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How can any proposition be necessarily true when there always seems to be a possible world in which it may be false?

For example the proposition that "4+4=8" is given to be a necessarily true proposition, but there could be some world where mathematics was defined differently rendering it to be a possibly true proposition?

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    Propositions and strings of symbols are two different things, the subjects of propositions are concepts regardless of symbols assigned to pick them out. One can redefine "4" to be 2, or "+" to be -, but that will have no effect on the proposition expressed by 4+4=8 in standard symbols, see SEP, Propositions. The same goes for "bachelors are unmarried man" or "water is H2O". To paraphrase Quine, changing the definitions is changing the subject.
    – Conifold
    Aug 16 at 22:51
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    In math theory such as PA and systems including PA a necessary proposition usually is interpreted as provable within the said system pioneered by Solovay and furthered by Boolos and surprisingly what's necessary (provable) doesn't always imply it's true due to Gödel's 2nd incompleteness theorem! Thus your example 4+4=8 happens to be both provable and true within PA and thus necessarily true in your term, but the consistency of PA aka Con(PA) isn't necessarily so in the above so-called Gödel-Löb sense... Aug 17 at 3:07
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    When you write, "there could be some world where mathematics was defined differently rendering it to be a possibly true proposition" this suggests to me that the possible world you imagine has a sentence formed from the same symbols, but this sentence would not express the same proposition since there is a shift in the meaning of the symbols. When we say that a proposition is necessarily true, we assume that the meanings (intensions) of the terms are fixed across worlds. Otherwise we would be evaluating different propositions in different worlds.
    – Avi C
    Aug 17 at 8:29
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    To make this concrete, assume that in @, ZFC accurately models reality, but in some other world w, INT accurately models reality. In that case, R has the truth value of Z at @, while having the truth value of I at w. Since these truth values differ, R would not be necessary. However, it might seem implausible that there could be this kind of variability across possible worlds.
    – Avi C
    Aug 17 at 18:37
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    @JKusin It's tough to say. The proper methodology for selecting foundational axioms and underlying logic from a realist perspective is tricky. Gödel had some interesting thoughts on this in terms of Platonism (see, A Logical Journey: From Gödel to Philosophy, by Hao Wang). Essentially he posits that Platonic mathematical concepts can be directly perceived by the human intellect, and formal systems are true if they represent these concepts. From this perspective ZFC is a good model because it formalizes much of how set theorists perceive concepts (although it is incomplete).
    – Avi C
    Aug 17 at 20:18

3 Answers 3

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Great question, and one in which you will find much dissent. There are several major positions. Two illuminating articles to give you background are:

The only book which I own which is a primary source is Kripke's Naming and Necessity and is considered an essential read if you defend a posteriori necessity, which is a form of metaphysical necessity.

Brief Background on Necessary and Certain Knowledge

Historically, the view was that necessary truths were primarily in the domain of a priori knowledge; philosophers in the olden days would say things like "2+2=4" must always be true! This appeal to a prioriticity is largely conducted by adducing logical and mathematical propositions as being true irregardless of the individual. So, this reconciles very strongly with the appeal of rationalism in the original sense that conceivability and introspection were certain forms of knowledge, and the products of those knowledge were irrefutable because the truths were objective. This complemented a very long standing school of thought that says there are certain things that are thought that are actually real and exist albeit in some removed plane not in contact with our physical plane but that our minds nonetheless have access to. Remember, in the time of Descartes, neurology hadn't even been invented, and it was quite a mystery what thought was. These sorts of positions are considered realist positions and the most famous of is Platonism (SEP).

In response to the notion that there is certainty in rational thoughts came the likes of Hume and others who formed the backbone of an empirical movement. These philosophers emphasized the fallibility of reason and thought in general. One of the oldest and most fundamental questions in philosophy, according to professional epistemologists like Robert Audi, is that of skepticism. How can we know anything for sure? Descartes's famous "cogito ergo sum" was a declaration that introspection was the source of certainty and thus necessity. The empiricists pushed back on that and argued that the senses and the mind could be deceived. Thus, to this day, there's a certain tension between belief that reason is certain, and that it is uncertain with rationalism on one side and radical skepticism arguing there's no knowledge at all. According to the IEP article above, most professional philosophers today accept a fallibilist notion of knowledge particularly since Gettier threw epistemologists into the lurch with his elegant problems that said knowledge is not true, justified belief.

Saul Kripke

Kripke is a giant in contemporary logic and philosophy. He did work on formal modal and semantic theory, and is known for his attack on a type of theory related to proper names, a topic that Bertrand Russell and others worked very hard to secure certain theory in which are related to the classical question of the nature of equivalence and identity. His attacks were tremendously influential according to WP:

Such arguments seem to have convinced the majority of philosophers of language to abandon descriptive theories of proper names.

As a byproduct of those attacks, he made a famous argument involving H2O that empirically speaking, certain things are certain by virtue of them being empirical facts, and thereby challenged the orthodoxy of a prioriticity being essentially a necessary condition of necessary claims.

Modal Rationalism and Modal Empiricism

According to SEP, the modern day landscape of positions on modality and possible worlds is very complicated and many philosophers, such as the famous David Lewis, took in all sorts of directions like his modal realism which basically claims that imagining of possibility equates to establishing the concrete reality of other worlds:

Most comprehensively in On the Plurality of Worlds, Lewis defended modal realism: the view that possible worlds exist as concrete entities in logical space, and that our world is one among many equally real possible ones.

