PREDICATE LOGIC

"Obviously, 'being' is not a true predicate, i.e., a concept of something that can be added to the concept of a thing" (Kant)."(Kant).

"It is for logic to discern the laws of truth" (Frege).

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Fundamental Concepts
First-order predicate logic (LP1) is an extension of propositional logic. The added features are:

• The use of predicates (or attributes) applied to individual elements, so that propositions have structure. The notation used is P(a), or simply Pa, with P being the predicate and a the element to which the predicate applies. For example, "Socrates is mortal" is encoded as:Mortal(Socrates).
  
  A predicate may affect or relate to nelements (a₁, a₂, ... , a_n (n-ary relation). In this case, the notation is P(a₁, a₂, .... , a_n). For example, "Everest is higher than Mont Blanc", where a₁ is "Everest", a₂ is "Mont Blanc" and P is "It is higher than". Therefore, it would be coded as: Is_higher_than(Everest, Mont_Blanc).
  
  
• The use of element variables. For example, P(x) indicates that an element variable x has property P
  
  
• The use of quantifiers. There are two standard quantifiers: the universal (∀) and the existential (∃). These quantifiers are linked to element variables. For example, "All men are mortal" is coded as.
  
  
  ∀x(Hx → Mx)
  
  where H is the predicate "man", M is the predicate "mortal" and the arrow (→) is the logical implication operator. The sentence expresses "To be a man implies to be mortal".
  
  The two quantifiers are dual. This property can be expressed in two ways (where "¬" is the "contrary" or "logical negation" operator):
  
  
  ∀xPx → ¬∃x¬Px
  (if every element has property P, then there is no element that does not have property P)
  
  ∃xPx→ ¬∀x¬Px
  (if there is an element that has the P property, then not all elements do not have the P property).
  
  An alternative notation for quantifiers, also widely used, is Peano's notation for the universal quantifier: (x). And either no alternative notation is used for the existential quantifier (having a derived character) or the form ∃x is used.

Problematic

• In the notation P(a) or P(x) there is no explicit operator (or descriptor) connecting an element (or variable) to the predicate, although it can be interpreted (as Frege did) as the application of a function: the name of the function is P, the argument is a and the result of the function is V (true). Moreover, the logical thing would be for the element to appear first and then the predicate, as in natural language, as in "Socrates is mortal".
  
  
• The notation P(a₁, a₂, ... , a_n) does not describe the structure of the relationship between the elements, i.e., it does not reflect the semantics. The interpretation is of functional applicative type (the P property is the name of the function and the elements are the arguments). For example, "Pepe is David's father": Father(Pepe, David).
  
  The notation P(a₁, a₂, ... , a_n) is only justified when the elements a₁, a₂, ... , a_n share with each other the P property. For example, "David and Eve are siblings": siblings(David, Eve).
  
  
• Variables are element variables only. There are no predicate variables.
  
  
• The quantifiers are very limited. They only allow expressing "All" and "Some", together with their opposites "None" and "not-All". It does not even allow expressing "Some" (i.e., more than one).
  
  In general, possible quantifiers can be classified (in addition to the universal and existential) into:
  
  
  • Concrete. These are the ones that refer to numbers. Such as "3 men", "12 women", etc.
    
    
  • Ambiguous or diffuse. Such as "Many men", "Few men", "Quite a few men", "Almost all men", etc.
    
    
  • Concrete-diffuse. Such as "Between 3 and 7 men", "An even number of men", "At least 3 men", "7 men at most", "All men except 2", "About 10 men", etc.
    
    
  • Proportional. They refer to a proportion. Such as "Half as many men."
    
    
  • Proportional-diffuse. Such as "More than half of the men", "About one-third of the men", etc.
  
  Logicians and linguists recognize the limitations of traditional quantifiers, so they have tried to generalize them, although they have done so with different criteria: associating them with higher-order properties, with relations between sets, etc. In general, generalizations have been made in a complex way, at a theoretical level and with little or no practical value. [see Applications - Logic - Generalized Quantifiers].
  
  
• Predicates of predicates (higher-order predicates) are not supported. For example, if A(e) is "The element e is blue", we cannot express that blue is dark (i.e., apply the predicate "dark" to the predicate "blue").
  
  Second-order predicate logic (LP2) admits higher-order predicates along with the possibility of quantifying predicates and using them as variables. But there should be predicates of any order.
  
