Does an implied premise mean a formal fallacy if used in deduction?

Question

Let's just say we have an implied premise:

2.a Socrates is a philosopher (implied premise), but not explicit

Then is the following a formal fallacy?

1. Socrates is a man.
2. All men are mortal.
3. Socrates is a mortal philosopher.

?

If it is, then you would have to say that the following is a fallacy too because some premises are implied and not express:

1. John likes playing football
2. John's friends also like playing football
3. John plays football with his friends.

Implied premises:

1. John plays football because he like to.
   
2. John's friends play football because they like to.
   
3. John and his friends play football together.
   

Or to make it simpler, the following would also have to be a fallacy:

1. My dog is happy when I look after him.
   
2. I always look after my dog.
   
3. Therefore my dog is happy.
   

2.a My dog is not happy when he injures himself (implied premise).

It seems to me that if implied premises don't count in the construction of a valid conclusion, then nearly every system of arguments is a fallacy because all information and premises are hardly ever given. So what about the Socrates example?

Edit: Sorry for making this long, but if implied premises can count, then there can't be fallacies:

1. If P then Q
2. Not P, therefore not Q.

Implied premises:

1a. If M, then Q.
1b. M.

Answer 1

See Enthymeme :

An enthymeme is a logical fallacy in which a categorical syllogism omits a premise that is necessary for the conclusion to be true or omits the conclusion itself. The missing proposition is considered to be implied.

The fallacy is a syllogistic fallacy and a formal fallacy.

Formal fallacy because

a formal deductive arguments is a set of sentences in which some sentences are premises and one is the conclusion, and the inference from the premises to the conclusion is guaranteed by the premises alone. Since enthymemes in the proper sense are expected to be deductive arguments, the minimal requirement for the formulation of enthymemes is that they have to display the premise-conclusion structure of deductive arguments.

Thus, in order to guarantee the formal validity of the argument, it is necessary to supply the missing premise.

See also The Concept of Enthymeme in Aristotle ans see Syllogism with an unstated premise :

An enthymeme (Greek: ἐνθύμημα) is a rhetorical syllogism (a three-part deductive argument) used in oratorical practice. Originally theorized by Aristotle, there are four types of enthymeme, at least two of which are described in Aristotle's work.

The first type of enthymeme is a truncated syllogism, or a syllogism with an unstated premise. Here is an example of an enthymeme derived from a syllogism through truncation (shortening) of the syllogism:

"Socrates is mortal because he's human."

The complete formal syllogism would be the classic:

All humans are mortal. (major premise – unstated)

Socrates is human. (minor premise – stated)

Therefore, Socrates is mortal. (conclusion – stated)

While syllogisms lay out all of their premises and conclusion explicitly, these kinds of enthymemes keep at least one of the premises or the conclusion unstated.

And see also : Roy Sorensen, Are Enthymemes Arguments?, NDJFL (1988) :

Although there is disagreement as to how 'enthymeme' is to be defined, there is a consensus that all enthymemes are invalid arguments.

Answer 2

The examples of reasoning you present are NOT properly formed arguments as written. The idea of hidden premises go back to the times of Aristotle and his encounters with Sophist. The techniques was very popular in what is now called Rhetoric. The fact is human beings were reasoning prior to Aristotle was born. When Aristotle formalized logic he did not do so to teach but to illustrate there are distinct techniques one can use. You can reason the way these Sophist guys reason OR you can reason THIS way. Mind you many Sophists were labeled as con men or deceivers because of their reasoning styles. Neither Aristotle, Plato or Socrates considered Sophists the same as Philosophers. The public history indicates could not distinguish philosophy from Sophism. This lead to everyone being called a “philosopher”. This move diminished the respect for the subject from way back then to present day. There is an implicit rejection of the comparison between Sophist and Philosophers. For political reasons people don’t say anything or people just are blind to the message. Don’t call a philosopher a Sophist and don’t call a Sophist a philosopher. They reason like x and we clearly do y.

There are rules in which to create syllogisms and it seems to me people just don’t care about rules and do what they want. The propositions you use have to be related. What does that mean? It means you must link the propositions in some manner and in syllogisms this is called the middle term. This is the term that repeats only in the premises and cannot appear in the conclusion. You probably are unaware of this rule and hence why you question the reasoning examples you provide. Strictly formal rules require an argument to have two propositions as premises whether you see them both or not. For every two propositions a conclusion can be deduced whether you SEE IT or not. You cannot have an argument with an odd number of premises for this reason without a hidden premise. Every two propositions I can deduce the third proposition. You adding some random sentence into an argument is not proper and not allowed. The proposition would have to have a common term with one of the other propositions presented. No random sentences. Once you add a proper middle term then you need a valid argument form expressed by the terms a MOOD and FIGURE in classical logic. Once you have correctly formed propositions and a correct figure and form the argument can have missing components technically. The reasoning can be evaluated much easier if everything was visible at once. It is the logicians job to extract the hidden things and then evaluate the argument. I agree it would be nice for every argument to be clearly expressed in written form but emotions or other factors get in the mind of the person who takes short cuts. Life is not that easy for us to expect people to put things on a golden platform just for us. Logicians can still deal with it when people are trying to issue shady reasoning using said techniques. There is no excuse to say some reasoning is bad because you don’t see all the premises or the conclusion is missing when you have two propositions.