Dave Gudeman comments:
I find the whole concept of modality very opaque. Sometimes modality just seems to be about a priori truths, a matter of conceivability, and I fancy that I grasp it. Then I read something like "if God exists He exists necessarily" . . . and I realize that I don't have any idea what that means.
I think there are two separate misapprehensions here. One is about apriority and conceivability. The other is about modality and existence. I'll leave the first misapprehension for later.
I exist. But I might not have existed. (Had my father been killed in the South Pacific in WWII, then I would never have existed.) To say that I might not have existed is to say that my nonexistence is possible. In 'possible worlds' jargon: Though I exist in the actual world, there are merely possible worlds in which I do not exist. But if this jargon is confusing, forget about it. It is not needed.
Are you with me this far, Dave? Now given that my nonexistence is possible, my existence is not necessary. Therefore, my existence is contingent. So we define:
X is a contingent being =df X is such that x is possibly nonexistent (if existent) and possibly existent (if nonexistent).
From this definition it follows that a noncontingent being is one that is either necessary or impossible. Now you will agree that the mere framing of a definition does not guarantee that there is anything answering to it. Still, is it not at least understandable that there should be some impossible beings, some necessary beings, and some contingent beings? And if you deny this, are you saying that nothing that exists is modally qualified, or are you saying that everything that exists is contingent?
Examples. Consider a plane figure that is both round and square (i.e., not round). Such a plane figure is impossible: not just nonexistent, but necessarily nonexistent. The number 9, however, is not only possible, but necessary: it exists in all possible worlds, not just in some of them. The moon is neither impossible nor necessary: it exists but it might not have existed.
Now Dave: where exactly is your difficulty? Is it epistemological? How do I know that this table before me is a contingent being? How do I reasonably believe that there is a possible world in which I am biking at time t rather than blogging at t? Or is it semantic? What is meant by 'necessary,' 'possible,' etc. in this discussion?
Are you perhaps vexed over the the question of the ontological grounds, if any, of modal truths? Is your question about the truth-makers, if any, of such truths as 'Possibly, I am biking now'?
Or perhaps you doubt that there are any modal truths in the first place. (If there aren't any, then they can't be known and don't need truth-makers.)
Or is it something else? Please let me know and then we can proceed.
Related Posts (on one page):
- Contingent and Noncontingent Existence and Nonexistence
- By Popular Demand: An Excursus into Modal Metaphysics
That's part of the answer. More in a separate post.
Maybe it would help if I explained how I view the problem in terms of model theory. In model theory (loosely speaking), a theory is composed of two parts, the logic, which is the base language and the set of rules for drawing inferences, and the model, which is an assignment of values to variables. Consequently, there are two kinds of symbols, those which take their meaning from the logic and those which take their meaning from the model.
Typically, for example, in a sentence like (P &~Q) the connectives &and ~ take their meaning from the logic and the propositional variables P and Q take their meaning from the model. The logic is considered fixed and the model is replaceable. For example if you use the model (P=false, Q=false) then (P &~Q) is false. But if you use the model (P=true, Q=false) then (P &~Q) is true.
Some things you can know to be true just from the rules of the logic alone. They are true in all models. For example ~(P &~P) is true in all models according to the normal logic of propositions. Clearly, there is an analogy between this and necessary/contingent. In the world of propositional logic, a proposition is necessary if it is a tautology and it is contingent if it depends on the model (for brevity, let's get rid of impossibility and absurdity by folding them into necessity and tautology in the usual ways).
But the logic isn't really a fixed quantity. You can create a logic with whatever rules you want. Some authors like to add the rules of equality to the predicate logic because they think (For all a . a=a) should be a tautology.
So I use this framework as a way to understand necessity. I'd like to create a logic such that the necessary truths correspond to tautologies in the logic. To do this, I have to, for example, add enough rules to make all apriori truths into tautologies (I am using the formalism here far beyond the study of reasoning, but that should not matter).
This "predicate logic+ the apriori" doesn't seem to cover what you mean, though. You are adding something more, and I don't know what that is.
Some possibilities occur to me for extending this logic. We might for example assume that physical laws are part of the logic and then "necessary/possible" corresponds to "required by physical law/allowed by physical law". Or we only make the most fundamental physical laws like conservation of energy and momentum be part of the logic and leave things like the speed of light and Planck's constant as part of the model.
Of course neither of those last two idea match what you are saying either. So it would answer my question if you could explain to me what the rules of the underlying logic are. Or failing that, some criterion for knowing whether a given bit of knowledge is in the logic or the model.
We agree on the following. In propositional logic, every proposition is either a tautology, a contradiction, or a contingency. A tautology is true in all models. A contradiction is true in no model. A contingency is true in some but not all models. I put it a little differently than you did, but it amounts to the same thing.
Every tautology is a logical truth, and is logically necessary. But surely there are necessary truths that are not tautologies. You mention a priori truths. Consider 'Every effect is an effect.' Clearly, this is an a priori truth. I know it to be true independently of experience. It has the logical form Every F is an F. But that form is not a tautological form, since in propositional logic it has the form, p, which is a contingent propositional form. In other words, there are logical truths that are not tautologies. So if you are trying to construe every necessary truth as a tautology, you will fail.
I distinguish different types of necessity: narrowly logical necessity, broadly logical necessity, nomological necessity. I would not call a law of nature "part of the logic" because the ground of its necessity does not lie in logic but in certain natural necessities. For excample, the necessity of 'It is either snowing or not snowing' is grounded in its logical form. But the necessity of the inverse square law is not grounded in its logical form.
