Further Modal Concepts: Consistency, Inconsistency, Contradictoriness, and Entailment

I argued earlier that the validity of argument forms is a modal concept.  But the same goes for consistency, inconsistency, contradictoriness, and entailment.  Here are some definitions. 'Poss' abbreviates 'It is broadly-logically possible that ___.' 'Nec' abbreviates 'It is broadly-logically necessary that ___.' '~' and '&' are the familiar truth-functional connectives. 'BL' abbreviates 'broadly logically.'

D1. A pair of propositions p, q is BL-consistent =_df Poss(p & q).

Clearly, any two true propositions are consistent. (By 'consistent' I mean consistent with each other.  If I mean self-consistent, I'll say that.)   But there is more to consistency that this.  It is a modal notion.  Consistency cannot be defined in terms of what is actually the case.  One must also consider what could have been the case.  As long as p, q are contingent, they are consistent regardless of their truth-values. If both are true, they are consistent.  If both are false, they are consistent.  If one is true and the other false, or vice versa, they are consistent.

D2. A pair of propositions, p, q, are BL-inconsistent =_df ~Poss(p & q).

D3. A pair of propositions p, q are BL-contradictory =_df ~Poss(p & q) & ~Poss (~p & ~q).

Note the difference between inconsistency and the stronger notion of contradictoriness.  If two propositions are inconsistent, then they logically cannot both be true.  If two propositions are contradictory, then they are inconsistent but also such that their negations logically cannot be true.

Example. All men are rich and No men are rich are inconsistent in that they cannot both be true.  But they are not contradictory since their negations (Some men are not rich, Some men are rich) are both true.  All men are rich and Some men are not rich are contradictory.  Some men are rich, Some men are not rich are neither inconsistent nor contradictory.

D4. P entails q =_df ~Poss(p & ~q).

Entailment, also called strict implication,  is the necessitation of material implication.  If '-->' stands for the material conditional, then the right hand side of (D4) can be put as follows: Nec (p --> q).

(Alethic) modal logic's task is to provide criteria for the evaluation of arguments whose validity or lack thereof depends crucially on such words as 'possibly' and 'necessarily.'  But if I am right, many indispensable concepts of nonmodal logic (e.g., standard first-order predicate logic with identity) are modal concepts. 

 

 

Comments

[Dominic Bnonn Tennant]

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

[Bill Vallicella]

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.

[Chad]

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 blog!

Chad

[Bill Vallicella]

Chad,

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.

[Chad]

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.

[Bill Vallicella]

Chad,

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?

[Chad]

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.