# 1.4.1: Predicates

- Page ID
- 9859

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A predicate is a kind of incomplete proposition, which becomes a proposition when it is applied to some entity (or, as we’ll see later, to several entities). In the proposition “the rose is red”, the predicate is is red. By itself, ‘is red’ is not a proposition. Think of it as having an empty slot, that needs to be filled in to make a proposition: “— is red”. In the proposition “the rose is red”, the slot is filled by the entity “the rose”, but it could just as well be filled by other entities: “the barn is red”; “the wine is red”; “the banana is red”. Each of these propositions uses the same predicate, but they are different propositions and they can have different truth values.

If *P *is a predicate and *a *is an entity, then *P*(*a*) stands for the proposition that is formed when *P *is applied to *a*. If *P *represents ‘is red’ and *a *stands for ‘the rose’, then *P*(*a*) is ‘the rose is red’. If *M *is the predicate ‘is mortal’ and *s *is ‘Socrates’, then *M*(*s*) is the proposition “Socrates is mortal”.

Now, you might be asking, just what is an entity anyway? I am using the term here to mean some specific, identifiable thing to which a predicate can be applied. Generally, it doesn’t make sense to apply a given predicate to every possible entity, but only to entities in a certain category. For example, it probably doesn’t make sense to apply the predicate ‘is mortal’ to your living room sofa. This predicate only applies to entities in the category of living things, since there is no way something can be mortal unless it is alive. This category is called the domain of discourse for the predicate.

We are now ready for a formal definition of one-place predicates. A one-place pre- dicate, like all the examples we have seen so far, has a single slot which can be filled in with one entity:

Definition 2.7.

A **one-place** predicate associates a proposition with each entity in some collection of entities. This collection is called the domain of discourse for the predicate. If *P *is a predicate and *a *is an entity in the **domain of discourse** for *P*, then *P*(*a*) denotes the proposition that is associated with *a *by *P*. We say that *P*(*a*) is the result of **applying** *P** *to *a*.

We can obviously extend this to predicates that can be applied to two or more entities. In the proposition “John loves Mary”, loves is a two-place predicate. Besides John and Mary, it could be applied to other pairs of entities: “John loves Jane”, “Bill loves Mary”,

“John loves Bill”, “John loves John”. If *Q *is a two-place predicate, then *Q*(*a*, *b*) denotes the proposition that is obtained when *Q *is applied to the entities *a *and *b*. Note that each of the ‘slots’ in a two-place predicate can have its own domain of discourse. For example, if *Q *represents the predicate ‘owns’, then *Q*(*a*, *b*) will only make sense when *a *is a person and *b *is an inanimate object. An example of a three-place predicate is “*a *gave *b *to *c*”, and a four-place predicate would be “*a *bought *b *from *c *for *d *euros”. But keep in mind that not every predicate has to correspond to an English sentence.

When predicates are applied to entities, the results are propositions, and all the op- erators of propositional logic can be applied to these propositions just as they can to any propositions. Let *R *be the predicate ‘is red’, and let *L *be the two-place predicate ‘loves’. If *a*, *b*, *j*, *m*, and *b *are entities belonging to the appropriate categories, then we can form compound propositions such as:

*R*(*a*) ∧ *R*(*b*) *a *is red and *b *is red

¬*R*(*a*) *a *is not red

*L*(*j*, *m*) ∧ ¬*L*(*m*, *j*) *j *loves *m*, and *m *does not love *j*

*L*(*j*, *m*) → *L*(*b*, *m*) if *j *loves *m *then *b *loves *m*

*R*(*a*) ↔ *L*(*j*, *j*) *a *is red if and only if *j *loves *j*

Predicate logic is founded on the ideas developed by Charles Sanders Peirce (1839–1914), an American philosopher, logician, mathematician, and scientist.10 Many of his contributions to logic were appreciated only years after he died. He has been called “the most original and versatile of American philosophers and America’s greatest logician.” and “one of the greatest philosophers ever”. As early as 1886 he saw that logical operations could be carried out by electrical switching circuits; the same idea was used decades later to produce digital computers, as we saw in Section 2.3. You can read about his colourful life at the link below.

Source: en.Wikipedia.org/wiki/Charles_Sanders_Peirce.