1. F → p
   A false proposition (F) materially implies any proposition (true or false).
   
   
2. p → T
   Truth (T) is materially implied by any proposition (true or false).
   
   
3. ¬p ≡ p→F
   The negation of a proposition p is equivalent to the material implication of that proposition to falsity (F).
   
   
4. ¬p ≡ p→p
   The negation of a proposition p is equivalent to its material self-implication.
   
   
5. Other tautologies.
   p→(q→p)
   ¬p → (p→q)
   (p→q)∨(q→p)

Any of these tautologies give rise to paradoxes when interpreted extralogically, i.e. when p and q are sentences of a semantic level higher than the logical scheme itself. For example, the form p→(q→p) is represented in "If 2+2=4, then if the world is square, 2+2=4".


The strict implication of Clarence Irwin Lewis

These problems have led to the need to revise material implication in its propositional logic aspect (not in its decision logic aspect) for true implication to exist. C.I. Lewis offered a solution with the so-called "strict implication", replacing material implication. There is strict implication when, in a conditional, the consequent is inferred from the antecedent, which reflects the natural concept of implication: p strictly implies q if the truth of p is inconsistent with the falsity of q. Lewis expressed it symbolically thus:

¬◇(p∧¬q) (it is not possible that p and notq)

The concept of strict implication was the driving force that inspired Lewis to create modern modal logic. The first formalization of strict implication appeared in the 1932 book "Symbolic Logic" [1959], written in collaboration with C.H. Langford. In this work he introduces the possibility operator (◇) as a primitive modal operator, and presents several axiomatic systems (S1 to S5) of modal logic using strict implication. That there are 5 axiomatic systems is because Lewis discovered that his concept of strict implication was not absolute; that there were five ways of understanding the relation between antecedent and consequent.

Nevertheless, Lewis did not succeed in eliminating paradoxes completely, for he held that it was not possible to eliminate them. For example: every necessary proposition is strictly implied by every other proposition. Saul Kripke, by means of his "semantics of possible worlds", also tried to eliminate the paradoxes of implication, but he did not succeed completely either.

Lewis's notation of the possibility operator (◇) is still standard, although the dual operator is usually used: the necessity operator (□), which is used as a primitive operator for two reasons:

1. Because strict implication is more easily defined: A strictly implies B is written as □(A→B) (it is necessary that A implies B).
   
   
2. Because the possibility operator can be defined by the necessity operator, and vice versa:
   
   ◇x ≡ ¬□¬x (it is possible that x is equivalent to "it is not necessary non-x")
   
   □x ≡ ¬◇¬x (it is necessary that x is equivalent to "it is possible that not x")

It was John Lemmon [1957], who defined Lewis axiomatic systems as extensions of classical logic by means of the necessity operator (□) and a set of axioms.


MENTAL, the Union of Implication and Conditional
MENTAL uses the mechanism of strict implication by means of the "Conditional" primitive, which has the basic form x→y (read "if x then y"), or else y←x (read "y if x"), and whose interpretation is operative: if x exists, the expression x→y evaluates to y; if it does not exist, it evaluates to the null expression (θ). Therefore, a function can be considered to take the value y or θ, depending on whether x exists or not. There may or may not be a relationship between antecedent (x) and consequent (y); it all depends on the specifier.

The "Condition" primitive can be used in two ways:

1. As a specific or particular condition, which implements the conditional to realize the decision logic. For example,
   
   (a>3 → ((b = 5) (x = x+1)))
   
   
2. As a generic condition (specified by a generic expression (parameterized or not), to perform the inference logic as the logic of necessity, which is performed automatically. Examples:
   
   
   1. ⟨( x/man → x/mortal )⟩
pepe/man // automatically generates pepe/mortal
      
      
   2. ⟨( a>3 → a=3 )⟩
(a = 7) // automatically, a takes the value 3
a // ev. 3

The "Condition" primitive of MENTAL can be simple or compound, i.e., be of the form x→y (if x exists, then y; otherwise θ, the null expression) or of the complete form (z ←' x → y) (if x exists, then y otherwise, z). These two forms can be used for decision logic and inference logic.

