Answering this seemingly simple question very much depends on just how the question is understood. Just exactly what is meant by a "convincing justification." Just exactly what is meant by "deductive reasoning." Let's tackle the second question first, since it is necessary to understand just what one is trying to justify before one evaluates whether the justification offered can be considered "convincing".
"Reason" is the label we give to our means of cognitive understanding. In this respect, it is to be contrasted with "emotion", "perception", "instinct", and perhaps "whimsy". Reasoning is that particular mode of thinking in which the thinker transitions from some beliefs that are presumed to be true (premises) to a conclusion which is judged to be true because the premises are true. Using our reason, being rational, being reasonable, means looking for reasons when making decisions, taking actions, and pursuing goals. We employ reasoning, we are rational, when we appeal to reasons as the justification for our beliefs -- whenever we justify our belief that B because of our beliefs that (A_{1}, A_{2}, A_{3}, . . . , A_{n}). By the very definition of rational and reasoning, we are being irrational and unreasoning when we form some belief B in the absence of any reasons (I will ignore, here, the causative theory of perceptual belief formation). Of course, in some cases our reasons for believing that B can be said to "work" in the pragmatic sense, in that our reasons for believing that B are in some way dependent on whether (A_{1}, A_{2}, A_{3}, . . . , A_{n}) are true. In other cases, our reasons for believing that B can be said not to "work", in that our reasons for believing that B are, for various reasons, more or less independent of whether (A_{1}, A_{2}, A_{3}, . . . , A_{n}) are true or not.
Philosophers have divided the cognitive practice of moving from believing (A_{1}, A_{2}, A_{3}, . . . , A_{n}) to believing B into a number of differently described "kinds" of rational thinking. "Deductive reasoning" is the label that is attached to one particular kind of rational thinking - the kind that (in theory at least, and somehow) guarantees that the conclusion (B) is true if the premises (A_{1}, A_{2}, A_{3}, . . . , A_{n}) are true. Inductive reasoning, reasoning by analogy, reasoning to the best explanation, and probabilistic reasoning are examples of other "kinds" of cognitive processes of getting from a belief in (A_{1}, A_{2}, A_{3}, . . . , A_{n}) to a belief in B.
The theoretical basis of the guarantee offered by deductive reasoning is provided by the rules of deductive logic. What is now called "deductive" reasoning was first explored by Aristotle, Pythagoras, and other Greek philosophers of the Classical Period (600 to 300 B.C.). In his writings, Aristotle formulated a series of rules for reasoning that would lead to indubitable conclusions. Aristotle's rules have since been added to by such notable philosophers as Frege, Russell, Kripke, among others. Today we have a number of highly complex formal languages of deductive logic, some of which claim to be incompatible with others. But all of which claim that their conclusions follow necessarily from true premises.
So given the preceding, I will understand the "deductive reasoning" mentioned in the essay's title question as meaning reasoning in a manner that the established rules of some system of Deductive Logic are intended to formalize. But this, in turn, gives us two different ways of understanding just what the "convincing justification" mentioned in the title question is intended to justify. It is unclear whether what is being referred to is "deductive implication" - the logical relations among syntactic forms within some system of logic; or "deductive inference" - the actual inferential moves made by a person in the course of reasoning, that happen to be consistent with the rules of some system of deductive logic. The question can thus be understood as inquiring into either -
(i) the justification of the formalization of deductive reasoning into some particular system of "Deductive Logic". What is it that justifies the formalized rules of inference and basic axioms of some particular system of deductive logic, and is that justification convincing?
or -
(ii) the justification of the actual reasoning that we do, that the formalized rules of deductive logic are intended to capture. What is it that justifies our confidence that our cognitive process of getting from (A_{1}, A_{2}, A_{3}, . . . , A_{n}) to B (in a "deductive" manner) gives us a reason to believe that B? Is our belief that B because of (A_{1}, A_{2}, A_{3}, . . . , A_{n}) convincingly justified?
Since it is the existence of the rules of Deductive Logic that define the scope of what is meant by "reasoning in a fashion that the formalized rules of deductive logic are meant to capture", I will address first the justification that exists for those rules. Only then will I deal with the justification that exists for the inferences we do in fact make.
