# 2.4.3: 2.4.D Tarski’s world and formal structures

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To help you reason about sets of predicate logic statements, or even arguments expressed in predicate logic, we often use a ‘mathematical structure’. For some of these structures a visualisation in the form of **Tarski’s** world can sometimes be useful.

Figure 2.9: An instance of a Tarski World.

What is truth? In 1933, Polish mathematician Alfred Tarski (1901–1983) published a very long paper in Polish (titled Pojeçie prawdy w jez̧ykach nauk dedukcyjnych), setting out a mathematical definition of truth for formal languages. “Along with his contemporary, Kurt Gödel [who we’ll see in Chapter 4], he changed the face of logic in the twentieth century, especially through his work on the concept of truth and the theory of models.”

Source: en.wikipedia.org/wiki/Alfred_Tarski.

In Tarski’s world, it is possible to describe situations using formulas whose truth can be evaluated, which are expressed in a first-order language that uses predicates such as Rightof(*x*, *y*), which means that *x *is situated—somewhere, not necessarily directly—to the right of *y*, or Blue(*x*), which means that *x *is blue. In the world in Figure 2.9, for instance, the formula ∀*x*(Triangle(*x*) → Blue(*x*)) holds, since all triangles are blue, but the converse of this formula, ∀*x*(Blue(*x*) → Triangle(*x*)), does not hold, since object *c *is blue but not a triangle.

Such an instance of Tarski world can be more formally described as a ‘mathematical structure’ (which we refer to as a formal structure occasionally). These structures allow us to evaluate statements in predicate logic as being true or false. To formalise a structure, we need to describe two things: the domain of discourse *D *of the structure and for all of the predicates, for which objects of the domain they are true. We do so using set-notationwhich we discuss in more depth in Chapter 4. The formal description of the structure *S *depicted in Figure 2.9 is:

• *D *= {*a*,*b*,*c*,*d*,*e*}

• *Blue \(^{S}\) *= {*b*, *c*}

• *Gray \(^{S}\)*= {*a*,*d*}

• *Red \(^{S}\)*= {*e*}

• *Square \(^{S}\)** *= {*a*}

• *Triangle \(^{S}\) *= {*b*}

• *Circle \(^{S}\) *= {*c*,*d*,*e*}

• *RightOf \(^{S}\) *= {(*b*,*a*),(*c*,*a*),(*d*,*a*),(*e*,*a*), (*b*,*c*),(*d*,*c*),(*b*,*e*),(*c*,*e*),(*d*,*e*)}

• *BelowOf \(^{S}\) *={(*a*,*c*),(*a*,*d*),(*a*,*e*),(*b*,*c*),(*b*, *d*), (*b*, *e*), (*c*, *d*), (*c*, *e*)}

• *LeftOf \(^{S}\) *= {(*a*,*b*),(*c*,*b*),(*e*,*b*),(*a*,*c*), (*e*, *c*), (*a*, *d*), (*c*, *d*), (*e*, *d*), (*a*, *e*)}

• *AboveOf\(^{S}\) *= {(*c*,*a*),(*c*,*b*),(*d*,*a*),(*d*,*b*), (*d*, *c*), (*e*, *a*), (*e*, *b*), (*e*, *c*)}

Notice that for the one-place predicates we have a set of objects for which this predicate is true (e.g., only *b *and *c *are blue) and such a set is denoted using ‘{’ and ‘}’ symbols, called‘curlybraces’orjust‘braces’.12 Forthetwo-place predicates we have a set of tuples that are denoted using ‘(’ and ‘)’ symbols, called ‘parentheses’ or ‘round brackets’. In this case, for instance, the fact that (*a*, *b*) is in the set *LeftOf **S *means that *LeftOf *(*a*, *b*) is true for this structure, i.e., *a *is left of *b*.

Such formal structures can also be defined to disprove arguments written in predicate logic, as we will see in Section 2.5.3.