What is important to know is that today, there are those who prefer the traditional theses that a priori reasoning is a firm and largely sufficient condition for establishing necessary truths. In fact, such a position seems a reasonable extension however of traditional philosophical positions still. But since Kripke certainly, there's been a push back with some philosophers taking the notion of an empirical basis of necessity and running with it (PhilSE).

So, what 'possible worlds' and 'necessary and contingent means' is a somewhat disputatious topic, and whether you're an anti-realist or realist at heart presages your interpretation of these things. That's why I pushed back on your question on Pegasus about necessary truths. Unfortunately, there are some philosophers who are unable to separate their personal beliefs from more descriptivist beliefs about right and wrong philosophy, and reject pluralism because their gods tell them, or they're too smart to be wrong, or they simply don't accept fallibilism as a basic premise of knowledge. Some philosophers reject the synthetic-analytic divide in the absolute, and other the a priori-a posteriori distinction.

What is a Necessary Truth in a Possible World?

There's no universal consensus on this, so you'll get more than one reasonable answer. From a position of mathematical constructivism or empiricism, both truths and truth systems are products of the mind and are constructed, perhaps using non-classical logics, like intuitionistic logic even formally. When you ask:

there could be some world where mathematics was defined differently rendering it to be a possibly true proposition?

It comes down to 'possible world' and 'true proposition'. For me personally, I go the route of the representational theory of mind (SEP) and rely heavily on nominalist and constructivist themes. Language is a product of the mind, and it describes internal experiences which are products of embodied cognition. Thus, a possible world is part of a truth-conditional construct that can be represented by a model-theoretic object. I'm a logical pluralist, so one needn't even require the Law of the Excluded Middle. Truth, logic, and possibility are ideas, concepts and thus there's no one-true-position on modality. Lewis's modal realism is a defensible argument with presumptions I just happen to reject. Could someone teach or design a computational agent to use and believe 2+2=5? Sure. It's a fact it can be done. The question then is more a question of utility. Would it be of any value? Certainly not if you're trying to build an accounting system or train a child to do arithmetic and function in the real world. In that sense, the a priorticity of "2+2=4" has build in normativity. There are no gods to hand down truths on high in these matters. We prefer 2+2 is 4 to is 5 because it's more elegant, and more importantly leads to closure of operations. Hence, in my view "2+2=4" is a contingent truth, one my intuition and reason defend because I want my number systems to obey closure. I could easily define '+' piecewise anyway I want and even be consistent with physicalist models.

So, you'll find lots of people who will give you pet theories or explicate the theories from the canon on these matters, but any argument put forth is just that, an argument. Whether or not you are moved by it is what philosophy is all about.

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"when there always seems to be a possible world" - the crux of your argument thus appears to be this: that concievability implies (metaphysical) possibility, ie existence of a possible world.

This is a well-trod out debate in the literature. Some key points

(a): It is clear that we will have to clarify our notion of concievability so that we might work with it. For example, any notion of agent relative concievability will face problems. To illustrate, suppose that you are bad at math, hence you can concieve of a largest prime number. Meanwhile, your professor, who is better, and not an ultrafinitist, knows that such a thing cannot exist. So agent relative conceviability, since relative to that agent's epistemic state, is tricky to argue from, since metaphysical possibility is not agent relative.

(b) Kripkean a posteriori necessities also throw a wrench in the matter. For example, one might have been able to concieve of water not being H20, but in fact (if you follow Kripke), this is an aposteriori necessity.

(c) Chalmers introduces a whole host of notions of conceivability, by his lights, only two survive the twin criteria of (a) and (b).

For a brief introduction as applied to one of the most famous arguments in phil mind, see https://plato.stanford.edu/entries/zombies/#ConcArguForPossZomb. And if you'd like to see it discussed with respect to theology, consider checking out the modal ontological argument for God (Plantinga, Oppy).

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How can any proposition be necessarily true when there always seems to be a possible world in which it may be false?

Propositions are understood as true or false of the real world. Thus, given one's assumption of what the real world is, some statements will be true of it and some statements will be false of it.

Necessity is logical necessity. If A is true, then A is (necessarily) true. If A is true, then B is possibly true and possibly false, though not possibly true and false.

Thus, given one's assumptions of what the real world is, some statements will be true of it, and therefore necessarily true, and some statements will be false of it, and therefore necessarily false.

Necessity only appears as a consequence of one's assumptions, which most of the time are left implicit, which explains that some people come to essentialise necessity.

For example the proposition that "4+4=8" is given to be a necessarily true proposition,

4 + 4 = 8 is regarded as necessarily true given our assumptions as to what the symbols '4', '+', '=' and '8' mean, as well as our assumptions about what the real world is etc.

Absent those assumptions, 4 + 4 may well get to be equal to the Eiffel Tower or whatever.

but there could be some world where mathematics was defined differently rendering it to be a possibly true proposition?

Once you start from different assumptions, you may end up with a different conclusion.

Some assumptions reasonable people always make is that when someone states 4 + 4 = 8, he or she is talking about the real world and he or she is using the usual meaning of the symbols involved in the statement. We all presumably live in the real world, so this is a good assumption, just to avoid wasting our time and energy.

If one wants to talk about alternative worlds that nobody knows that they exist, then ones makes clear the relevant assumption: In some alternative world that I don't know that it exists, 4 + 4 doesn't make 8. Good to know.

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