  
• There is no standard notation for contravariant predicates. For example, "This object is not blue." Not having a predicate is not the same as not having a predicate as the contrary predicate.
  
  
• The subject of predicates is contemplated in the framework of logic, when it is something absolutely general, i.e. a predicate should be able to be applied, not only to logical expressions, but to all kinds of expressions: functions, rules, structures, etc.
  
  
• No distinction is made between intrinsic and extrinsic predicates. Only extrinsic predicates are contemplated, which are attributes or qualities (blue, rich, tall, good, deadly, etc.). Intrinsic predicates are those that an entity possesses by itself. For example, the set {a, b, c} has the intrinsic property of having 3 elements.
  
  
• The scope of a quantifier is sometimes ambiguous, especially when several quantifiers are used in the same expression. The problem lies in the fact that the scope of each identifier is not formally delimited.
  
  
• The LP1 language is descriptive. Since it is not operational, logical expressions are static, they cannot be modified dynamically. Functions and recursive expressions are also not contemplated.
  
  
• LP1 language (like propositional logic) is a particular language, which is not integrated into a general language or a universal language.

MENTAL, a Generalized Predicational Language
Predicative Expressions

• With an argument: P(a) is encoded as a/P (the element a is particularized by the predicate, attribute or quality P This notation corresponds to that of natural language (as in "Socrates is a man"): Socrates/man The "Particularization" primitive is generic; it is applicable to any expression. Its use as an element/predicate is only a particular case.
  
  Functional notation can also be used, but whose result is not V (true), but an element or a set of elements. For example,
  
  
  (Father(David) = Pepe)
Children(Pepe) = {JoseJuan Natalia Eva David}
  
  
• With several arguments: P(a₁, a₂, ... , a_n) is encoded by a multi-level particularization structure with higher-order predicates. Examples:
  
  
  1. "A dark blue object": object/(blue/dark)
     
     
  2. "David and Eva are siblings": {David Eva}/siblings
     
     
  3. "John gave a book to Mary" is a ternary relation (between the elements John, book and Mary), which we can express as a certain event e consisting of four attributes:
     
     ( e/((subject/John) (action/give) (object/"a book") (receiver/Mary)) )
     
     In this case, the attribute has the structure "name/value".
  
  
• The predicate opposite of p is p' (because of the general semantics of the opposite operator). For example, a/(blue') (a is not blue).
  
  
• Recursive expressions are allowed. For example:
  
  x is ancestor (Ant) of y if x is parent (Pro) of y, or else if there is a z that is ancestor of y):
  
  
  ⟨( x/(Ant(y)) ↔ ((x/(Pro(y)) ∨ ⟨z/(Ant(y))⟩ )⟩

Quantifiers

With MENTAL you can specify several types of the quantifiers mentioned above. For example:

• Concretes. They are specified as magnitudes.
  "3 men": 3*man
  
  "3 tall men":
  3*(man/tall)
  
  
• Concrete-diffuse.
  "At least 3 men":
  ( {⟨( x ← x/h )⟩}# ≥ 3 )
  
  "Between 3 and 7 men."
  ( {⟨( x ← x/h )⟩}# ∈ {3...7} )
  (with h being the predicate "man").
  
  
• Proportionals.
  "Most men smoke" (h: predicate "man", f: predicate "smoke", f': predicate "non-smoke"):
  
  ( {⟨( x ← x/h ← x/f )⟩}# &⟨( {⟨( x ← x/h ← x/(f') )⟩}# )
  (The number of men who smoke is greater than those who do not smoke).

Quantifiers restricted to a scope

• Universal quantifier.
  Universal quantifiers are expressed as parameters of generic expressions (which are encoded in bold) and their scope is clearly delimited by the symbols "⟨" and "⟩".
  
  Example: any element x belonging to a set C, satisfies property P
  
  
  Traditional notation:
  ∀x(x∈C → Px).
  
  MENTAL notation:
  ⟨( x∈C → x/P )⟩
  
  
• Existential quantifier.
  Existential quantifiers are expressed as the number of elements of a set that satisfy the property of being greater than zero or being distinct from the empty set.
  
