1

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?

Improve this question
3
  • 1
    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
    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
    Jan 3, 2016 at 5:24
  • 1
    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 ⊃ Px, namely that if x is S then it is also P.
    – nwr
    Jan 3, 2016 at 5:49

1 Answer 1

Reset to default
1

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 Sx ⊃ Px 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) Sx ⊃ Px is true, while the statement (x) Px ⊃ Sx 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.

Improve this answer
2
  • 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
    Jan 3, 2016 at 20:58
  • @LePressentiment No problem. Both the edits are improvements on the originals. Thanks. N
    – nwr
    Jan 4, 2016 at 1:10

You must log in to answer this question.