Deductive reasoning
At the beginning of this blog, we spend some time thinking about proper ways of reasoning. We need to make sure that we are going to think and reason in a way that won’t lead to too many errors. What good would it be to spend a lot of time thinking about various issues, and drawing conclusions, if we made mistakes along the way? We want to make sure that our arguments are correct.
Let’s first look at some examples. “If I don’t fill the tank, I will run out of gas before I get back home” would be a very simple, mundane argument, that we might not even utter consciously. At the other end of the spectrum, here is one of many arguments for the existence of God: “I can conceive of a supremely perfect being ; existence is a perfection ; therefore God exists” (this is adapted from Descartes’ ontological argument, in his Fifth Meditation).
We notice that these series of statements, or propositions, rely on logic and have a purely formal structure or pattern, something like: “if A then B; A is true; therefore B is true”. Logicians have systematically codified the formal structure of many types of arguments. Syllogisms, for example, are arguments like: “All humans are mortals, Socrates is human, therefore Socrates is mortal”. This argument has the form: “All As are Bs, C is an A, therefore C is a B”, which turns out to be a “valid” form of argument. Logicians talk about “valid” and “invalid” argument forms in a precise way. An argument form is valid if and only if it is not possible to instantiate it (replace A, B, C by actual propositions) in such a way that the premises (what comes before “therefore”) would be true, yet the conclusion (what comes after the “therefore”) would be false. If an arguments has a valid form, if you grant the premises (if the premises are true), you have to grant the conclusion (the conclusion is necessarily true). In the argument about Socrates just above, you cannot escape the conclusion if you grant the premises, just because of the logical form of the argument. Here is an argument with an invalid form, therefore an incorrect reasoning: “John owns a house ; therefore John owns a house and a piece of land”. Can you reconstruct the formal structure of that argument, and see why it is not valid?
Correct arguments need to have logical consistency, but that is actually not enough for an argument to be convincing. For example, “All animals with wings fly ; penguins have wings ; therefore penguins fly” is a valid argument, the logic is correct, yet it is ultimately false, because one of the premises says something incorrect about the world. For an arguments’ conclusion to be true, we require that the premises be true on top of the argument being logically valid. We want the premises to say something true about the world, or at least something that we all agree on, or that we think is true, if we can trust our senses … (philosophical rabbit hole in sight!). This is referred to as the semantics of the argument. A correct, true, convincing argument need to have both a valid logical structure, and true semantics. Logicians referent to that last requirement as “soundness”.
If an argument is valid, we know that whatever the truth value of the premises, we correctly calculate the truth value of the conclusion from that of the premises. Validity only ensures that we calculated right, without regard to whether the premises say something true or not. If an argument is valid and its premises make some claim that we regard as true about the world or the universe of discourse, then it is also sound, and the conclusion makes a substantive, true claim about the world. Note that it is not always straightforward to retrieve the logical form of an argument, and that even listing all the premises can be difficult - humans are very good at suppressing premises they find obvious. Suppressing premises is probably also a practical necessity, we cannot enumerate all the facts of human experience in each of our arguments.
From a knowledge point of view, and if we were to separate a subject and an object, the subject being me, who thinks right now, and the object being outside of me in some sense, like a tree I’m thinking about, roughly, validity would reside on the side of the thinking subject, whereas soundness would reside with the object under consideration. What I mean is that, we humans, the subjects making a claim, require logic, but it does not follow that the world around us (the objects) need to behave according to that logic. Validity alone does not tell us anything about the world, whereas the soundness requirement brings semantics into the picture, that is, meaning about facts of the world. I am fully aware that talk of “subject” and “object” is fraught with many philosophical questions, that I will take a look at later in this blog.
There is more to logic and deductive arguments (which have been studied since Antiquity), but this simple presentation should serve us well enough for now. In the next post, I will talk about another very important type of reasoning: induction.
References - Understanding Arguments; An Introduction to Informal Logic, by Walter Sinnott-Armstrong and Robert J. Fogelin (Cengage Publishers; 9th ed. Concise Version and 9th ed. Complete Version, 2014)