# Propositions that are always true, but aren't tautologies?

Consider the following statements:

1. "Snow melts during the day in the Sahara"
2. "A human will die without oxygen"
3. "Photons have no rest mass"

These are statements that are always true, not because of the rules of logic but because of the laws of science. They are different from statements like "A parent has passed on his genetic material" or "The bachelor doesn't have a wife".

What type of statement are 1,2,3?

• Regarding 3, that's for rest mass. The in practice relevant mass, the mass that attracts gravitationally, is relativistic mass, and photons (and all other kinds of particle) have that. One doesn't need advanced math to see that. E.g., as a gedanken experiment, place a heavy hydrogen bomb inside a perfectly reflective shell, with a little particle X in orbit around that. Let the bomb detonate, converting part of its mass to photons. The gravitational attraction of the sphere doesn't change, X continues orbiting. So. May 19, 2015 at 21:09
• I stand corrected. May 19, 2015 at 21:13
• I am not sure what "always" here refers to, usually "always true" refers to "true in all possible worlds", which these propositions are not. It can't refer to time either, snow did not melt in Sahara during the ice age. One can even imagine that in some distant future humans will alter their physiology to survive without oxygen. These propositions are simply true statements about our world, and now. May 19, 2015 at 21:55
• @Conifold: I think "always" here refers to "under all reasonable interpretations". That's in principle a vague thing, but so is the concept of entropy. For example, interpreting a sentence written in the present form about Sahara, written in 2015, as if should be referring to the situation a million years hence, isn't reasonable. It is the kind of statement that can go into a dead wood textbook. Because it's likely to remain true longer than the textbook lasts. May 20, 2015 at 0:50
• Would they not just be called facts?
– Alex
May 20, 2015 at 10:19

## 4 Answers

Perhaps what you're looking for is the distinction between a contingent and a necessary truth. In order to distinguish between the two, it is best to think in terms of "possible worlds" (conceiving of some world or reality where things could be different than they are in our world, usually as a result of altering the laws of nature or the chain of historical events).

Contingent truth: a proposition that is possible, rather than necessary. A proposition that is possible is neither necessarily true (a proposition that is true in all possible worlds) nor necessarily false (a proposition that is not true in any possible worlds). If there is a proposition that is true in our world, yet one can conceive of a possible world where that proposition would be false, then it is a contingent truth. For example, the proposition "If I throw this baseball, then it will fall to the ground" is contingently true because of the presence of gravity. One can easily conceive of a world where gravity does not exist (i.e. that world's laws of nature differ from our laws of nature) and therefore the proposition would not be true in that world. This is the idea behind the modal operator for possibility - a proposition preceded by such a modal operator is true if it holds true in at least one possible world. It would seem that your three propositions are contingent truths, and therefore by definition not tautologies.

Necessary truth: A proposition that is true in all possible worlds, meaning it is impossible to conceive of a possible world where the proposition is not true. For example, the proposition "P v ~P" (It is always the case that either P or ~P) is a necessary truth. It is [widely believed to be] impossible to conceive of a possible world, regardless of the nature of that possible world, where "P v ~P" does not hold. These types of truths essentially will always hold true even if humans didn't exist. (I've been told before that mathematical and logical truths are not identical, but putting that aside for a moment, think about 2 + 2 = 4. Even if there were no objects to count or humans to compute basic arithmetic, 2 + 2 will never equal another number - keep in mind this is not meant to bring into discussion the nature of perception and reality.) This is true of all theorems of logic. In propositional logic, a theorem is a proposition, or conclusion, that does not rest on any previous assumptions - it's inherently true.

Great question and props for noticing that there is something fundamentally different between the two types of propositions you quoted in you question.

In logic, a tautology (from the Greek word ταυτολογία) is a formula that is true in every possible interpretation. – Wikipedia

There is an 'interpretation' possible in which snow does not melt during the day in the Sahara / a human lives without oxygen / photons have no mass. That is because these statements can only be verified with a posteriori knowledge.

Tautologies are always true a priori. For example, (P ∨ Q) → (Q ∨ P) is true under every possible interpretation of P and Q because of truth tables, and is hence a tautology.

