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yes, but not if it is illogical.

Q: Is a Conjecture accepted without proof in a logical system?

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No, theorems cannot be accepted until proven.

Such terms are called axioms, or postulates.Exactly which terms are defined to be axioms depends on the specific system used.

Axioms and Posulates -apex

Because mathematics is a axiomatic system so that every new statement remains a conjecture until it is proved.

I don't know what you mean by solving logical deduction. Do you mean how do you tell, given an allegedly logical deduction, whether it really is logical? Or do you mean, given a theorem, how do you logically prove it, that is, prove that it logically follows from the axioms? The last question is very complicated. Some theorems have taken centuries to prove (like Fermat's last theorem and the independence of Euclid's Parallel Postulate), and some have not yet been proven, like the Goldbach conjecture and Riemann's hypothesis. The first question is much simpler, but to describe exactly how to verify the validity of a deduction, we would need to know what kind of deduction it is. For example, a deduction involving only logical connectives like and, or, if-then, not can be verified with a truth table. Those involving quantification or non-logical symbols like set membership require looking at the proof and seeing that each step can be justified on the basis of the axioms of the system, whether it is the system of Euclidean Geometry, of the field of real numbers, or of Zermelo-Frankel Set Theory, etc.

Related questions

Postulates and axioms are accepted without proof in a logical system. Theorems and corollaries require proof in a logical system.

yes

No, theorems cannot be accepted until proven.

axioms

Axioms, or postulates, are accepted as true or given, and need not be proved.

Such terms are called axioms, or postulates.Exactly which terms are defined to be axioms depends on the specific system used.

Axioms and Posulates -apex

Postulates and axioms.

An axiom is a statement that is accepted without proof. Proofs are based on statements that are already established, so therefore without axioms we would have no starting point.

The question asks about the "following". In those circumstances would it be too much to expect that you make sure that there is something that is following?

The statements that require proof in a logical system are theorems and corollaries.

The statements that require proof in a logical system are theorems and corollaries.