1

Source: p 456 (chart), 457 (Quote), A Concise Introduction to Logic (12 Ed, 2014) by Patrick Hurley

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In these examples, the expressions [...] [that I coloured in green], and so on are called statement functions. A statement function is the expression that remains when a quantifier is removed from a statement. It is a mere pattern for a statement. It makes no definite assertion about anything in the universe, has no truth value, and cannot be translated as a statement.

  1. How are the 2 italicised sentences correct? They appear to debase and denigrate the green which (to me) appear truly important, because:

  2. am I correct that the green expressions match, and so assert, the red ('S are P')?
    I know that the purple (and not the green) explains the Universal Quantifiers 'all' and 'no'.

2

It is an interesting observation that you make, since, on the surface, the two "statement functions" appear to express contradictory things.

Your headline question "How can a statement function make no definite assertion about anything?" is answered with the obvious response : because that is how the author has defined a statement function in the quoted text. When one reads a statement function such as Sx ⊃ Px, one needs to ask "what x?" before one can impose any meaning, and this is what the quantifiers do.

As the author states, the green expressions (statement functions) cannot be translated as a statement, so according to the author's intentions it is not correct to say than the green expressions "match, and so assert" the statement "S are P". Indeed, according to the quoted text, a statement function makes no assertion.

The author's example shows how quantification of a "statement function" can impose meaning onto the resulting statement, and that different quantifiers can yield quite different symbolic translations and meanings.

EDIT

If you are familiar with the concept of function in mathematics, then just as a mathematical function has no meaning without a domain, in logic a statement function has no meaning until its domain is specified by quantification.

  • Thank you for your answer. Sorry to bother you, but this subsequent of mine may matter because the author subsequently seems to say that Statement Functions DO mean something, in a subsequent chapter (unfortunately) about which I could not have known when I wrote the question above. – Greek - Area 51 Proposal Jan 3 '16 at 3:25
  • +1. Thank you. Answer your edit at the bottom: Yes; I know the basics about a Function in maths, but I am no set theorist. – Greek - Area 51 Proposal Jan 3 '16 at 20:54

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