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I cannot understand what an argument is with a necessarily true conclusion. Could you explain to me what it is and write here some examples?
Moreover, what is the meaning of “necessarily true” and “necessarily false” in classical logic?
I am a bit confused by the word “necessarily”.
Maybe it is useful to recall the basic definition.
A deductive argument is valid :
if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.
In other words, for the validity of an argument is necessary that the truth of the premises implies the truth of the conclusion.
A simple example of valid argument is the following syllogism :
All A are B;
All B are C;
Therefore, all A are C.
The "necessity" of the entailment relation typical of valid deductive inferences is here expressed by the fact that we cannot [it is impossible] find examples such that both premises are TRUE and, at the same time, the conclusion is FALSE.
Having said that, if the conclusion of an argument is a statement that is always TRUE, like e.g. "Every raven is black or not Every raven is black", applying the above definition we may easily check that this type of argument is always valid.
If you want to manage "modal" operators attached to single statements, see Modal Logic.
In section 3.2, after defining "valid" the authors of forallx define other logical notions: necessary truth, necessary falsehood, contingent sentence, necessarily equivalent, jointly possible and jointly impossible. These concepts may be what you are looking for in classical logic. Together they help to clarify each other.
Here is how they describe necessary truth:
A necessary truth is a sentence that must be true, that could not possibly be false.
They provide this example on page 13:
Either it is raining here or it is not.
You do not need to look outside to know that it is true. Regardless of what the weather is like, it is either raining or it is not. That is a necessary truth.
In chapter 38, they also discuss modal logic where necessity and possibility are two modalities.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf
What do you MEAN by "Classical Logic"? In today's society many people think classical logic means symbolic logic aka Mathematical logic. If you mean math you should SAY or Express you mean MATHEMATICAL reasoning which differs from pure deductive reasoning. Math is a sub set of deductive reasoning. 150 years ago if you would have asked this same question people would not think math but Aristotelian logic.
The concept of necessity can come about in more than one way. One way is answered above by Mauro. He described the definition of validity which necessitates the conclusion being true if the premises are true. That is, you can make the conclusion false no matter how hard to try once the premises are true and related in a logical manner. It is indeed impossible to do. If you meet someone who claims they can do it watch the relationship of the premises and conclusion. Chances are there is no logical relationship, the propositions being used as premises are not properly formed, or something linguistic is being swapped from the real topic. A logical relationship is necessary when the result could not possibly go another way. Sometimes this is built into language such as all triangles have three sides. There are also objective truths that are neccessary such as all women are human beings. This latter example is not linguistic as I do not refer to the term used but the actual beings physically. Even if you want to play word games all you would be doing is changing on word for another that expresses the exact same idea. This is why propositions are NOT sentences. The idea being expressed counts not the specific terms used. Propositions EXPRESS an idea and are not sense verifiable themselves. We make the sentence to Express the proposition to others. So no nit picking on words being used in an argument--that is a psychological effect not a logical one.
In summary, a necessity is a relation where the result is unavoidable. Keep in mind the human factor: people can make false claims out of ignorance. So because I say you is p occurs then q occurs DOES NOT mean that it is true in reality. A necessity is a case where the result is forced in a sense. This also reflects the notion of objective knowledge as the result cannot vary truth values but remain constant forever. For example, in the game of Chess there are unavoidable checkmate patterns. That is if you or anyone falls into a mate pattern you WILL LOSE objectively once the correct moves are played. The words smartest computer would not be able to avoid the mate NOR would the Chess World Champion regardless of who that is. What happens if the premise is necessarily false? Well from a practical standpoint the argument will not likely be persuasive. From a technical standpoint the argument can still be valid. I would discourage the thought or IDEA in your mind that if I argue with you and my premises are blatantly false you can claim my argument is not GOOD or that my conclusion must be false. Do not do that move. That is fallacious thinking --a logically bad thing to do.
What is an argument with a necessarily true conclusion?
An argument typically has premises and a conclusion. An argument which is such that once you assume the premises true then the conclusion can only be true is said to be logically "valid".
For example:
Ex. 1 - Aristotle is not here but he could have given you the answer you want; therefore, Aristotle could have given you the answer you want.
Ex. 2 - It rains and the sun is shining, therefore it rains.
"Necessarily true" just means that given the truth of the premises, it is not possible that the conclusion be false.
The fact that the premises may in fact be false themselves is irrelevant. "Assuming the premises true" means that you choose to accept they are true irrespective of whether they are actually true.
For example:
Aristotle was Russian and all Russians ate caviar for breakfast; therefore, Aristotle ate caviar for breakfast.
The conclusion is presumably false. And so are the premises. However, once you choose to assume the premises true, then the conclusion can only be true as well. Said differently, if the premises were true, the conclusion would have to be true as well.
The fact that the conclusion is sometimes necessarily true given the truth of the premises comes from the form of the argument. It is the form of the argument that makes the truth of the conclusion "follow from" the truth of the premises.
The notion of "necessity" here is crucial. It conveys the idea that once someone has accepted that the premises are true, they are compelled to accept the truth of the conclusion, and this merely by virtue of the form of the argument that makes the truth of conclusion follow from he truth of the premises.
This is crucial also because this is what ensures that we can agree on some conclusions. If we both accept the truth of the premises and if we can both understand that the truth of the conclusion follows necessarily from the truth of the premises, then we will agree that the conclusion is true.
