Just recently saw the following argument in a logic book: all lions are herbivores all zebras are lions ————– therefore all zebras are herbivores this seems to be logically valid syllogism, but it is disturbing.

I have been reading your site, and there is somewhere you said a conclusion can logical valid but unsound. Is the following argument valid but unsound? I am not sure about what unsound arguments mean? Can you please clarify this for me. – Johnson Mafoko

Yes, that is a valid, unsound argument. The structure is good, but the content is bad. This is the case even though the conclusion is correct.

The way it works is this:

Invalid means the structure is bad. There are no benefits to an invalid argument, the premises have no meaningful connection to the conclusion.

Valid means the structure is good. If an argument is valid, it means the conclusion is at least as good as the premises. So if you put in true premises, you get a true conclusion. However, it doesn’t mean that if you put in false premises you get a false conclusion. In logic, false premises can lead to any conclusion, even when the argument structure is valid.

Sound means that the argument is valid and that the premises are true. A sound argument will guarantee a true conclusion. It is the only type of argument that guarantees a true conclusion.

Please note that only “formal” arguments –the kind of very artificial, highly structured arguments found in logic books and dealing only with unambiguously true or false statements –can be either valid or sound. (Different terms are used for less formal arguments).

Is this statement a tautology: “If there were no opportunities there would be no crimes” ?

It depends on how strictly you want to define the word “tautology”.

A) FORMAL: If you wanted to evaluate it this statement as a formal tautology you would have to rewrite it as a formal statement first. In the form

(a=opportunities, b=crimes)
it is not a tautology, but in the form

(a=opportunities, b=actions and opportunities + actions=crimes)

it is a tautology, because no possible assignment of a and b makes the statement as a whole false.

Although, taken literally, it seems to verge on a tautology in a rhetorical sense, you could reasonably argue that it functions rhetorically as a stand-in for the substantive claim “preventing opportunities is the best way to prevent crimes”.

Is this sentence a question or a metaquestion?

It’s both. It has the form and function of a question, so it is a question, but it is about the questioning process, so it is also a metaquestion. Formal languages such as first or second order logic have paradox-avoiding restrictions that force an either/or choice between using regular langauge or meta langauge, but not both, but English, being a natural language, has no such constraints.

Would you agree with those such as Alister McGrath that Christianity is rationally defensible, or would you say that the rational aspect is unimportant? If the latter, how would you respond to charges of thoughtless fanaticism in your religiosity?

This is a fantastic question.

I begin my answer by noting that rationality is overvalued and its capacity overestimated. The ability of the human mind to apprehend what it considers is vast, but not unlimited. Not everything, therefore, can be understood in ways that make apparent sense and align with all the other things that we know. In particular, God would not be God if He could be fully comprehended. For this reason, I side with those who call faith unreasonable.

As a student of Kierkegaard, however, I also note that the central paradox of Christianity, of God present with us, is no more of a paradox than the paradox of existence itself. Why should there be something rather than nothing? Why does our existence mainifest in the shape that it has, rather than in some other form? Why does each of us individually and idiosyncratically exist, and why are we bound by space and time? These are questions that have no rational answers, yet we live with the paradoxes they imply because we lack the ability to do otherwise.

This leads me to what I take to be the key Kierkegaardian insight: The mystery of Christ is not only on a par with the mystery of existence, it is in fact the same mystery. The mystery of why God would enter the universe and suffer and die is the very same mystery as why that universe would exist at all, and why there would be suffering and death within it in the first place.

All this having been said, however, I think there’s a danger in dismissing faith as merely or dogmatically irrational. The believer, I would claim, is not simply a believer in defiance of all evidence –which would indeed make him the thoughtless fanatic of your query.  Speaking as a believer, I would say that God has demonstrated His existence to me with evidence that is plentiful and personally compelling –yet not of a sort that lends itself to conclusive depersonalized proofs.

My aim in making such a claim is not to present a case for God’s existence capable of convincing the non-believer, but to advance the argument that the intrinsic irrationality (or what we might call the “transrationality”) of faith does not necessarily imply that the person who embraces faith must do so in an irrational manner. One may safely assume that the person who believes does so for personally valid reasons, even if those reasons are not easily understood by the non-believer.

This, it seems to me, is the best way to approach the ontological proofs of theological rationalists like Aquinas and Descartes, the apologetics of someone like C.S.Lewis, or the calculated wagers of Pascal and his ilk –not as attempts to equate faith with reason, but rather as ways of demonstrating that faith and reason are at least compatible with one another; and therefore that the embrace of one does not necessitate the destruction of the other.