When philosophers have examined the way in which we actually do reason, they have identified specific patterns that emerge. These patterns are called "logical forms". The logical form of an argument is to be distinguished from the content of the argument. Just as a wine glass has a distinctive shape and is distinguished from what you put in it (the wine), so argument forms are identifiable and not to be confused with the actual premises and conclusions used. Logicians focus on the pattern - the "form" - of the reasoning, and abstract away from the specific content of the reasoning. The reasoning patterns captured by logicians are therefore said to be "Topic neutral".
Aristotle's logical works contain the earliest formal study of reasoning that we have. The ancient commentators grouped together several of Aristotle's treatises under the title Organon ("Instrument")^{(1)} In his writings, Aristotle identified four "logical forms", and formulated a series of rules for reasoning that would lead to indubitable conclusions.
The four "forms" identified by Aristotle were - (i) the categorical syllogism that organizes thinking by categories or concepts; (ii) the hypothetical syllogism that employs the "if-then" construct - "modus ponens" and "modus tolens"; (iii) the alternative syllogism that employs the "either-or" construct; and (iv) the disjunctive that employs a "can't be both" construct. If the major premise and the minor premise are both true and the structure of the argument correctly maps to one of Aristotle's logical forms, then the conclusion must necessarily follow and be true.
Modern formalizations of deductive reasoning into a system of deductive logic define formal expressions called formulae, or forms, which are strings of symbols in a formal language that has a precise syntax. In this aspect the formalizations of deductive reasoning that are Deductive Logic are like programming languages. The first order quantified predicate logic of Frege and Russell is one example of such a language.
A "Deductive Logic" is therefore based on a set of carefully specified axioms and rules of inference. By applying the rules appropriately, one can derive (infer, deduce) conclusions from a specified set of premises. A "Deductive Argument", then, is a specific piece of reasoning with one or more premises and a conclusion arranged in a particular "form". (Within the rules of deductive logic, an "argument" is to be distinguished from a disagreement. One may use a deductive argument, in the logician's sense, in order to win an argument, in the everyday sense of a dispute.)
A formalization of deductive reasoning into a system of Deductive Logic allows three key technical concepts to be defined -
Completeness - If a formalization of deductive reasoning is powerful enough to derive every valid logical consequence, it is called complete. In particular, a complete specification of deductive logic can derive every tautology. The soundness of a deductive logic can be guaranteed, and formally proved, given -
So what is it that justifies the particular forms that logicians select as constituting "deductive logic"? Nelson Goodman, in his Fact, Fiction, and Forecast^{(2)}, argued that the rules of deductive logic are justified because they properly describe the reality of how we reason -
"Principles of deductive inference are justified by their conformity with accepted deductive practice. Their validity depends upon accordance with the particular deductive inferences we actually make and sanction" [Goodman, pg 63]
But Goodman has the flow of justification backwards. He suggests that the inferences we do accept justify the rules of deductive implication that we dream up. But he misses the fact that it is the existence of rules of reasoning that we label "deductive" that results in certain of our actual inferences being called "deductive inferences". The rules of deductive logic are norms that propose to tell us how we should be reasoning when we reason deductively. Goodman misses the critical distinction between the inferences we call "deductive" and the inferences we call "inductive". What justifies the rules we dream up and call "deductive logic" is the fact that they are intended to preserve truth. In every instance, what makes it a rule of "deductive" reasoning is that the conclusion is intended to be guaranteed to be true if the premises are true.
As Lachs and Talisse^{(3)} argue -
"Contrary to Goodman, the rules of deductive logic are not justified because they adequately describe our deductive practices. They do not. The rules of deductive logic are justified relative to the goal of arguing truth preservingly, i.e. in such a way that the truth of the premises guarantees the truth of the conclusion. The results that provide the justification are known as soundness and completeness. Soundness says that every argument we obtain from the rules of deductive logic is such that truth is preserved when we go from the premises to the conclusion. Completeness states the converse. Every argument that has this property of truth preservation can be obtained from the rules of deductive logic. So the rules of deductive logic are justified relative to the goal of truth preservation. The reason is that they further this goal insofar as all and only deductively valid arguments are truth preserving." [Lachs, Pg 477]
In his article, The Justification of Deduction^{(4)}, Michael Dummett presents the suggestion that a convincing justification for the rules of deduction need not be a suasive argument, merely an explanatory one. And hence, he continues, the fact that all so far offered justifications for deduction have been ultimately circular - employing at least one rule in the justification of the rules of deduction - is acceptable. But as Susan Haack^{(5)} has replied, "The trouble with a circular argument is not just that it is not persuasive . . . it is also that it is indiscriminating".[Haack Pg 220-221] For example, the gambler's fallacy inference can be convincingly explained (and hence, by Dummett's reasoning, judged convincing) if the gambler's fallacy is permitted as part of the justification. So Dummett's approach to the problem is fundamentally flawed. Moreover, the ambiguity inherent in the essay's title question is mirrored by the ambiguity that Dummett encompasses in his article. According to Haack, "The tension [that Dummett finds] between the necessity and informativeness of deduction is generated by [his] failure to observe the distinction between deductive implication and deductive inference." [Haack, Pg 238]
The rules that constitute a system of deductive logic are defined so as to ensure that if the premises are true, then the conclusion is true. It may certainly be true that a logician might inspect a number of examples of how we actually reason, observe a common "form", notice that when examples fit the "form", when premises are true, the conclusion is also true - and then reach an inductive conclusion that for an argument of this form, this will always be the case. But once that inductive conclusion is formatted as a rule (say for example "P → Q", "P", therefore "Q") then conclusions reached by using the rule are true by definition. Of course, once we substitute a specific real world example for the variables in the rule (for "P" and "Q" in this case), we can inspect the reality of whatever "Q" stands for to see if indeed it is true. To do this is to verify that the rule works as intended. It is not to verify that the rule describes how we reason. This gives us two ways to examine the defined rules of any particular system of deductive logic. We can examine the syntax of the system. And we can examine the semantics of the system.
To examine the syntax of the system, we examine the rules and ignore the content. (We look at the glass, and ignore the wine, to use an earlier analogy.) It is here that the technical concepts of soundness and completeness come into play. A soundness proof assures us that the deductive system in question is consistent, i.e. that we will never be able to prove a contradiction in it. This certainly is a desirable feature of a system of logic, to say the least. A completeness proof assures us that within the deductive system in question every properly formatted valid expression is provable. Completeness is not a necessary feature of a useful system of logic. The lack of a completeness proof would mean that there are syntactically valid statements that one cannot demonstrate are true. However useful soundness and completeness are, however, they cannot offer any justification for the deductive system -
"An obvious shortcoming of soundness and completeness proofs as Ã¢‚¬Ëœjustifications' of the logical systems for which they are available is simply this: there are too many of them. Soundness and completeness proofs establish something important about the internal cohesiveness of a logical system. But there are different logical systems, systems which their proponents take to be rivals of each other, each of which can be shown to be sound and complete." [Haack Pg 224]
To examine the semantics of a system of deductive logic is to examine just how the syntax gets translated into English, and then to examine whether in actual fact, the conclusion (B) is true. Because even though the syntax of the system may be sound and (possibly) complete, and the form of the argument may be valid, and the premises (A_{1}, A_{2}, A_{3}, . . . , A_{n}) when understood in English may all turn out to be true, the only way to verify that the defined rules have resulted in a true conclusion as was intended, is to go and look and see.
So after all this analysis, what is it that justifies the formalized rules and basic axioms of some particular system of Deductive Logic, and is that justification convincing? The answer is rather straight forward - the rules of inference and the basic axioms of any system of Deductive Logic intended to guarantee that the conclusion (B) is true if the premises (A_{1}, A_{2}, A_{3}, . . . , A_{n}) are true. That is what makes the rules and axioms define a system of "Deductive" logic, rather than some other sort of logic. Of course, the only way to know whether that guarantee has been actualized in some particular real world interpretation of the syntax, is to go and look and see.
