# Validity of a modal argument?

Would you say that the following argument is valid?

And if not, why exactly?

For all we know, A may be the state of B;

What C does is determined by the state of B;

Therefore, for all we know, what C does may be determined by A.

A, B and C should be understood as referring to "things".

Thanks. EB

• Could you give an example for A, B, and C to clarify? – Eliran Jan 12 '19 at 17:39
• What does it mean "to be the state of" ? What does it mean "is determined by the state of " ? These expression are not "formalizable" with usual logical constants. – Mauro ALLEGRANZA Jan 12 '19 at 17:51
• @Eliran - I would rather not. As soon as you give particular instances, people focus of whether the premises are true or not. But, you can pick up any set of three things, as long as it says something meaningful, for example, A = Water, B = H2O molcules; C = Sea. But this shouldn't be necessary. The argument is valid or it isn't, irrespective of what A, B and C are taken to mean, as long as it makes sense. EB – Speakpigeon Jan 12 '19 at 18:14
• @Mauro - We could perhaps reduce it to that: For all we know, A may be S; What C does is F(S); Therefore, for all we know, what C does may be F(A). – Speakpigeon Jan 12 '19 at 18:18
• Isn't the argument valid on the face of it as interpreted in everyday language? – Speakpigeon Jan 12 '19 at 18:20

Since you have chosen to present the problem in abstracto, I am assuming that you are not so much worried about the exact nature of determining.

The first, most simple formalisation that comes to mind is:

``````s(x) ... the state of x.
D(x) ... x determines what c does.
``````

Then we would write

``````◊a = s(b) ... possibly a is the state of b
D(s(b))   ... the state of b determines what c does.
◊D(a)     ... possibly a determines what c does.
``````

So does the last sentence follow from the first two? No it does not, one has to replace the second sentence by

``````□D(s(b))     ... it is known that the state of b determines what c does.
``````

Now the third one will follow, at least in, say, S4 which was the epistemic system that Hintikka settled for. If am not mistaken, the reasoning should go through in K already.

• OK, that's very helpful, thanks. I have to correct one bit: It's not "the state of a and b are the same", it's "a is the same as the state of b". I'm not sure this affects the meaning or validity, but it's the original idea. As to the necessity of premise 2, the absence of the "as far as we know" phrase was meant to convey that premise 2 is accepted as "known to be true", and perhaps therefore as necessarily true? "Necessarily true" I think is less easily understood as "known to be true", so if known implies necessarily true, we could have it both ways, logic and understandability. – Speakpigeon Jan 13 '19 at 13:44
• And knowledge is the only certainty of truth we... know, so it would be hard to do without it. Is there a way to try and keep it? – Speakpigeon Jan 13 '19 at 13:55
• Thank you, I corrected the part about the states. Regarding the other issue: since the modality in question is knowledge, I have corrected that as well. The second translation seems to be perfectly reasonable to me, in natural language "we know that..." is mainly used for emphasis, so often dropped, even if a sentence is claimed to be known. – Jishin Noben Jan 13 '19 at 14:02
• That's just perfect! I was rather convinced the thing was valid but I still don't understand how it could possibly be formally proven that it is valid, even once it is properly formalised as you did. Any advice on this? – Speakpigeon Jan 13 '19 at 19:30
• Are you looking for a proof in an axiomatic system, or a formal proof? – Jishin Noben Jan 18 '19 at 13:07

Would you say that the following argument is valid?

And if not, why exactly?

It is not. The connection between A and what C does is too accidental to allow an inference of A's [possible] determination of the latter. Thus, one can speak in terms of coincidence, not causation.

The non-conclusive premise A may be the state of B gives a connotation of possibility of alternatives. To the extent that the state of B could be something else, A's capacity to "influence what C does" is weakened. This applies regardless of whether A is just a value/constant or a variable on its own (apropos of your clarification).

Consider C(B)=B^2, whence B determines what C does. The state of B may be A=3, but the value 3 itself is not what controls C. The rule of correspondence is between C and B, not between C and one of the possible values of B.

• If C(B)=B^2 then B does not determine what C does. What C does is squaring, and what B determines is what square you get. That is a category mistake. – Jishin Noben Jan 13 '19 at 14:28
• @JishinNoben "That is a category mistake" ... only if one takes the position that the premise "What C does is determined by the state of B" means that each different input x (or sets thereof) ought to override/replace/redefine f(x). Also a piecewise function would overcome your remark, but it would still miss the point I am making. Given the context of my answer, I think it is clear that by the language "what C does" I mean "what number C produces" in response to the input. – Iñaki Viggers Jan 13 '19 at 17:51
• @Jishin - Thanks, but I don't actually understand your point! If C(B)=B^2 and B determines what C does but the state of B does not determine what C does then premise 2 is just false. You would need to find another example where the state of B does determine what C does. Also, as implied semantically by premise 1, A may well not be the same thing as the state of B, so A indeed may not determine what C does, but this is implied by the "may" of the conclusion. And if, however, A is indeed the state of B, then, obviously, A determines what C does, granted premise 2. So, I disagree with your point. – Speakpigeon Jan 13 '19 at 19:55
• @Speakpigeon A is not what determines "what C does". The determination or dependence relation lies in B only (see your premise #2). The possible instance where the state (or value) of B is A does not imply that A itself causes C's specific response in that scenario. – Iñaki Viggers Jan 13 '19 at 20:20
• I disagree. Premise 1 semantically implies that if (not "when") A is indeed the state of B, as it may well be, it is always the state of B. There is in this case no temporal restriction, and no notion of any "instance" applicable. A is just the state of B. The two are the same thing under a different label. So, granted premise 2, I don't see how the conclusion could be false, and in fact could possibly be necessarily false. – Speakpigeon Jan 13 '19 at 20:49