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Contingent sentences can always be true
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Problem
I have a question about contingent sentences. To my knowledge, contingent sentences are sentences that are neither tautologies nor contradictions.
In a textbook I read that a sentence might always be true and still be contingent.
The example given was as follows: If there never were a time when the universe contained fewer than seven things, it follows that the sentence "At least seven things exist" would always be true. Yet the sentence is contingent; its truth isn't a matter of logic. There is no contradiction in considering a possible world in which there are less than seven things.
Could someone please explain why this is the case? As well as explain the example to me?
In a textbook I read that a sentence might always be true and still be contingent.
The example given was as follows: If there never were a time when the universe contained fewer than seven things, it follows that the sentence "At least seven things exist" would always be true. Yet the sentence is contingent; its truth isn't a matter of logic. There is no contradiction in considering a possible world in which there are less than seven things.
Could someone please explain why this is the case? As well as explain the example to me?
Solution
In mathematical logic - the discipline of logic that has interested computer scientists the most - there is a difference between proof-theoretic truth and model-theoretic truth.
Beyond metaphor (of which we are by no means innocent), neither has anything to do with reality, experienced or imagined, and with the idea of truth that is dealt with by empirical scientists, natural philosophers and historians, to which you allude in your question.
Contingent is, to us, another name for a sentence that is satisfiable, but not a tautology. This has to do with its truth equivalence being dependent on the axioms and rules of inference available (proof-theoretic aspect), or on the valuation of its symbols (model-theoretic aspect).
Gödel showed us that these two aspects do not necessarily go together very well, though. Caution is advised.
Beyond metaphor (of which we are by no means innocent), neither has anything to do with reality, experienced or imagined, and with the idea of truth that is dealt with by empirical scientists, natural philosophers and historians, to which you allude in your question.
Contingent is, to us, another name for a sentence that is satisfiable, but not a tautology. This has to do with its truth equivalence being dependent on the axioms and rules of inference available (proof-theoretic aspect), or on the valuation of its symbols (model-theoretic aspect).
Gödel showed us that these two aspects do not necessarily go together very well, though. Caution is advised.
Context
StackExchange Computer Science Q#66834, answer score: 5
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