Gillian linked^{*} to a paper by Gil Harman and Sanjeev Kulkarni, which contains this nice explanation of the distiction between inference (reasoning) and implication (what follows from what):

A related problem with the traditional picture is its treatment of deductive principles like (D) as rules of inference. In fact they are rules of about what follows from what. Recall what (D) says:

(D) From premises of the form “all F are G” and “a is F” the corresponding conclusion of the form “a is G” follows.

(D) says that a certain conclusion follows from certain premises. It is not a rule of inference. It does not say, for example, that if you believe “All F are G” and also believe “a is F” you may or must infer “a is G.” That putative rule of inference is not generally correct, whereas the rule about what follows from what holds necessarily and universally. The alleged rule of inference is not generally correct because, for example, you might already believe “a is not G” or have good reason to believe it. In that case, it is not generally true that you may or must also infer and come to believe “a is G.“

Perhaps you should instead stop believing “All F are G” or “a is F.” Perhaps you should put all your energy into trying to figure out the best response to this problem, which may involve getting more data. Or perhaps you should go have lunch and work out how to resolve this problem later!

From inconsistent beliefs, everything follows. But it is not the case that from inconsistent beliefs you can infer everything.

Deductive logic is a theory of what follows from what, not a theory of reasoning. It is a theory of deductive consequence. Deductive rules like (D) are absolutely universal rules, not default rules, they apply to any subject matter at all, and are not specifically principles about a certain process. Principles of reasoning are specifically principles about a particular process, namely reasoning. If there is a principle of reasoning that corresponds to (D), it holds only as a default principle, other things being equal.

Another thing to be careful about in logic class. It took a while to train myself to avoid pronouncing “p → q” as “p implies q,” which is a very common thing to do in math/CS logic circles. I’m now very careful to distinguish the conditional from implication, and try to get my students to do the same.

* Gillian’s post happened a while ago, I know. I’m slow. Do go and read it and the comments.

I’ve thought about that distinction between implication and the conditional, but haven’t come up with a good way to pronounce things that lets me deal with nested conditionals. Have you found a good way? Posted by Kenny Easwaran

“If p then q” or “p arrow(s) q”?”If p then: if q then p” is less ambiguous than “p implies q implies p,” which might as well be parsed as “If: if p then q, then p”. Of course, if you insist on “implies” or use “arrow” you can do “air brackets”: “p arrow (waves right hand in semi-circular motion) q arrow p (waves left hand in semi-circular motion).” Posted by Richard Zach

I have to register a protest at the statement you quote from Harman and Kulkarni: “Deductive rules like (D) are absolutely universal rules, not default rules, …” Firstly, in what sense are they universal? There has been discussion in the anthropolohical literature about whether modus ponens and other “standard” rules of deduction are adopted by all societies, and my reading of that literature is that they are not. Secondly, even if some rules of deduction were universal across cultures, that does not mean that we should adopt them without due consideration. But this may not be possible for all rules. The philosopher Susan Haack has shown the inherent difficulty of justifying modus ponens. If you try to justify it with examples of its application, then you are using induction to justify a deductive rule. If you try to justify it using truth tables applied to generic variables, then you risk using modus ponens itself in the justification. Neither solution is satisfactory. Thirdly, even if we decided, after due consideration and on balance, that particular deductive rules were justified, that would not mean they would be appropriate to all applications or in all domains. The best example I know is Reductio Ad Absurdem (RAA), arguing from a contradiction. There are many domains, eg computer science, where a reasonable person may insist on constructive proofs for results. I certainly would feel unsafe flying in an aircraft whose software systems had been shown to be failure-free using RAA. Posted by Peter

simply put:” I inferred from your implication”…Meaning:this is what I got out of what you were saying.Inference:”inference”-one’s deduction ( referring to)Implication: Implication in my thought.( you’re implying) ok? yes, no? makes Latin sense

“…I inferred from your implication…”The inference is from the recipient of the implication.One implies a thought and the recipient infers from it.

Bulletin of Symbolic Logic. 12(2006) 353-354.John Corcoran, Meanings of Inference, Deduction, and DerivationPhilosophy, University at Buffalo,Buffalo, NY 14260-4150E-mail: corcoran@buffalo.eduAbstract: The verbs ‘infer’, ‘deduce’, and ‘derive’ used in logic are ambiguous; logicians use each with multiple normal meanings. Several of their meanings are vague in that they admit borderline cases. This paper juxtaposes, distinguishes, and analyzes several senses of these three-place action verbs, focusing on a constellation of recommended senses. In the sense to be recommended, the verb ‘infer’ is used for the epistemic activity of [a person] judging a proposition to be true by determining that it is a consequence of given propositions known to be true. The verb ‘deduce’ is used for the epistemic activity of determining that a proposition is a consequence of given propositions. Aristotle discovered that the same process of deduction used inferentially to draw a conclusion from premises known to be true is also used non-inferentially to draw a conclusion from premises whose truth or falsity is unknown, or even from premises known to be false. Applying his grasp of this point in the first few pages of Prior Analytics, he distinguished demonstrative from non-demonstrative deductions. He wrote: “Every demonstration is a deduction but not every deduction is a demonstration.” Inference, or demonstration, producing knowledge of the truth of its conclusion, and deduction, producing knowledge that its conclusion is logically implied by its premises, must both be distinguished from derivation, which consists in arriving at a string of characters by means of rule-governed manipulations starting with given strings of characters. Derivation by itself does not and cannot produce knowledge. No proposition can be inferred from a contradiction because no contradiction is known to be true. Every proposition can be deduced from a contradiction. And what can be derived from what depends upon which character-manipulating rules are allowed. This paper treats the relevant views of Aristotle, Boole, Frege, Lewis, Hilbert, Tarski, Church, and others.