# Is deductive logic, and more specifically propositional logic, ultimately derived from induction?

Is the system of propositional logic itself induced, beyond its manipulated assumptions? It seems to be the case that we use propositional logic on the basis of two facts - firstly that it seems to work when applied correctly and with sufficient knowledge, which is an a postiori justification seemingly from induction, and secondly that the specific rules of inference can be induced easily. Conditionals are induced from the fact that certain events seem to connect to one other predictably, the "and" function is induced from the observation that certain events seem to connect to one another necessarily to create a valid conditional, the "or" function is induced from the observation that certain conditionals can be valid in multiple distinct or combined cases, and so on. It seems then that our use of propositional logic is itself validated only by induction. Another way to phrase this question is through a thought experiment. If a human lived in a world in which events connected randomly, where the connection of events did not seem to follow deductive rules, would they be able to imagine up deductive reasoning at all? If so, how would they arrive at this thought? If not, doesn't this demonstrate that propositional logic is justified only by induction? Keep in mind that claiming that "our world does not function this way" does not answer why a human in such a world would not be able to arrive at the idea of deduction, if deduction is not based on induction. Likewise, claiming that "deduction is useful" seems to be an argument from induction of the results of using deduction.

• NO................................. Commented Jan 1, 2021 at 14:07
• @MauroALLEGRANZA What do you mean by "no"? And if you're refuting my argument, please explain why. Commented Jan 1, 2021 at 14:22
• The term INDUCTION expresses that the famous 5 senses are the foundation of knowledge: sight, taste, touch, smell, & hearing. This is also known a SENSE VERIFICATION. All SCIENCES are grounded by INDUCTION processes. Many human experiences in our lives involves at least one of the senses. This may be why you think the way you do here. Deductive reasoning expresses I DO NOT NEED my senses to derive new information: I use only my THOUGHTS in a particular way. I may write out a proof for other to see the thinking but this is FOR OTHERS to see the thinking. I don't NEED to use any of my senses. Commented Jan 1, 2021 at 15:06
• @Logikal you're misunderstanding my argument. How do you know what the functions of propositional logic mean without witnessing with the senses the connections between events? That is, if you were, as my example said, born into a chaotic world without rules, how would you derive the idea of a conditional? No events would seem to be connected to one another, so how, specifically, would you go about coming up with conditionals? The point isn't whether it's thought or felt, the point is that you can't think about conditionals without experiencing sensed connections between events. Commented Jan 1, 2021 at 15:18
• "a world where the connection of events did not seem to follow deductive rules" NO; the events in the world follow (maybe) laws: physical, biological, social, psychological. There is no "deduction" in events: deductions occur in human argumentations. Commented Jan 1, 2021 at 17:16

Potentially, you could be asking one of two different questions. One is how did our ability to do logic get started? i.e. a question about origins. The other is how do we consider logic to be grounded or justified? i.e. a question about the epistemology of logic.

In the case of the former question, we really just don't know. We know from studies of animals that some species have a basic ability to reason, and some are able to count. Mothers with a litter of six young usually know when one is missing. But how this developed in humans we don't have much idea: it happened in prehistoric times and we don't have records, so we can only guess. I think we can safely say though, that by the time the stoic philosophers in the third century BCE were devising the logic of propositions (i.e. 'and', 'or', 'if') we were a long way past the simple idea that we are just generalising from experience.

A comparison with counting is perhaps appropriate: parents teach their children to count by showing them real things such as stones or trees or ducks and getting them to count them. But once we progress beyond the nursery stage, we don't need to count ducks to know how arithmetic works, nor do we believe that arithmetic is correct just because it works on ducks, no matter how many times we practice counting them. Similarly, the words 'and', 'or' and 'if' are part of the English language, and we learn how to use them when we learn our first language. The examples that we were presented with when we learned how to use them are no longer relevant to our understanding.

In the case of the second question, there is much disagreement about the epistemology of logic. Some positions are that it is an innate and privileged form of a priori knowledge, that it is grounded in the grammar of the language we use, that it is derived from the inferential relationships that we consider semantically compelling, that it is grounded in a theory of meaning, that it is justified by its close relationship with the concept of computation, that it is justified by virtue of being the product of natural selection, that it is justified in an indirect way by the contibution it makes to our scientific knowledge.

However, the idea that we simply proceed by induction from observations is implausible. For one thing, propositional logic is concerned with the connection between the truths of propositions, not between events. There is nothing intrinsically causal about the connections. For another, once a logical connection is correctly grasped, further examples become irrelevant, except for illustrative purposes. By contrast, with inductive support, the more data you have the better, because the data is needed to support the proposed connection.

As to your thought experiment, I would be inclined to say that a world that is so chaotic that no connections of any kind are observable would be a world in which I, or any living thing, could not exist, so the question is moot. The most that you might say is that since we use logic to organise information, if we have no information there is nothing to organise, so logic would be a pointless exercise.

• Very detailed answer, thank you. I take two primary issues with it, however. Firstly, I said "connections," not "causations" purposefully, I didn't mean that the connections were causal. Deduction is also certainly concerned with the relationships between truth values generally, and the fact that you admitted simple examples are used to illustrate the basic concept before the general one points toward a form of induction. If logic doesn't work, it serves no purpose, because it isn't sound in its construction, meaning the chaotic world is still relevant. Commented Jan 3, 2021 at 4:08
• Secondly, math is not exempt from this seeming issue. Using examples is originally useful, even if it is later unnecessary - because you understand the mental motion from repeated example to generalization. With a sufficient number of axioms, math works logically far beyond possible examples, but this does not prove that the examples were not necessary to start. Further, geometry is idealized, and useful, but appears to come from the fact that we live in a world where space exists. The extrapolation of this gives the idea of higher spatial dimensions, so this isn't an issue for my argument. Commented Jan 3, 2021 at 4:13
• Essentially, I'm trying to dig away at what you said about the uncertainty of origins. I'm arguing that a blank-slate theory of mind can still entirely account for the deductive concepts that we have. We obviously learn from the structure of our language and through the knowledge passed down to us from our various teachers, but being given knowledge that makes no sense to us from experience is jarring, and would change how we look at logic. So, the person in the chaotic world would not discover the concept of logic, since if the idea occurred it would be discarded as silly. Commented Jan 3, 2021 at 4:16
• Given the uncertainty of origin, and because the blank-slate theory of mind is a separate topic, I will accept this answer. Thanks for your help. Commented Jan 3, 2021 at 4:17