predicates in natural languages

see also: semiformal and formal descriptions of a firstorder predicate logic. informal, semiformal and formal descriptions of propositional logic. 
A predicate is a feature of language which you can use to make a statement about something, e.g. to attribute a property to that thing.
If you say "Peter is tall", then you have applied to Peter the predicate "is tall".
We also might say that you have predicated tallness of Peter or attributed tallness to Peter.
A predicate may be thought of as a kind of function which applies to individuals (which would not usually themselves be propositions) and yields a proposition. They are therefore sometimes known as propositional functions. Analysing the predicate structure of sentences permits us to make use of the internal structure of atomic sentences, and to understand the structure of arguments which cannot be accounted for by propositional logic alone. 
Predicates (or relations):

There are very very many predicates in natural languages, most of them of no special interest to logic.
Unlike propositional logics, in which specific propositional operators are identified and treated, predicate logic uses arbitrary names for predicates and relations which have no specific meaning (until an attempt may be made to apply the logic). 
Though predicates are one of the features which distinguish firstorder logic from propositional logic, these are really just a bit of extra structure necessary to permit the study of quantifiers. The two important features of natural languages whose logic is captured in the predicate calculus are the terms "every" and "some" and their synonyms, whose analogues in formal logic are called the universal and existential quantifiers. These features of language refer to one or more individuals or things, which are not by themselves propositions and which therefore force some kind of analysis of the structure of "atomic" propositions. 
Where a logic is concerned not only with sentential connectives but also with the internal structure of atomic propositions it is usually called a predicate logic.
The most well known, and probably the simplest of these logics is known as classical or boolean, firstorder, predicate logic or, perhaps more appropriate but not so often used, quantifier theory. 
The "classical" or "boolean" bit says that propositions are either true or false (there being no third possibility).
The "firstorder" bit says that we consider predicates (or relations) on the one hand, and individuals on the other; that atomic sentences are constructed by applying the former to the latter; and that quantification is permitted only over the individuals. 
Firstorder logic permits reasoning about the propositional connectives (as in propositional logic) and also about quantification ("all" or "some").
A classic, if elementary, example of what can be done with the predicate logic is the inference from the premises:
