I wanted to get your thoughts on what I perceive to be an overlap in deductive and inductive reasoning and what practical importance, if any, this might have. To start with some simple definitions, deductive reasoning is, these days, defined generally as a form of reasoning in which the conclusion is guaranteed by the premises based on the form of the argument; whereas inductive reasoning is defined as a form of reasoning in which the conclusion is inferred from or warranted by the connection between the premises. These definitions are different from the older definitions of reasoning from general to particular (for deduction) and reasoning from particular to general (for induction). Moreover, today deductive arguments are analyzed in terms of validity and soundness, where validity has to do with the form of the argument and is truth preserving rather than truth making, and soundness has to do with both validity and the truth of the premises (so all sound arguments are also valid). Inductive arguments, on the other hand, are analyzed in two analogous terms: strength and cogency. A strong argument is one in which the premises strongly warrant the conclusion (without respect to their truth value), while a cogent argument is a strong one that also has true premises. There's a lot more that could be said about that and I have some questions even about those distinctions and some of their more precise relations, but I want to press on to get to the question that really interests me.
Inductive reasoning is not, we are told, a matter of form, but there are, still, certain common forms. The typical form a simple induction is something like:
1. Proportion Q of known instances of population P has attribute A 2. Individual I is a member of P 3. Therefore I probably has attribute A
An instance of this type of argument might be:
1'. Young men who live in the southeastern United States and who own trucks usually like country music 2'. John is a young man who live in the southeastern United States and who own trucks usually like country music 3'. Therefore, John probably likes country music
And this is said to be an inductive argument because the conclusion isn't guaranteed; after all, John might not like country music after all. Similar forms and instances could be made for generalizations, arguments from analogy, arguments from enumeration, etc. But now I have to wonder . . . isn't this example deductive and even necessarily true after all? Suppose it turns out that John does NOT like country music. Does it follow that 3' is false? On the one hand, it could seem obvious it is. If John doesn't actually like country music, then there is no probability that he does. I can say there is a 1:6 chance that when I roll this dice that it will land on a 2, but I have to say that before I roll it. Having rolled it, and seeing it land on a 2, I can't say it has a 1:6 chance of actually being a 2. I can say it had a 1:6 chance of being a 2. I can say it has a 1:6 chance of being a 2 on the next roll. But I must say it has a 1:1 chance of being a 2 now that actually is a 2, and a 0:6 chance of it being a 1, 3, 4, 5, or 6. Construed this way, again, 3' is false because there is no probability that John likes country music.
But doesn't 3' really just mean exactly what it says? There is a probability that John, as a member of a particular population, likes country music. There is a sense in which that probability remains regardless of whether John does or doesn't like said music. Statisticians, in fact, need that data and would build population distributions based exactly on that. In other words, we could interpret the syllogism along these lines:
1''. 84% of members of population that are {male; between the ages of 1636; live in the states of Georgia, Alabama, Mississippi, South Carolina, North Carolina, Tennessee, or Arkansas; and are truck owners} [let this population be P] are also members of the population that {like country music} [let this population be Q] 2''. John is a member of population P 3''. Therefore, there is an 84% chance that John is a member of population Q
Now, technical matters aside on the relation between samples and population and how you actually figure out probability (I know I've overly simplified and misrepresented that part of the analogy) . . . grant that John is not, in fact, a member of population Q (he doesn't like country music). That doesn't chance the statistical fact that he still had a statistical probability of being with said group. The dice that landed on 2 still had a 5:6 chance of landing on some other number.
If this is correct, then I finally come to my question: is there not a real sense in which this argument is, in fact, a real deduction? That is, the conclusion is necessarily true, for even if John turns out NOT to like country music, the conclusion, as stated remains true, and that necessarily so. And if is in fact the case, then it seems to me that ALL induction is of this type, that ALL induction relies on a previous deduction.
Take this argument by analogy
1. I have a 1999 Toyota Camry 2. Tonya has a 1999 Toyota Camry, and her car gets 30+mpg 3. My car will get 30+mpg
Perhaps this isn't a strong argument, but it is still necessarily true as stated, so long as we interpret it probabilisticly. What this really means is that there is some statistical chance that my car gets 30+mpg. It seems to me that, so long as we are interpreting inductive arguments with reference to strength, then we necessarily are making them statistical and thus, in principle, a very specific form of deduction.
I think my confusion is rooted in the two ways to interpret the conclusion of the inductive argument (so see my comments on the dice example). But I wonder if this isn't getting at an error, or at least an imprecision, in the definition proposed for "inductive reasoning." For it truly is necessarily true that some event has a certain probability of obtaining, and that whether it does or does not. We are deducing that probability. Is induction, then, merely taking the trivial step of predicting what will obtain based on those percentages (whether or not we've done the work of actual calculating the numbers or just going with our general perception)? Or have we gotten off at the very beginning of all this?
Sorry for the length, but those of you who have studied or even read anything on the philosophy of logic or know how modern logic compares to classical logic, if you could enlighten me on this and how to think about it, I would be appreciative.
Thanks
_________________ Indeed, the Lord Jesus, when He prayed to the Father, "that all may be one. . . as we are one" (John 17:2122) opened up vistas closed to human reason, for He implied a certain likeness between the union of the divine Persons, and the unity of God's sons in truth and charity. This likeness reveals that man, who is the only creature on earth which God willed for itself, cannot fully find himself except through a sincere gift of himself. ~ Pope Paul VI, Gaudium et Spes 24.3
