# What 'certain liberties' inject meaning to Statement Functions?

Source: p 465, A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

[...] 5. Py ⊃ Cy    3, 4, Hypothetical Syllogism
6. (x)(Px ⊃ Cx)     5, Universal Generalisation [...]

As noted earlier [on p 457 which I questioned earlier], the expressions in lines 3, 4, and 5 are called statement functions. As such, they are mere patterns for statements; they have no truth value and cannot be translated as statements. Yet if we take `certain liberties`, we might characterize line 5 as saying
“If it is a P, then it is a C["],
where “it” designates any item at random in the universe.
Line 6 can then be seen as reexpressing this sense of line 5.

1. What does the author mean by `certain liberties`?

2. I am most confused; does the above quote appears to contradict p 457?
What is the final answer on whether Statement Functions mean anything?

• Thanks for your comment on my previous answer. As per your request, I have posted some thoughts on this question in my answer below.
– nwr
Commented Jan 3, 2016 at 5:06
• @NickR +1. You are most welcome. It is I who must thank you. To fortify my comprehension, please allow me to reread both your answers tomorrow.
– user8572
Commented Jan 3, 2016 at 5:24
• The thing to keep in mind here is the distinction between the formal and the informal reading of statement functions. Even not knowing what S and P denote, we can assign some informal meaning to Sx &sup; Px, namely that if x is S then it is also P.
– nwr
Commented Jan 3, 2016 at 5:49

When the author suggests that "we take certain liberties", he means "if we drop the rigours of our logical formalism and look at statement 5 in a non-rigorous, hand-wavy, informal way", then we can translate statement 5 accordingly.

To best understand the formal concept of a statement function, the key word to focus on here is "function". As I noted in my previous answer, a function is only fully defined when we have assigned it a domain (i.e., a universe of discourse).

For example, if S denotes "is divisible by 4" and P denotes "is divisible by 2", then SxPx can only be assigned a meaning when we know the domain/universe from which x is chosen. Otherwise, from a formal point of view, x is just a typographical symbol and the statement function is just (what the author refers to as) a pattern. If the universe is the set of whole numbers then the meaning is obvious and the statement (x) SxPx is true, while the statement (x) PxSx is false. Of course, from an informal point of view, we can assign a meaning once we know what S and P denote.

Therefore, the example given here does not contradict the formal statement of page 457, since the treatment given here is an informal one.

• I modified your post marginally and am sorry for any offense; I thought to link to the answer aforesaid. Please feel free to refine.
– user8572
Commented Jan 3, 2016 at 20:58
• @LePressentiment No problem. Both the edits are improvements on the originals. Thanks. N
– nwr
Commented Jan 4, 2016 at 1:10