  Examples:
  
  
  1. There exists at least one element x belonging to a set C that satisfies the property P
     
     Traditional notation:
     (∃x)(x∈C ∧ Px)
     
     MENTAL notation:
     {⟨( x∈C ∧ x/P )⟩ }≠∅
     or
     ({⟨( x∈C ∧ x/P )⟩ }# > 0)
     
     
  2. For every natural number n, there exists a larger m number.
     
     Traditional notation:
     ∀n∈N ∃m∈N | m>n
     
     MENTAL notation: ⟨( n → {⟨m>n⟩}≠∅
     
     
  3. Between two distinct real numbers, there is an intermediate number.
     
     Traditional notation:
     ∀r1∀r2(r1∈R ∧ r2∈R) → ∃r3 | r1>r3 ∧ r3<r2.
     
     MENTAL notation: ⟨( r1≠r2 → {⟨( r>r1 ∧ r<r2)⟩ }≠∅

You can simplify the notation by defining (if x is a set);

⟨( ∃x = x≠∅ )⟩ or ⟨( ∃x = (x# & > 0) )⟩

For example, example notation 1 would be: ∃{⟨(x∈C ∧ x/P)⟩}


Generalized quantifiers

Traditional quantifiers are first-order only, i.e., they refer to a number of elements of a set that satisfy a qualitative type property (all or some). In MENTAL, quantifiers can be generalized by the number of entities (elements of a set, sets, sets of sets, etc.) that satisfy a certain selection criterion.

In the following examples, S is a set of sets:

1. Some sets of S have more than n elements.
   
   {⟨( C∈S → (C# > n) )⟩ }>1
   
   
2. All sets of S contain some element with property P

⟨( C∈S → {⟨( x∈C ← x/P )⟩} = C# )⟩
   
   
3. The number of sets containing more than n elements is m

{⟨( C# > n )⟩} = m )⟩

Properties

1. If all elements have property P, then there are elements that have property P

∀xPx → ∃xPx
( ⟨x/P⟩ → {⟨x/P⟩}≠∅ )
   
   
2. If all elements have property P, then there are no elements that do not have property P; and vice versa.
   
   ∀xPx ↔ ¬∃x¬Px
( { ⟨x/P⟩} ↔ {⟨(x ←' x/P)⟩}= ∅ )
   
   
3. If there is an element that has property P, then not all elements do not have property P); and vice versa.
   
   ∃xPx ↔ ¬∀x¬Px
( {⟨(x/P)⟩}≠∅ ↔ {⟨(x ←' x/P)⟩}≠∅ )

Conclusions

MENTAL, as a universal formal language, entails great advantages over traditional predicate logic notation:

• It provides a homogeneous and coherent notation, in an integrated language, which is descriptive and operational.
  
  
• Predicates can be applied to any expression.
  
  
• The universal quantifier is a parameter in generic expressions, so the notation is simplified.
  
  
• Quantifiers and higher-order predicates can be formally specified, using the generic resources of the language.
  
  
• Generalized quantifiers can be defined.

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Addenda
Frege's Conceptography

Frege published in 1879 his revolutionary work entitled "Begriffsschrift" (Conceptography) in which he laid the foundations of modern mathematical logic, ushering in a new era in this discipline that had remained virtually unchanged since Aristotle. In this work he made a decisive contribution: predicate logic, including the quantifiers "All" and "Some" (in modern notation, ∀ and ∃, respectively).

Frege's quantifiers acted on an expression as a totality. Today, it is considered a "scope" on which quantifiers act.

Frege discovered that Aristotle's quantifiers were not relations between elements but between sets of elements. They were dyadic (of two arguments), but that it was possible to define them as monadic (of one argument) using logical implication. For example, in ∀x(Ax → Bx) the operator ∀ is monadic.


Bibliography

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• Groenendijk, J; Stokhof, M. Dynamic Predicate Logic. Linguistics and Philosophy, 14, 39-100, 1991.
  
  
• Lamiquiz Ibáñez, Vidal. La cuantificación lingüística y los cuantificadores. Universidad Nacional de Educación a Distancia, 1990.
  
  
• Leonetti Jungl, Manuel. Los cuantificadores. Arco Libros, 2008.
  
  
• Krynicki, M; Mostowski, M.; Szczerba, L. (edits.). Quantifiers, models and computation. 1995.
  
  
• López Palma, Helena. La interpretación de los cuantificadores, aspectos sintácticos y semánticos. Visor Libros, 2000.
  
  
• Montague, Richard. The proper treatment of quantification in English. Disponible en Internet.
  
  
• Pospesel, Howard. Introduction to Logic: Predicate Logic. Prentice Hall, 2002.