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For an example, it looks like I have an example of an argument which is both deductive and informal:

Gabriel is a wolf

Gabriel has a tail.

Therefore, Gabriel’s tail is the tail of a wolf

I consider it informal because there don't seem to be formal logical system that allow for such structure.

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Your argument is indeed valid, but it can readily be formalised using first order predicate logic, so it would not qualify as an example of an argument that is deductively valid and informal. First we should note that the second premise strictly means that Gabriel has at least one tail, i.e. it does not exclude Gabriel having more than one. Obviously our background knowledge of wolves is that they have only one tail unless they are strange mutants, but in logic we prefer to be strict about such things. The conclusion however, uses a definite description, which indicates a unique tail. Therefore, to make the argument more accurate, we should either amend the second premise to "Gabriel has one and only one tail", or amend the conclusion to "Gabriel has a tail that is a tail of a wolf".

The second option is simpler, but either would work. We can formally write the premises and conclusion as:

P1. Wolf(gabriel)

P2. (∃x)(TailOf(x, gabriel))

C. (∃y)(∃x)(TailOf(x, gabriel) ∧ Wolf(y) ∧ TailOf(x, y))

The conclusion is provable from the premises by conjoining the premises and then using existential generalisation.

A more general answer to your question is that it depends on what you are willing to count as a deductively valid argument, and there is, perhaps surprisingly, rather a lack of consensus on this issue. Some logicians, in a tradition that goes back to Abelard, distinguish formal and material validity, but count both as types of logical validity. On this position, arguments such as "Edinburgh is east of Glasgow; therefore, Glasgow is west of Edinburgh" would count as materially valid, since the conclusion follows from the premise, but not formally. Other logicians reject the concept of material validity and would treat such an argument as enthymematic, i.e. as having a hidden premise to the effect that: for any x and y, if x is west of y then y is east of x and vice versa.

Another issue that is relevant to your question is how one chooses to treat identity. George Orwell is identical with Eric Blair, but does the argument "George Orwell wrote 1984; therefore, Eric Blair wrote 1984" count as valid? It is not formally valid, but on a popular view of necessity, if the premise is true then the conclusion follows by necessity. Again, we could go either way and say on a broad view that the argument is valid because it is necessarily truth preserving, or we could restrict our understanding of what counts as logic to that which is formal, and say that the argument is only valid if we treat "George Orwell = Eric Blair" as a hidden premise.

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    The issue here with the reasoning is that the conclusion is almost identical to the premises you began with. This is circular reasoning by definition. Formally circular reasoning is VALID. By definition of validity, ciircular arguments cannot have true premises & a false conclusion.If you disagree how would you counter this objection to the reasoning being VALID.People must understand the term VALID has undergone CHANGES from Philosophy to Mathematics. I am showing there are issues in reality with the reasoning. Philosophy did not concern with VALIDITY when SOUNDNESS is a higher principle.
    – Logikal
    Aug 6 '21 at 15:51
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    Just to be clear here: formally VALID ARGUMENTS can be fallacious in reality. Consider the following syllogism which you can translate into Mathematical logic if you desire: All empty sets are mammals. All unicorns are empty sets are mammals. All unicorns are mammals. Mathematics say logic is about validity & then says logicians are not concerned with the truth of the premises. This is a immediate inconsistency. Why is SUBALTERNATION FALSE again? Because of existential import correct? You would say All unicorns are mammals is true but then say SOME unicorns are mammals is FALSE.
    – Logikal
    Aug 6 '21 at 16:23
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    If the logician doesn't concern himself with whether premises are true or false how would existential import arise? Consider this: If all boxes have four sides. This is a box. Therefore this object has four sides is VALID. BUT when you ask what is inside the box that is not about FORM OR VALIDITY. The Mathematical people say there Are no existing UNICORNS in the real world so there being at least one unicorn is FALSE. BUT math people maintain logic is not about CONTENT but VALID FORM at the same time they don't care if the premises are true or false. They seem to clearly care about UNICORNS!
    – Logikal
    Aug 6 '21 at 16:29
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    Make that make sense. A circular argument would be formally valid but fallacious in the real world: consider all unicorns are white mammals. Therefore unicorns are mammals. Seems impossible to have a false conclusion if the premise is true BY DEFINITION. In the real world we would not accept this reasoning. Thus we have cases where formal validity hinders true knowledge or useful knowledge we can apply to the real world as human beings. Make it make sense: how can reasoning be valid & also be a fallacy? Fallacy means there is at least one center example that can be shown to be false.
    – Logikal
    Aug 6 '21 at 16:34
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Not necessarily.

A deductively valid argument is one for which, given that the premises are true, the conclusion can't be false.

A formal proof is one constructed according to a set of formal string-rewriting rules, such as ZFC, or one that can be translated in a straightforward way into such rules.

An informal argument, by contrast to a formal one, is an argument that is not constructed according to a set of formal string-rewriting rules, and can't be translated in a straightforward way into such rules.

