Now let us formally present the syntax of MPL.

MPL contains the following symbols:

1. Predicate letters : A, B, C, ... Z. If a subscript is added to a predicate letter, the result is a predicate letter, e.g. A1 , B274, etc.

2. Constants : a, b, c, ... t. If a subscript is added to a constant, the result is a constant, e.g. a1, b24, etc.

3. Variables : u, v, w, x, y, z. If a subscript is added to a variable, the result is a variable, e.g. x1, y24, etc.

4. Five sentential connectives :

5. Open and close brackets : ( )

An expression of MPL is a string of one or more symbols of MPL.

The syntactic rules, or formation rules, for MPL are as follows :

Rules of formation:
  1. Any predicate letter followed by a constant is a wff.

  2. If φ is a wff, then ~φ is a wff.

  3. If φ and ψ are wffs, then (φ&ψ), (φvψ), (φ→ψ), (φ↔ψ) are also wffs.

  4. If φ is a wff that contains any constant ω, and β is a variable which does not occur in φ, then the resulting expression that is obtained by replacing one or more occurrences of ω with β and then attaching "∃β" or "∀β" to the front, is also a wff.

  5. Nothing else is a wff.
Rule #1 tells us that expressions such as "Pa", "Qb", "Gc" are wffs. Rules #2 and #3 are just like rules from SL. We can apply rules #2 and #3 to form new wffs such as "~~Pa", "~((Pa&Sc)↔(~Fb∨Ka))".

Rule #4 is a little complicated. It applies only to wffs that include at least one constant. So it applies to expressions such as "Qa", "(~Pa→Qb)", "(Pa&Qa)", but not "∃xFx". The rule says that we can pick a constant in a wff, replace one or more occurrences of that constant by a single variable, and then add either an existential or universal quantifier to the front. Here are some examples:

From "Qa", we can generate wffs such as "∃xQx", "∀yQy".

From "(~Pa→Sa)", we get "∃y(~Py→Sy)" or "∃x(~Px→Sx)" or "∃x(~Px→Sa)" or "∃x(~Pa→Sx)", or "∀x(~Px→Sx), or "∀y(~Py→Sa)" etc.

From "(~Qa→Qb)", we get "∃x(~Qx→Qb)", "∀z(~Qa→Qz)" (But not "∃x(~Qx→Qx)"!)

Exercises

Question 1 : Are these expressions wffs of MPL?
  1. (P&(P&~R))
  2. ∃x(Mx↔~Qx)
  3. ~∃x~(Ma↔~Qx)
  4. ~(~∀xKx∨∃y~Qy)
  5. ∀x∀y(Qx&Py)
  6. ∀x(Px&∀y~Ly)
  7. ∃x(P→Qa)
  8. ∀x∀yFxy
  9. ((∀xKx&∀yQy)∨Ma)
  10. (∀xKx&((∀yQy)∨Ma))
  11. (∀xKx&∀yQy∨Ma)
  12. ~~~∃x~~~Gx
  13. (~~~∃x~~~Gx&Hx)
  14. ~~∃x(~Gx&Hx)
  15. ∀x((Ga∨Bx)↔(Kb&Gx))
  16. ∀x((Gx∨Bx)↔(Kx&Gx))
  17. ∀x((Ga∨Ba)↔(Ka&Ga))