What would be the relation between logic and philosophy?
To quote Wikipedia:
Logic is the use and study of valid reasoning. The study of logic
features most prominently in the subjects of philosophy, mathematics,
and computer science.
Philosophy is the study of general and fundamental problems, such as
those connected with reality, existence, knowledge, values, reason,
mind, and language. (Wikipedia)
Therefore philosophy is not primarily the art of persuading; rhetoric is:
Rhetoric is the art of discourse, an art that aims to improve the
capability of writers or speakers to inform, persuade, or motivate
particular audiences in specific situations. (Wikipedia)
As for your example
i.e. Wouldn't you say that if I like people I should like animals
To me that form of argument seems like an analogy which is also a form of logical reasoning whose validity is however not necessary, and therefore doesn't qualify as a definite argument or statement in scientific or philosophical discussions, nevertheless, as we'll see later, it can serve as a compelling subject for logical examination (scientific and/or philosophical).
That's because since in science and (especially in) philosophy we are pursuing definite and certain knowledge about beings, only arguments that are logically necessary and do not entail uncertain implications are recognized as acceptable arguments. And since the strongest form of argument is deduction, they are the primary form of argument in philosophy.
Additionally to ascertain that beyond the formal validity of arguments, their constituent premises (axioms) are also true all statements must be reducible to the most basic and evident truths whose validity is 100 percent certain and definite. These self-evident truths or principles include the law of identity and non-contradiction. See this article for primer discussion. Every other premise beyond them must be ultimately traced back to them via logically valid line of analysis.
But if your axioms have not been directly or indirectly inferred from the first-evident principles, through a logically valid line of reasoning -- where only deduction yields 100 percent certainty -- then it means that you have simply no philosophy. In other words if your axioms are 'shaky' then it means that the entire system of statements inferred from them will be consequently shaky; a house of cards that will partially or entirely collapse once some or all of your premises are negated.
Example of Applied Inductive and Deductive Reasoning
Now as an example of applied deductive and inductive reasoning, let’s take your own argument and see if we can verify it through these modes of reasoning:
We first form a problem: why do humans love animals?
Then form a general hypothesis: Humans love things with which they have something in common.
We prove it through induction: We love our spouses, our siblings, our colleagues, our fellow nationals with all of whom we have something in common.
We form a second hypothesis: Humans have a lot in common with animals too.
We prove it through induction: Animals are animate and have emotions, like us.
Therefore it follows by deduction that: we love animals because they have a lot in common with us.
The deduction as laid out in common pedagogical form:
Humans love things with which they have something in common.
Humans have certain things in common with animals.
Therefore Humans love animals.
However if any of our premises are disputed, our conclusion may be either entirely undermined or only modified. For example if we argue that: Humans do not necessarily love things with which they have something in common under certain circumstances. For example they do not love thieves so long as they want to steal from them. And they do not love predatory animals so long as they want to attack them.
However as you can see the above challenges can at best only limit the implications or conditions of validity of our conclusion which can be easily sorted out by applying slight changes to our premises and conclusion. But at times they can entirely undermine the conclusion. This can be done for our example if one can completely disprove that humans love things with which they have something in common.