## Friday, February 20, 2009

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Bill, I'm sorry to ask such a silly question in the comments here; I've tried searching online in numerous places but just can't find anything. What does =df indicate? Is it just definition; another way of writing := ? I'm not really familiar with first order logic, but it seems to be used a bit differently.

Regards,
Bnonn

Hi Bnonn,

'=df' indicates that the expression on the left hand side (the definiendum) is being defined by the expression on the right hand side (the definiens). Definitions specify necessary and sufficient conditions for the application of a concept. They thus have the form of biconditionals: if LHS, then RHS & if RHS, then LHS.

Bill,

It is arguably not possible that (grass is green and some water contains no hydrogen). That, at any rate, is a widespread view since Kripke. But surely the propositions that grass is green and that some water contains no hydrogen are broadly logically consistent.

Thanks for your comment. Perhaps you could tell me what you mean by broadly logically consistent. On my understanding of the term, the two propositions you mention are not BL-consistent.

Let me give you 'BL-consistent', 'BL-possible', and the like. I'll define a different term, which I'll call 'consistency*'. Here is my definition:

P and Q are consistent* iff some proposition having the same logical form as the conjunction of P and Q is BL-possible.

That's how I'd define it. But consistency* is one that I think I have an intuitive grasp on, and this definition is a substantive analysis of that intuitively grasped notion. Notice that consistency* is a modal notion in the sense that it is defined in terms of BL-possibility. I just think that it involves the notion of logical form as well as the notion of BL-possibility.

We could also define 'consistent*' in terms of the ordinary concept of validity that most of us learned (and have taught) in introduction to philosophy:

P is consistent* with Q iff the argument "P, therefore ~Q" is not valid.

Notice that this is not a valid argument:

Grass is green.
Hence, all water has hydrogen in it.

So, by this definition, these two propositions are consistent*.

Your notion is perfectly intelligible and certainly important in philosophy. The notion I'm describing is just broader and, I think, more central to logic proper.

I think what you are defining is narrowly-logical consistency. It could be put as follows: p, q are NL-consistent =df the logical form of (p & q) has some true substitution-instances. This looks to be equivalent to your defn. of consistency*.

Now consider *Grass is green* and *Some water contains no hydrogen.*I say they are BL-inconsistent but NL-consistent. They are BL-inconsistent because the second prop is nec false (given Kripke). But they are NL-consistent since the logical form of their conjunction is (p & q), a form which admits of true substitution-instances.

Does this make sense?

Makes sense. I agree with everything you've said, then. I was just thrown by your use of 'broad' and 'narrow'. I was thinking that the broader notion of consistency is the one that counts more pairs of sentences as consistent.

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