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I know that a predicate is also called a propositional function, since it accepts one or more entities as its argument, and return a proposition. For example, let $P(x,y)=``x\text{ is the father of }y."$, then $P$ is a predicate, and meanwhile the propositional function, since when we substitute Adam and Jeff into $P$, then $P(\text{Adamm},\text{Jeff})=``\text{Adam is the father of Jeff.}"$, which is a proposition.

Now, in the propositional logic context, the logic connectives($\wedge,\rightarrow$, etc) are essentially functions that take some propositions(not entities this time) as their arguments, and also return the propositions. For example, $\wedge(1<5,2+2=4)=(1<5)\wedge (2+2=4)$, which is also a proposition. So do we, or can we call these logical connectives the propositional functions? Why or why not?

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  • $\begingroup$ They are ussually called Truth functions. $\endgroup$ – Mauro ALLEGRANZA Dec 2 '16 at 13:00
  • $\begingroup$ The issue is that the term propositional function has already an "established" meaning in modern logic. $\endgroup$ – Mauro ALLEGRANZA Dec 2 '16 at 13:05
  • $\begingroup$ @MauroALLEGRANZA I think the sentence in wiki: "a truth function is a function from a set of truth values to truth values." is a little weird if the author at the same time considers the logical connectives are of truth functions. Since, as I know, connectives (in propositional context) take proposition(s) and return a proposition, rather then take or return $T$ or $F$. The corresponding truth value of the result of the connectives, is always gained by applying a valuation function on the propositions; that is, a connectives never directly returns a $T$ nor a $F$. $\endgroup$ – Eric Dec 2 '16 at 13:15
  • $\begingroup$ In the "syntax view" we have propositional letters (or symbols) : $p_i$: in the "semantic view" we have truth values : $\{ 0, 1 \}$. When you speak of "propositions" are you referring to "content" (i.e. to meaning) or to expressions (strings of symbols) ? $\endgroup$ – Mauro ALLEGRANZA Dec 2 '16 at 13:22
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    $\begingroup$ One problem is that usually functions are defined (in terms of sets which are defined) in terms of logical operators. So "truth function" is about like defining graphs as "trees with cycles" or defining real numbers as "one dimension of a complex number" or defining an n-dimensional point as "the n-dimensional square with zero volume". Not wrong, but completely backwards. That isn't to say that there aren't some logics that define functions before they define operators (lambda logic for example). $\endgroup$ – DanielV Dec 2 '16 at 15:08
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Sure, we could call logical connectives functions. It's perfectly reasonable, and it fits the definition of "function" just fine. The more common terminology is operator, but function works just as well.

However, it's useful to distinguish between connectives and predicates. Connectives are truth-functional, which means that the truth of their output is dependent only on the truth of their inputs; predicates aren't, because "truth" doesn't even make sense for the argument of a predicate. Much more can be said about truth-functional operators than about "propositional functions" in general.

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  • $\begingroup$ Yes, I know that logical connectives are " stronger "(means the former is a special case of the latter) then (a general) propositional functions. But if just talk about the correctness, is it correct to call logical connectives a type of propositional functions.? $\endgroup$ – Eric Dec 2 '16 at 13:08
  • $\begingroup$ @Eric As far as I know, "propositional function" does not have a standard meaning. If it has been formally defined for you, then it depends on whether it has been defined as "a function that outputs propositions" (in which case yes, connectives are) or "a predicate" (in which case they aren't). Either option is perfectly reasonable, so it depends on exactly which source you're working with, $\endgroup$ – Reese Dec 2 '16 at 13:16
  • $\begingroup$ Thanks! I missed to notice that if the propositional function is simply defined as "a function that outputs propositions", then propositional functions will no longer be equivalent to *predicates*(whose arguments definitely can't be the propositions), which would be a problem if most people tends to consider these terminlogy equal(though I'm not sure whether so). All things are clear now, thanks for your help!! $\endgroup$ – Eric Dec 2 '16 at 13:31
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can we call these logical connectives the propositional functions?

The issue is legitimate, but "history" followed a different path and now we have a quite established convention.

For the source, see :

  • Alfred North Whitehead & Bertrand Russell, Principia Mathematica to *56 (1st ed.1910, 2nd ed.1927); see Introduction (to the 1st edition), page 1-on:

[page 5] Variables will be denoted by single letters [...] $p, q, r$ will be called propositional letters, and will stand for variable propositions [sic! meaning that they are variables for propositions].

[page 6] An aggregation of propositions [...] into a single proposition more complex than its constituents, is a function with propositions as arguments. [...] there are four special cases which are of fundamental importance, since all the aggregations of subordinate propositions into one complex proposition which occur in the sequel are formed out of them step by step. They are (1) the Contradictory Function, (2) the Logical Sum, or Disjunctive Function, (3) the Logical Product, or Conjunctive Function, (4) the Implicative Function. [...]

The Contradictory Function with argument $p$, where $p$ is any proposition, is the proposition which is the contradictory of $p$, that is, the proposition asserting that $p$ is not true. This is denoted by $\sim p$.

[page 7] These four functions of propositions are the fundamental constant (i.e.definite) propositional functions with propositions as arguments, and all other constant propositional functions with propositions as arguments [...] are formed out of them by successive steps.

[page 14] Let $\phi x$ be a statement containing a variable $x$ and such that it becomes a proposition when $x$ is given any fixed determined meaning. Then $\phi x$ is called a "propositional function"; it is not a proposition, since owing to the ambiguity of $x$ it really makes no assertion at all.

Principia Mathematica has been blamed for this kind of sloppiness regarding the sintactical specifications of the language.

Thus, the two different usages of "propositional function" has been subsequently replaced by connective and open formula respectively.

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