It should not need elaborating that the vast majority of people, the vast majority of the time, do not consciously employ any specific rules when they infer B from (A_{1}, A_{2}, A_{3}, . . . , A_{n}). Only the rare individual trained to some extent in the rules of some system of logic, actually consciously attempts to employ those rules when passing from premises to conclusion. So any explanation as to why and how we come to believe B when we believe (A_{1}, A_{2}, A_{3}, . . . , A_{n}) has to rest not on the rules that have been established for some system of logic, but on a psychological analysis of the way that the human mind generally works. This conclusion is reinforced by the many observations by psychologists that the average person does not in fact reason in a deductive way most of the time. A Google search of "common fallacies" yields a plethora of examples of just how non-deductive common every-day reasoning happens to be. The Internet Encyclopedia of Philosophy lists 177 common fallacies of reasoning^{(6)}.
So let's focus on that tiny portion of common reasoning that is in fact the kind of reasoning that formalized rules of deductive logic are intended to capture. Is there any justification at all that can be offered, that would explain why reasoning in a particularly "deductive" fashion is preferable to the other ways that we reason? Is there any convincing justification for using deductive reasoning rather than, say, any of those 177 forms of fallacious reasoning?
A belief is a relationship between concepts that you think accurately represents the facts of the matter. A belief is to be distinguished from an opinion, thought, surmise, hypothesis, or wish, etc., in that a belief is intended and expected to be true. A belief aims at truth. (A figure of speech of course. "Beliefs" are not things that can really "aim" at anything.) Opinions, thoughts, surmises, hypotheses, wishes, and so forth, do not aim at truth. They are not presupposed to be true, and are not expected to be true. To believe something, is to believe that it is true. When it comes to our struggle for survival, it pays (on average, with only very rare exceptions, and certainly in the long run) to believe the truth about where the tiger lurks. To base our survival on whimsical beliefs, opinions, thoughts, surmises, hypotheses, wishes, and so forth, is to run a greater risk of becoming lunch rather than enjoying lunch. For reasons of survival, therefore, we need our beliefs to be true, on average and in the long run, more often than not, because true beliefs are more likely to be useful - more likely to yield the expected results when relied upon - more likely to offer a survival advantage. Truth, therefore, becomes the fundamental standard of correctness for belief. Any rule or norm of reasoning that promises that true conclusions will result from true premises (more often than not) is thereby convincingly justified.
As we saw in the previous section, what distinguishes "deductive" reasoning from all other sorts of reasoning, is that - by the very definition of what "deductive reasoning" is - deductive reasoning attempts to guarantee that the conclusion is true if the premises are true. Hence if one reasons using deductive forms of argument, then one can confidently infer B from (A_{1}, A_{2}, A_{3}, . . . , A_{n}) because the rules of deductive reasoning are set up so as to guarantee (i.e. much more often than not) that if (A_{1}, A_{2}, A_{3}, . . . , A_{n}) are all true, then B is true as well. This is what constitutes convincing justification for deductive reasoning. Or rather, it constitutes convincing justification for reasoning in accordance with the rules of deductive logic.
What this line of thought leaves an open question, of course, is the justification we have for believing that the rules of deductive logic guarantee the truth of the conclusion. How can we be justified in following the defined rules of deductive logic? In Lewis Carroll's famous piece What the Tortoise said to Achilles^{(7)}, Achilles presents to the Tortoise three propositions -
(A1) Things that are equal to the same thing are equal to each other.
(A2) The two sides of this triangle are things that are equal to the same thing.
(B) The two sides of this triangle are equal to each other.
The Tortoise accepts (A1) and (A2) but does not accept (B). The Tortoise refuses to accept B even when Achilles adds the additional premise
(A3) If A1 and A2 are true, then B must be true.
There has been much debate about what the moral of the tale was supposed to be. But one of the more interesting (and germane to this essay) interpretations is that the Tortoise is demonstrating that the defined rules of deductive logic contain no motive force. Despite the definitions of logical implication involved, and the purported guarantee that when the premises are true, the conclusion is true, there is nothing inherent in a supposedly true conclusion that would force the mind to accept (believe) the conclusion. Accepting the conclusion of a sound and valid deductive argument remains a matter of voluntary choice - a choice that cannot be forced. Especially if the mind also contains other premises with which (B), perhaps erroneously, conflicts.