You want "some criterion for knowing whether a given bit of knowledge is in the logic or the model." Take 'Every effect has a cause.' That is a bit of knowledge. The ground of its truth is not in logic but in the concepts involved. So it's in the model. What's the criterion? Roughly, if a proposition is true, but not true in virtue of its logical form, then its truth is gounded 'in the model.'
But I have the feeling that this won't satisfy you.
I deleted Craig's comment since (1) he is violating my rules by not using his full name (which cannot be found on his blog either); (2) his comment shows a lack of understanding of the issues. For example, he fails to realize that propositional logic treats whole propositions as units without concern for their internal (subpropositional)logical structure. Other comments of his are equally substandard, and he is sometimes uncivil to boot.
The point is that rules of inference are interchangeable with propositions. We know what typical rules of inference look like, for example:
P &P --> P
where P is a propositional variable. But you can also express rules with propositional constants and many authors do something like this:
P --> T
P --> ~F
to make the propositions T and ~F into tautologies. In predicate logic, there is no reason that you can't add rules like this:
p(x) --> x=x
T --> N(0)
N(n) --> N(s(n))
along with everything else needed for Peano arithmetic. Arithmetic and equality then become part of the logic itself and the truths of arithmetic become tautologies in this expanded logic (which is no longer predicate logic).
I meant to suggest the use of this formalsm as a way of explaining necessity. In other words, I wasn't trying to "construe every necessary truth as a tautology" according to any pre-existing logic. I was suggesting that one can in principle design a logic L such every tautology in L expresses a necessary truth and every necessary truth that is expressible in the language of L is derivable in L (I am disregarding difficulties of completeness but this is only intended to be a pedogigical aid, not a complete logic of necessity).
This logic L would not be very much like any traditional logic because it would contain many rules of inference that would normally be considered part of the model. Your objection that there are apriori truths that are not tautologies would not apply to L, because by construction, if p is any kind of necessary truth expressible in the language of L, then p is derivable in L.
The point of this construction was to try to explain my confusion about necessity. I don't understand what criteria you would use to decide what goes in L and what is part of the model.
I guess I'm asking for some more rigorous explanation of the difference between a necessary and a contingent truth. You are trying to explicate a rule by providing examples, but your examples are not enough to let me infer the rule.
Sorry to be such a pain. I'm not just feigning confusion as a rhetorical ploy; I really do not get how philosophers use the word "necessary". And frankly that's unusual. I usually get things pretty quickly.
You are a computer scientist, and you may be operating with a different terminology and notation. I use 'p' 'q' 'r' and so on to denote propositions. I use 'T' and 'F' to denote truth values. I am assuming bivalence. So there are exactly two truth values. A truth value is not a proposition. So although
1. (p &p) --> p
is a well-formed formula (wff) in the propositional calculus, and is indeed a tautology,
2. p --> T
and
3. p --> ~F
are not wffs. I don't know that they mean. What authors are you reading?
There is probably no need to bring in the predicate calculus since I think the question you are asking can be asked in terms of the simpler logic of propositions.
(1) above is a tautology and therefore a necessary truth. Why? Because it is true in virtue of its logical form. It doesn't depend on content. In other words, (1) comes out true whether p is true or p is false. But p by itself can be either true or false. Therefore p is contingent. It depends on the way the world is.
Why are you not satisfied with the following explanation:
In the propositional calculus, a necessary truth is identical to a tautology; therefore, the difference between a necessary truth and a contingent truth is identical to the difference between a tautology and a nontautological truth?
Why can't emprical truths be packed into the logic? Because they are not true in virtue of their logical form. But I many not be understanding you.
The set of well-formed formulas is defined as follows. If S2 and S2 are well-formed formulas, then so are
P, where P is any propositional variable
T
F
~S1
S1 &S2
The rules of inference include (among others, I don't claim this is a good set...)
~~S1 -> S1
S1 -> T
F -> S1
S1 &~S1 -> F
You can see that in this formulation, T and F are indeed propositions. Or more properly, T and F are 0-ary operators which are applied without an operand to produce a proposition.
I'll try to get to your other objection tomorrow. It's getting late...
You say T and F are operators. No problem. But then how can they be propositions?
In a bivalent logic, propositions are either true or false. But it makes no sense to me to think of truth and falsehood as propositions. They are either properties of propositions, or operators upon propositions, or perhaps relations. But to think of truth as a proposition makes as little sense as to think of negation as a proposition.
2. Disallowing comments from a particular person, or deleting an offensive, off-topic, or otherwise substandard comment, has nothing to do with censorship. People who think otherwise confuse censorship with lack of sponsorship. I am under an obligation not to interfere with anyone's exercise of legitimate free speech rights. But I am not under any obligation to aid and abet anyone's exercise of free speech rights, legitimate or illegitimate.
3. The Comments area is not an open forum for anyone to say anything about any topic. As the name implies, it is primarily for commenting on the author(s)' posts. But to comment on them, one must have read them. And if I have spent three hours on a post, a reader will not understand it in thirty seconds. Secondarily, the Comments area is to facilitate civil discussion between and among commenters as long as the discussion remains on-topic.
4. Some undesirables: The skimmers, those who cannot read but only read-in. The sophists who, abusing argument, argue for the sake of argument. The ideologues, those who are out for power, not truth. The uncivil. The illogical. The politically correct. Worst of all, perhaps, are those who exemplify the anti-Socratic property: those who think they know what they don't know. If Socrates was famous for his learned ignorance, these types are marked by their ignorant unlearnededness.