I am following along the course "Language, Proof, and Logic" from Stanford on EdX.
I was confused by the question of the distinction between tautology and logical truth (aka logical necessity).
One particular explanation from a lecture, and then a lecture from the textbook cleared up a lot for.
I'm going to try to give some intuitive ideas here, that may be vague in certain respects (because I am not an expert in logic), but which helped me to understand the distinctions of the terms we are discussing.
logically possible claim: a claim is logically possible if it could be (or could have been) true, at least on logical grounds. By "logical grounds" is meant "that satisfies basic logical axioms."
For example, it is logically possible for a space ship to travel faster than the speed of light. It is not physically possible, but there is no logical axiom that is broken by something traveling faster than the speed of light.
An object not being identical to itself is not, however, logically possible. This would violate the meaning of identity, which is part of some basic axiom of logic.
Another way to think about a claim being logically possible is: a claim is logically possible if there is some logically possible circumstance (or situation or world) in which the claim is true.
Next we have the terms logical truth and logical necessity. As far as I understand these are the same thing. A claim is a logical truth aka logical necessity if it is true in every logically possible circumstance or situation; it is a sentence which cannot be false.
Consider the (atomic) sentence a=a. If we build a truth table for this atomic sentence, it has two rows, one for T and one for T.
By the definition of tautology, this is not a tautology: the sentence is not true for every possible truth value of its constituents (and note that there is only one constituent, the sentence itself).
However, it is not logically possible for a=a. Therefore, despite not being a tautology, a=a is a logical truth, aka a logical necessity.
Finally, let me show the diagram. The diagram includes some information which is specific to the course, in particular this thing called "Tarski's World". You can ignore it for the purposes of this discussion. You also don't have to know what the predicate Tet() is. The important thing is that any claim of the form P & ~P is a tautology.
Seems like a tautology is a claim that is true for whatever configuration of the atomic sentences it is composed of, and this means that the actual structure of the claim makes it so.
However, there are structures of claims that when you build out the truth table you see that the claim isn't true in all configurations. However, there is some other stronger constraint on what is possible than just the structure of the connectives of the claim. Such a stronger constraint is, for example, what I am conjecturing to be fundamental axioms/laws of logic.
This is what is exemplified by a=a or, as in the diagram above, also something like ~(Larger(a,b) & Larger(b,a)). If we were to use some other predicate instead of Larger(), we might get something which weren't a logical necessity. For example ~(Likes(a,b) & Likes(b,a)). It is totally conceivable that a likes b and b likes a, making the whole claim false. But it is not possible for b to be larger than a, and a to be larger than b. This particular predicate is admittedly very vague, and maybe there is some esoteric field which allows two objects to each be bigger than the other. If a and b represent physical objects, I believe it would probably be justified to say that if Larger means having more mass, then the claim in question, ~(Larger(a,b) & Larger(b,a)), is a logical truth.