Now, certainly the vast majority of deductively valid arguments are also formal proofs. Mathematics and logic are the only reliable ways we have devised of producing deductively valid arguments, and the methods in these fields can normally be formalized.

But there are possible exceptions. Consider Gödel's first incompleteness theorem. Let S be a Gödel sentence for arithmetic. It is often claimed that S is true, but unprovable in arithmetic. If you hold that S is necessarily true, then your argument is informal - not based on strictly formal string rewriting rules in arithmetic - but you claim it is valid. The same goes for Gödel sentences in any other formal system.

For another exception, consider the halting problem. Some programs halt, some do not. Additionally, some programs do not halt and can be proved not to halt, and some programs do not halt but can't be proved not to halt. (This follows because if you could formally prove for every program whether it halts, you would have solved the halting problem, which we know cannot be done.) Now, let P be a program that does not halt, but for which no proof of its non-halting can be obtained. The argument, that takes as premises the rules of Turing machines, and gives as a conclusion, "P does not halt," cannot be a formal one, so we can only say it is an informal one. But it is a deductively valid argument, since P in fact does not halt; the conclusion cannot be false while the premises are true.

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I want to write an answer here because some of the comments I added above were modified by intellisense. Therefore some statements will likely appear unintelligible to some readers. Hopefully I can clarify some misunderstandings here.

There are multiple types of reasoning. Let us get some pure facts out of the way. We know all reasoning is not deductive. We know there is something called INDUCTIVE REASONING, ABDUCTIVE REASONING and so on. This then begs the question how are we to distinguish these different types of reasoning.

The OP specifically mentions DEDUCTIVE REASONING. So some distinctions are in order. What makes deductive reasoning unique is that this form of knowledge if done correctly is ABSOLUTE. That is, if you actually follow the rules literally it will be impossible to get the conclusion wrong. Another way to say this is of you get the wrong conclusion you had to violate at least one of the many rules. There is no such thing as you followed ALL of the rules and you end up with the wrong conclusion. Even in other subjects this rule holds: 5 x 5 = 25 or too must have violated at least one rule of mathematics. You can't follow all of the rules and get another answer than 25.

This brings up categorical syllogisms because there were times there was no such thing as Mathematical logic as we know today. Aristotle did not have it nor did medieval philosophers. So how would they recognize deductive reasoning as opposed to INDUCTIVE REASONING for instance? Well inductive reasoning does NOT guarantee if all the premises are true then the conclusion must also be true. Inductive reasoning defines the conclusion as POSSIBLE. That is if you follow all directions and rules you may still get a different answer than you expected. This is how all sciences operate. There is no science that claims their knowledge is absolutely true under any and all circumstances. [For instance, no science offers absolute truths: NOT Physics, not Chemistry, not Biology, etc.] By definition, all sciences must allow statements that are falsifiable. This is a reason why there can be no automatic proof of God. We can't falsify an assumption which has zero evidence.

Deductive reasoning requires that propositions can be broken into classifications. They used to be described in philosophy as logically necessary and self contradictory. The terms that were used when I learned them were ANALYTIC and SYNTHETIC. No I did not mean how Kant defined these terms. This is context is independent of the philosopher KANT. [See philosopher Wesley Salmon as an example.] Whether you agree with the term names here is irrelevant. My description of the categories will be hard to knock. Logically necessary means that the proposition is impossible to be false. Mathematics calls this a TAUTOLOGY. First instance, all bachelors are unmarried men is true by the nature of how we define words alone. We don't literally have to have worldly experiences to understand this knowledge. With zero experience of running into single males I can grasp the concept of a bachelor. I can say the same thing about triangles. If a geometric shape has exactly three sides then it is a triangle must be true by definition. There are no triangles out there with more than three sides or less than three sides. No worldly experience required even if I have never seen a triangle personally. My knowledge did not come from my personal experience or beliefs. This knowledge came from another source clearly.

Inductive knowledge does exactly come from world experience. By definition alone of words I am not sure if an oven is hot. I am not aware that all birds can't fly by the definition of the word 'Bird'. We can see induction has little to do with SEMANTICS or the field of LINGUISTICS. Linguists governs how human beings use a particular language. Semantics would be a smaller circle inside the larger circle of Linguistics. Induction seems world experience based while the first type is more about language and NOT personal experience. In this way, the deductive reasoning process is not identical to that of the inductive reasoning process. If you have no absolute conclusion you have what we call PROBABILITY. By probability I mean from 1 percent to 99 percent likely to be correct. Deductive reasoning is 100 percent correct if done without error. There is no shot of going 98 percent. We can only use Zero percent or 100 percent. There is no middle ground or option with deductive reasoning. If the premises are true (and RELATED TO EACH OTHER CORRECTLY) then it is impossible for the conclusion to be false. There is no way possible for the conclusion to be false by definition. So if I (or you for example) end up with the wrong conclusion then we know I (or you) had to violate a rule somewhere along the way. Here are the examples given by the OP:

Gabriel is a wolf.
Gabriel has a tail.
Therefore, Gabriel' s tail is the tail of wolf.