What needs to be provided is a justification for thinking that any given rule (or theorem of some formalized language of logic) universally guarantees that true conclusions will result from true premises. What is it that justifies us believing that the rule of modus ponens, for example, will in all circumstances yield a true conclusion from true premises? Even though the rules are defined with the intention of generating this result, how do we know that the intent of this definition is attained? What needs to be provided to the Tortoise to demonstrate that in the interests of personal survival it is better to believe (B) than not, even if it conflicts with other beliefs? There are two threads to this justification - one based on an inductive generalization from experience; and the other based on an analysis of the concepts employed in the rules themselves.
Reality (the actual world) consists of nothing but individual particular existents. Our experience of the world is experience of those particulars. To those experiences and those particulars we apply our conceptualizations (categorizations). Through our cognitive processes of counting, generalization, and abstraction we form inductive generalizations about the particulars in the world. Our experiences tell us that in all cases of examined particular examples of, say, modus ponens reasoning, the conclusion is true when the premises are true. Hence we draw upon the "inductive premise" (that the future will be like the past) to reach an Inductive Generalization that in modus ponens reasoning, when the premises are true the conclusion will necessarily be true.
In many cases, our analysis of the concepts being employed in reasoning tell us that in such circumstances the meanings being employed will ensure that the conclusion will be true if the premises are true. It is not necessary to examine a large number of particular examples. The nature of the concepts employed ensures that the conclusion is true. (We could get into a lengthy discussion of the nature of concepts here, but I will leave that for another essay.)
The end result of this lengthy analysis is that deductive reasoning is a special kind of reasoning - a kind that ensures sufficiently more often than not that true premises will result in true conclusions. To distinguish deductive reasoning from other forms of reasoning, we have defined formalized rules. Reasoning that properly adheres to those rules is called deductive reasoning. Reasoning that does not employ those rules is not classed as deductive reasoning. (Reasoning that attempts, but fails, to properly adhere to those rules is invalid deductive reasoning.)
The formalized rules that have been defined for deductive reasoning, have been so defined as to ensure that if the premises of the arguments are true, then the conclusion of the argument is true. Truth is employed as the standard of correct reasoning because of the survival significance of true beliefs.
The belief that the rules of deductive reasoning do in fact guarantee the truth of their conclusions in any application of them is justified on the basis of both an inductive generalization of the experiences of particular examples of their use, and an analytic (and thus necessarily somewhat circular) analysis of the meanings of the concepts employed in those rules.
That this justification of deductive reasoning is convincing is demonstrated by the universal, and so far successful, reliance on deductive reasoning in all aspects of living. As long as one adopts an empirical stance towards knowledge - maintaining that all knowledge is ultimately based on inductive generalizations of experience, and denying the possibility of innate or intuitive sources of knowledge - inductive generalizations with as much of a track record as the rules of deductive reasoning have acquired is as good as it gets in the way of convincing justification.
(1) Smith, Robin, "Aristotle's Logic", The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.), forthcoming URL=<http://plato.stanford.edu/archives/win2008/entries/aristotle-logic/>.
(2) Goodman, Nelson; Fact, Fiction, and Forecast, 2^{nd} Edition, The Bobbs-Merrill Company, Inc., Indianapolis, Indiana. 1965. LCCN:65-17597.
(3) Lachs, John & Talisse, Robert; American Philosophy: An Encyclopedia, Routledge, New York, New York, 2008. ISBN 0415939267
(4) Dummett, Michael; "The Justification of Deduction", in Truth and Other Enigmas, Harvard University Press, Cambridge, Massachusetts. 1978. ISBN 0-674-91076-1
(5) Haack, Susan; "Dummett's Justification of Deduction" in Mind, New Series, Vol 91, No. 362 (Apr, 1982), Pg 216-239
(6) Dowden, Bradley; "Fallacies" in Internet Encyclopedia of Philosophy, California State University, Sacramento, 2008. URL=<http://www.iep.utm.edu/f/fallacy.htm>
(7) Carroll, Lewis (Charles Lutwidge Dodgson ); "What the Tortoise Said to Achilles", Mind Vol 4, No. 14 (April 1895): Pg 278-280.