Known self-evident unproven logical truths

Is there any authoritative source for all known self-evident logical truths that most specialists would agree are true although they can't be proven?

There are many different axiomatic systems, and sometimes it's not even clear that the axioms they admit are taken as self-evident or are regarded by most specialists as self-evident.

I'm only interested in valid inferences involving three or four terms, for example:

(A → B) ∧ (B → C) ⊢ A → C Hypothetical Syllogism

(A ∧ B) → C ⊢ A → (B → C) Exportation

((B → A) ∧ (C → A)) ∧ (B ∨ C)) ⊢ A

((A → B) ∧ (C → D)) ∧ (A ∨ C) ⊢ B ∨ D Constructive Dilemma

• What makes a logical truth "self-evident"? I am not sure exportation and dilemma are so "self-evident". Gentzen's natural deduction rules are probably better at "self-evidence" than most. – Conifold Feb 12 at 21:02
• See Logical Truth for an intro to the debate about logical truth. – Mauro ALLEGRANZA Feb 13 at 9:20
• @Conifold By self-evident, I mean not necessarily obvious but you can see the formula is true just by looking at it and taking the time to understand how it works. – Speakpigeon Feb 13 at 14:38
• With exportation and dilemma I found myself automatically running a quick natural deduction in my head to check, hence the comment. Incidentally, "understanding how it works", i.e. understanding the connectives since this only gives the logical form, amounts to mastering Gentzen's introduction/elimination rules for them, at least on the intuitionistic view, see Garson's Natural Semantics. – Conifold Feb 13 at 18:42

It is contentious even to suppose that logic is concerned with being 'self-evident' at all. The old-fashioned idea that logic represents the immutable laws of thought that hold everywhere for all rational beings has fallen by the wayside. Logic has to do with accounting for how it is that some propositions follow from others, or how some combinations of propositions are inconsistent. Formal logic represents our best efforts to codify these relationships.

As our knowledge of logic grows, we may come to revise our grasp of these relationships, or codify them in a different way. Also, we have learned how to apply logic to different semantic properties or modalities. We may have a logic of provability, or a logic of obligation, or a logic of uncertain belief, and the rules for these will differ from those of simple truth and falsehood. As a result, there is no single logic that applies everywhere, let alone a self-evident one.

You say of the four sentences that you give that they cannot be proven, but they can be proven in the propositional calculus: all of those are theorems provided we understand the → to be material implication. If we read → instead to be some kind of generic conditional in a natural language such as English, then none of them are universally true.

• Could you give an example of a natural conditional sentence falsifying (A → B) ∧ (B → C) ⊢ A → C? – Speakpigeon Feb 13 at 13:19
• By self-evident, I mean the traditional distinction that although not necessarily obvious you can still understand the formula is true just by looking at it and taking the time to understand how it works. – Speakpigeon Feb 13 at 13:22
• How even one proposition could be taken to follow from another without assuming first some axiom or inference rule as self-evident? – Speakpigeon Feb 13 at 13:24
• If you use the material implication, please prove first that it is at all a logical implication without relying on some prior assumption involving a self-evident logical truth. – Speakpigeon Feb 13 at 13:26
• 2. You seem to be grasping after the concept of a priori knowability. Some philosophers of logic understand logical validity in terms of the a priori, but it is contentious. Also, a great many people have historically claimed that some proposition was self-evident when it was just false. Relying on human intuition is apt to lead you astray. – Bumble Feb 13 at 15:46

What exactly are "logical truths"? Truths can be objective or contingent. Inference rules like the ones mentioned in your post are DEFINED by the connectors. The main connector in particular determines how the truth values will follow from the proposition in question. The truth table proves the inference rules to be objective truths for propositions. You should know propositions are not defined as sentences. Many objective truth are not currently provable to science. You must understand PROOF is a scientific term. PROOF expresses that there must be sense verification by a human being for some human being to believe x is true. This is why atheism exists. These human being refuse to accept anything that does not conform to science which is ultimately sense verification. Sense verification is defined by using at least one of the following human senses: sight, hearing, taste, smell or touch. In Mathematical logic you WRITE proofs which use touch. You also SEE the writing be it on paper, black board, white board, computer screen, etc. Can you provide an example of a proof without scientific sense verification? Why does truth have to meet this requirement. Objective truth has nothing to do with sense verification directly. Sure it may be the case that x can be verified by senses but what makes x objective is the truth value is permanent. Objective truths don't " DEPEND on . . . " Contingent truths depend on circumstances and can alternate from true to false at various times. Truth tables do not change do they once we identify the main connector or do you have an example to show truth table values change with specific connectors? Let me be clear. Because you may not be aware of a truth value does not mean there is NO TRUTH value. Many people fall for the "if I dont know x (i am unaware) then x must not be knowable. I am suggesting truth and proof are NOT tied together as you seem to think. Many people believe if there is no proof there is no crime. Similar to if you cant prove x then x has no truth. Your tone makes a reader think without proof there is no truth. You did not make this clear. Do you believe ther is no truth without a proof? You also did not make truth without proof as a possiblility.

• By "logical truth", I mean the dictionary definition, i.e. a logical truth is necessarily true given what it means alone, without reference to any empirical fact of the physical world. – Speakpigeon Feb 13 at 13:36
• The terminology is awful and incorrect. The proper term is LOGICALLY NECESSARY or in Mathematical Logic specifically they use the term TAUTOLOGY. So P is P is a tautology as it can only be true. A triangle has three sides is another example. So the answer still seems to involve truth tables if you are going the Mathematical route. The truth table proves the rules of inference directly. Are you trying to put this to practical use in the real world and that is confusing you? – Logikal Feb 13 at 13:43
• Actually, a really good example of objective empirical evidence falsifying not science but your own notion of "scientific sense verification" is indeed the 2,400 years long history of a number of logical formulas accepted by most thinkers as self-evident logical truths. Simplification, Modus ponens, reductio, etc. – Speakpigeon Feb 13 at 13:44
• Why do you keep using the word OBJECTIVE? You are using it wrongly. If x is objective then x must automatically be universal. That is like me saying I draw squares with four corners. Square implies having four corners doesn’t it? Objective implies universal truth & not some local & temporary truth. An objective truth holds everywhere on Earth. You keep indicating that there’s an objective truth here and not over there or over there the objective truth differs from over here. Are you using the term “Objective” to express the knowledge is independent & not personal opinion? – Logikal Feb 13 at 13:53
• I only used "objective" once, you used it 13 times! And, I used it advisedly. Ah, "universal truth", good! If you know what a universal truth is, then you know what a logical truth is. Same thing. Problem solved. – Speakpigeon Feb 13 at 14:08