Question

Deductive philosophical arguments are often presented semi-formally as a list of premises and the conclusion (and sometimes combinations of such sub-arguments). What is virtually never stated are the rules inference, probably because they're assumed to be uncontroversial.

E. g. in this argument:
first premise: if there is a chance of snow this weekend, I will not go out.
second premise: there is a chance of snow this weekend.
conclusion: I will not go out
the inference is just made by modus ponens, which can be easily omitted.
But e. g. in this rendering of Descartes' argument for the existence of the soul (Richard Swinburne: Are We Bodies Or Souls?, p. 72f) :
first premise: I am a substance which is thinking.
second premise: it is conceivable that ‘I am thinking and I have no body’.
third premise: it is not conceivable that ‘I am thinking and I do not exist’.
lemma: I am a substance which, it is conceivable, can exist without a body.
conclusion: I am a soul, a substance, the essence of which is to think.
the rules of inference are not so simple. Certainly what's applied here (modal logic, identity of indiscernibles?) is more powerful and controversial than modus ponens.
But in a highly formal context the rules of inference are always explicitly stated. And there's a trade-off between
large number of axioms and weak / simple rules of inference (Hilbert-style)
few axioms and powerful rules of inference (natural deduction)
Is it possible to construct an example roughly analogous to these two approaches but for semi-formal philosophical reasoning?

Answer 1

Deductive arguments in philosophy necessitate valid inferences where the conclusion must be true if the premises are. A semi-formal philosophical reasoning can be structured to mirror either Hilbert-style or natural deduction, focusing on the trade-off between axioms and inference rules. Understanding the balance between these elements is key to assessing the soundness of philosophical arguments.

Step-by-step explanation:

When constructing deductive arguments, philosophers often use a semi-formal style involving clearly stated premises and a conclusion. However, the underlying rules of inference may not always be explicit, as they are considered to be self-evident. In more formal contexts such as mathematics and logic, these rules are meticulously outlined. Philosophical reasoning can employ various logical structures to ensure the validity of deductive arguments. Validity implies that the conclusion must be true if the premises are true, irrespective of the actual truth of the premises. Three common valid argument structures are disjunctive syllogism, modus ponens, and modus tollens.

Creating an example analogous to the systems of Hilbert-style or natural deduction can be achieved by focusing on the balance between number of axioms and the strength of the rules of inference. A semi-formal philosophical argument akin to Hilbert-style would involve a large number of axioms with relatively straightforward inference rules. Conversely, one that mirrors natural deduction might utilize fewer axioms but apply more powerful and comprehensive rules of inference to achieve its conclusions.

An important aspect of testing these deductive inferences is to provisionally assume the truth of the premises and then scrutinize whether the conclusion logically stems from them. In philosophy, this is crucial for understanding deductive reasoning and assessing the soundness of arguments. Even though inductive reasoning doesn't guarantee the truth of conclusions, it can still generate reliable knowledge, affecting how we form beliefs about the world. By carefully examining the evidence, premises, and logical structure of arguments, we can gain insights into the nature of our knowledge and the world.