Second example, found in the comments section.

No human is a non living thing. No living thing is a non mortal. Thus all humans are mortal

The OP also states he is using a textbook by Patrick Hurley as extra information. What is wrong with the above reasoning? Let's list the issues. For one in categorical syllogisms there is a RULE stated in the HURLEY textbook about the term that repeats in the premises NOT APPEARING IN THE CONCLUSION. Term repeating in the premises is called the middle term and by definition must not appear in the conclusion. The middle term by definition is the term that repeats in the premises. In the example above Gabriel appears how many times? Perhaps you think I am being pedantic here. There is a reason to do so here because not being strict allows ERRORS to creep in. Secondly in categorical syllogisms there exists only 4 types of propositions that can be used (A, E, I, O) and NO MORE THAN THOSE. Singular subjects such as personal names are translates as A propositions. Your text says this doesn't it. This is not Psychology nor Rhetoric. The point here with the philosophy perspective is to eliminate emotive terms, vague terms, ambiguous terms, equivocal terms, etc. So NO you can't just go willy nilly with your wording of the premises. Premises any kind of way you FELL like writing them is not acceptable nor is it uniform for studying arguments. You must change your MODERN TERMINOLOGY to fit within A propositions, E propositions, I propositions, and O propositions without question. Yes we are all aware people don't typically read, write or speak in that fashion. The point is to have a universal system of evaluating arguments without having to get used to each writer's personal writing style. Does that make sense or no? Every premise must fit into the language of only A, E, I, or O propositions or we are allowing rhetoric and psychology into the reasoning which can mislead us. Furthermore if we review multiple arguments the universal formation of premises can be scanned very quickly. An out of place premise will STICK OUT to whomever is reviewing the argument. Why do we not want psychological or rhetorical content ? Because our senses and emotions can decieve us. In addition to being easily deceived there is no way to formally or mechanically trace where we made an error. By following the STRICT --but pedantic-- rules is a process that allows us to examine how we make mistakes in reasoning, EXACTLY WHERE we made the mistake at which step and finally avoid those mistakes. We can catalog mistakes and these patterns are called FALLACIES.

So the reasoning is not purely deductive by definition. There is no logical necessity for the premises to be true in the first example. The propositions offered could be false. Propositions that are deductive must be derived from experience or language alone. This means you should technically be dealing with true propositions to begin with. If not then you are obligated to state it's an hypothesis, axiom, or an assumption. Next there is no middle term used that is NOT found in the conclusion. If I rewrite the same sentence three times us that an argument by definition? The answer is no. Apparently some people think propositions are sentences. Propositions are not sentences. Rewriting or rewording of the exact same idea does not count as two or three different ideas. "You are fired!" expresses the same proposition as "Your employee services are no longer required by the company at this time. Please do not return for work." These are clearly different wording but Express the SAME IDEA and it is counted as one proposition. So Gabriel is a wolf. Gabriel has a tail could really be the start of an awsome poem or perhaps a future work of prose but this doesn't even meet the definition of ARGUMENT in the OP's text book written by Hurley. The second example was a comment above the answers. The OP violates the rules again. You can't have two negative premises. If you do so don't call this deductive reasoning. The conclusion will not be 100 accurate. That is, if I took another argument and followed the same steps will I ever get a false conclusion? If the answer is YES then a mistake was made. The OP must understand the context of ordinary English speaking can differ in different subject matters. Here this word means one thing in another college course of a different topic the same phrase or word means something else. This is common actually. Lawyers can use the word 'concert' completely different from me saying let's go to a Metallica concert. The domain we use the same words in can clearly differ in other domains. The examples provided show the reasoning is a bit more than just deductive reasoning. At best there is a mix of reasoning in the examples. It is not just deductive reasoning alone. I will end here unless more details or clarification is needed.

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  • That was interesting but the many typos make it tougher
    – Al Brown
    Aug 7 '21 at 9:57
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    I had to edit my answer because some of the words were accidentally changed by the intelisense feature. So I corrected some errors. If one re reads the answer now it should be an easier read with less errors.
    – Logikal
    Aug 7 '21 at 20:30
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    I would also recommend not using all caps for emphasis so much (or at all), and keeping it shorter and more directly relevant to the question. These style issues overshadow the points you are making.
    – causative
    Aug 7 '21 at 20:53
  • "You can't have two negative premises." But you can! math.stackexchange.com/a/2888561/161005
    – user161005
    Aug 8 '21 at 3:24
  • Just read this: math.stackexchange.com/questions/2888508/… TLDR; the logic is of example from comments is correct, it's deductive reasoning. But it doesn't conform to rules of syllogistic logic because it's not a categorical syllogism in the first place.
    – user161005
    Aug 8 '21 at 3:29

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