# Is the set of all true contingent propositions equal to the set of all true propositions?

Since the conjunct of a true contingent proposition and a necessary proposition is contingent and hence it is contained in the set of all contingent propositions. Does that mean that the set of all contingent propositions the same set as the set of all true propositions?

• I wouldn't have thought so. – Mozibur Ullah May 3 '18 at 20:36
• "True" where? Truth is relative to the possible world the proposition is evaluated on. Only necessary propositions are true in every possible world, contingent ones are true somewhere and false somewhere, impossible ones are false everywhere. – Conifold May 3 '18 at 21:43
• Truth is not relative. Something True may be relative. Truth is an universally objective thing. If it's not like that, it's not the truth, but at most something true. – Overmind May 4 '18 at 5:12
• @overmind, you are committing a hasty generalization. There are clear distinct TYPES of truth. Some truths are forever true which are objective truths; some truths about claims are NOT constants. That is, sometimes the claim is true & sometimes it is false: i.e., it is raining right now where I am. It is not always raining where I am now. There are semantic truths such as triangles have three sides or a bachelor is an unmarried male. There are necessary truths such as all women are human beings. You need to be clear which type of TRUTH you mean. Do not lump them together as this is confusion. – Logikal May 4 '18 at 22:20
• A truth that is not objectively universally true is not a truth. – Overmind May 7 '18 at 7:08

No. Call T the set of true propositions and T* the set of true contingent propositions. Let P be a necessary proposition.

All necessary propositions are true, so P in T.

But no necessary proposition is contingent, because contingent just means neither necessary nor impossible. So P not in T*. Then T /= T*.

Let Q be a true contingent proposition. Obviously Q in T, Q in T*. Now consider the proposition "P and Q". Now "P and Q" is a true contingent proposition. So "P and Q" in T, "P and Q" in T*.

But "P and Q" is not the same as P, nor as Q. (The use of "the sky is blue and a = a" is nothing like the use of either "the sky is blue" nor "a = a".) So saying "P and Q" in T* does not imply P in T*.

So, in short, the answer is no, but I can understand the intuition behind this. The issue comes from the nature of each of these things; truth and possibly.

What is possible to tends to be divided into three branches:

1) that which is logically possible

2) that which is metaphysically possible

3) that which is physically possible

Whilst, on the other hand, truth is generally taken as a relation between language (or propositions) and reality, depending on which view you're taking. The more popular views on truth tend to be correspondence or deflationary views which equate the truth of a truth-bearer with that truth-bearing expressing something and that thing is the case. In each possible world, the set of true propositions will be specific to that world; each proposition will be true with respect to the reality that is the case in that possible world.

The sets of each of these things will be distinct, as the set of all possible propositions is far greater than the set of all true propositions. This is because the set of all contingent/possible propositions contains all the true propositions, as they must be possible if they are true. But the set also contains non-true propositions; i.e. "the Eiffel tower is in London" is possible on all three versions of possiblity mentioned above, yet it fails to fall into the set of true propositions.

It would be correct, concerning conjunction with necessary propositions, to claim that the set of all contingent propositions contains the set of all necessary propositions. But it wouldn't be the case that the set of all true propositions is identical to the set of all contingent propositions - the latter set is far greater.