Example of decision logic:

(b=0 ← a>3 →' b=5)

Example of inference logic:

⟨( (a = 0) ←' (a > 3) → (a = 5) )⟩
(a = 4) // ev. a=5
(a = 2) // ev. a=0

In both types of logic there can be higher-order conditions, and all kinds of expressions can be specified in the antecedent and consequent.

In MENTAL there are no truth values (true, false), there is evaluation of expressions and existential values: existential expression (α) and null expression (θ). With MENTAL it is emphasized that it makes no sense to apply a truth value to a condition or a logical implication.

With MENTAL there are no logical paradoxes and there is no need to eliminate them, since they are a superficial phenomenon. Interpreted through the primitives, they take the form of fractal expressions. From the deep, paradoxes disappear.

With MENTAL the problem of implication and the duality "Implication vs. Conditional" is clarified and unified:

• "Conditional" is always interpreted as "If.... then...".
  
  
• Implication and conditional are forms that depend on their use within a generic or specific expression, respectively.
  
  
• Everything is based on the primitives "Condition" and "Generalization".
  
  
• Implication is a particular case of condition generalization.
  
  
• Material implication may be called "surface implication".

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Addenda
Historical background

• The truth table of material implication is also called "philonic table", because material implication was already known by Philo of Megara (4th century BC). It was widely used by the Stoics and by some medieval logicians. Then it fell into oblivion, until it was used by Frege and Peirce.
  
  
• George Edward Moore, in his 1920 paper "External and Internal Relations" proposed the term "entailment" to refer to a logical or strong implication, as opposed to the material or weak implication employed by Russell and Whitehead in Principia Mathemathica. This proposal of Moore's was purely conceptual and he did not go so far as to formalize it.
  
  
• One of the serious analyses and efforts to define the concept of entailment is perhaps due to Georg Henrik von Wright [1962], who attempted to define this concept on the basis of the notions of demonstrability and possibility.
  
  
• Another attempt to formalize entailment is due to Alan Ross Anderson and N.D. Belnap, who in 1966 in their "Pure Calculus of Entailment" developed an axiomatic system, managing to eliminate from the theorems the paradoxical formulas of material implication.
  
  
• Using the same concept of strict implication, there was another attempt to avoid the paradoxes of material implication, due to David Hilbert and Wilhelm Ackermann, who in their 1928 work "Elements of Theoretical Logic" [1962] succeeded in eliminating the paradoxical theorems by certain formal adjustments in the axioms.

Bibliografía

• Bochenski, Joseph M. Historia de la Lógica Formal. Editorial Gredos, 1985.
  
  
• Castrillo Criado, Pilar. La estructura de los condicionales, la implicación material y sus alternativas. UNED (Universidad Nacional de Educación a Distancia), 1999.
  
  
• Ferrater Mora, José; Leblanc, Hugues. Lógica Matemática. Fondo de Cultura Económica, 1962.
  
  
• Hilbert, David; Ackermann, Wilhelm. Elementos de la Lógica Teórica. Tecnos. Madrid, 1962.
  
  
• Lemmon, John. New foundations for Lewis modal systems. Journal of Symbolic Logic, 22:176-186, 1957.
  
  
• Lemmon, John; Scott, Dana. An introduction to modal logic. Blackwell, 1977.
  
  
• Lewis, C.I.; Langford, C.H. Symbolic Logic. The Century Company, 1932 (2a. ed., Dover, 1959).
  
  
• Lewis, C.I. The Calculus of Strict Implication. Mind, vol. 23, no. 90, April 1914, pp. 240-247.
  
  
• Moore, Geoge Edward. External and Internal Relations. Proceedings of the Aristotelian Society. 1919-1920. London, pp. 291-295 y 301.
  
  
• Wright, Georg Henrik von. Logical Studies. Routledge and Kegan Paul, London, 1962.