Or, as you said, "These are statements that are always true, not because of the rules of logic but because of the laws of science."

Or, in even other words: it is not impossible to imagine a cold Sahara, a human who doesn't need oxygen, or a photon that has no mass; but it is impossible to imagine P and Q such that (P ∨ Q) → (Q ∨ P) is false.

I am not aware of a special name for propositions that are always true in our world.

• Re "a special name for propositions that are always true in our world", those are called "facts". :) But upvoted anyway. May 19, 2015 at 21:11
• @Cheersandhth.-Alf the word 'fact' is a little complicated: en.wikipedia.org/wiki/Fact#Fact_in_philosophy; but yeah, something along those lines.
– user2953
May 19, 2015 at 21:13
• Really old question/answer, but it might be helpful to note that "true in every possible interpretation" requires holding certain meanings fixed across interpretations (usually the meanings of the logical connectives and other items in the "signature" of the model). While I'm sure you know this, it's an easy stumbling block for novices since there are competing logics that reinterpret, e.g., negation, and strip classical tautologies like the law of excluded middle of their tautological status. Jul 5, 2017 at 21:24

Kant described a typology of propositions before embarking on his critical philosophy.

He divided them between synthetic and analytic propositions which is essentially a grammatical distinction: the predicate is contained within its subject; examples of this are your counter-examples - the bachelor without a wife etc.

The second distinction is a priori and a posteriori; where the truth of the proposition relies on experience.

All of the propositions in your question are synthetic and a posteriori.

• Hm. I'm sorry to learn that synthetic versus analytic has so shallow a meaning, as you say "essentially a grammatical distinction", in philosophy, just as I learned in other commentary here that "fact" has been endowed with muddying-the-waters meanings in philosophy. Anyway, in the real world, as an example, it's very much easier to synthesize speech than to analyze it. And that means that e.g. Stephen Hawkings can easily generate speech, but can not so easily (I doubt he has it) get people's speech up as text on a screen. May 19, 2015 at 22:58
• Something is muddled in the second paragraph. Analytic and synthetic judgments differ conceptually and when we are lucky grammar also maps that distinction. Analytic judgments can be reached merely by understand a single concept and working out its corollary. Synthetic judgments require us to combine multiple claims to reach a conclusion (plato.stanford.edu/entries/analytic-synthetic). May 19, 2015 at 23:35
• @virmaior: there's undoubtedly more subtleties in the synthetic-analytic distinction than I've made out; when I said 'grammatically' I didn't mean in a specific determinate manner, but more in line how Kant explicated it; in the SEP entry you quoted, he is quoted as writing 'Either the predicate B belongs to the subject A as something that is contained (covertly) in the concept A; or B lies entirely outside of the concept A, though to be sure it stands in connection with it'. May 20, 2015 at 0:05
• It's that description I was referring to as 'grammatical'; and that because he was using the notions of subject and predicate; it's not 'pure' grammar in the way say Panini might have systemised Sanskrit - there is a semantic range or concept involved too - which is why I suppose you referred to them as concepts too. May 20, 2015 at 0:13
• @Cheersandhth.-Alf: see above + Kants not particularly interested in the distinctions outlined above for themselves: what he was after was the possibility of synthetic a priori judgements; which on the face of it is paradoxical possibility - but he goes on ahead to justify this possibility. May 20, 2015 at 0:16

The formulation of a proposition as property is not tautological, but its reformulation as definition (and anyway a definition as such) is tautological.

Propositions which are formulated as properties (e.g. 1,2,3, — 3 with the correction in the comment above) are always true. As propositions,°) properties are not tautological because they are empirical results.

In contrast to this, “A parent has passed on his genetic material” is the reformulation of a property (namely: “A parent my be identified by DNS.”) as definition. Albeit “The bachelor doesn't have a wife”, is anyway a mere definition. Definitions are always tautological.

Footnote:

°) But note, when not just used as isolated propositions, i.e. when practically applied as deduction, properties are used tautologically, because then they are merely reproducing circularly the result of the earlier induction by which they have been established.