Do you think that the following argument is a good one : X says that a certain metaphysical view is correct. X says that if this view M is correct, it should produce certain effects in the psychology of people that realize it. (Make them super-smart, or super-kind for example). It is observed that X has this type of psychology. Thus the metaphysical view M is true.

I am part of a minority of philosophers who believe that some variant on the argument you proposed can be incorporated into a strong inductive argument –in fact I’m currently at work on just such a project, named “Pragmatic Metaphysics”. However, the argument as so stated is an example of the formal fallacy “affirming the consequent”.

P1. If Metaphysical Claim then I am Super Kind
P2. I am Super Kind
C. Therefore Metaphysical Claim

One problem is that you haven’t explained why we believe P1 is true, But this argument fails even if P1 is true because we have never established “If Super Kind then Metaphysical Claim”, and the relationship is not reversible.

The danger with affirming the consequent is this: If you start with a true and known statement “B”, such as “2 + 2 = 4”, then every conditional of the form “If A then B” will automatically be true, regardless of the nature of “A”. Thus, “If it is sunny tomorrow, then 2 + 2 will equal 4.” “If it is not sunny tomorrow, then 2 + 2 will equal 4.” “If the Yankees win the World Series then 2 + 2 will equal 4.” “If the devil beats his wife, then 2 + 2 will equal 4.”

Because of this, the type of argument you outlined can never be formally sound. However, it is possible to make a similar strong inductive argument, given the following conditions:

  • “A” (i.e. Metaphysical Claim) must be a reasonable explanation for many observable things, not just “B” (Super Kind). So “If A then B, C, D, E, & F”, where B through F are all true.
  • “A” must not have further implications that are false. So it must not be the case that “If A then G” where G is false.
  • “A” must have predictive value. So “If A then H tomorrow,” where the prediction comes before H is observed, but where H confirms the prediction.
  • “A” must not have stronger rivals. There must not be an A´ or an A´´ that explains the same data in a simpler and better fashion.

Even given all this, we do not prove “A,” we can only say that there is reason to believe A, or that A offers good explanatory value. For instance, if we say it is a natural law all planets move in ellipses, there is as sense in which this is a metaphysical claim (since we can observe the planets but not the law in itself). We accept it as true because it has many true implications, no important false implications, excellent predictive value, and no stronger rivals.

1) The claim “There is extraterrestrial life in the universe, because my father said so” is an example of an appeal to authority. But it can be viewed as an enthymeme, where the hidden assumption is that “my father is always right”. In such a case, there is no logical problem with the argument. Do you agree? 2) Do you think that to say that : “Person A is biased , therefore\what he says is wrong” is fallacious? It can be interpreted as “person A is biased, therefore his information cannot be trusted. Therefore what he says is wrong”. 3) Are errors of logic errors of psychology as well? Or perhaps, only errors of psychology? Appeal to authority besides being a logical fallacy, has a whole psychology and sociology besides it.

1) We must always be careful to distinguish between arguments that are valid and those that are sound. A valid deductive argument has the proper logical structure (where true premises guarantee a true conclusion) but that is of little use if the premises themselves are false or unsure. Your expanded version of this argument is indeed valid, but the hidden premise cannot possibly be true (since human fallibility is unavoidable), so the argument is still unsound.

2) Your fallacy is the assumption that untrustworthy information is always wrong. In some ways reliably wrong information could be nearly as useful as reliably right information –it would at least enable the ruling out of some possibilities. In this case, Person A is biased therefore his information is not reliably right –but it is not reliably wrong either. Therefore, his opinion tells us nothing about the rightness of his position.

3) As we’ve discussed before, formal deductive logic is an artificial system designed to extend and improve upon (and theoretically to perfect) the conclusions reached by raw human intuitions (while inductive logic might perhaps be described as the mid-ground between the two). The tendency to confuse the rules of logic with the intuitions that inspired them does indeed have psychological and sociological aspects –but the same could be said for nearly all human practices.

Inductive arguments establish objective facts, so how can they be considered subjective?

Let’s accept for the moment the idea that there are objective facts about the world, things that are right or wrong, true or false independent of any observer. If so, those cannot be “established” by arguments. All that can be established by arguments is the grounds for believing that certain claims (statements we hold as true) are factual (objectively true).

An argument is always a conditional. In a deductive argument, the only question is whether or not the premises are true. If they are, then the conclusion must also be. But in an inductive argument, there is an additional amount of indeterminacy added by the inability of the premises to absolutely guarantee the condition.

Where a deductive argument can be absolutely and objectively classified as either valid or invalid, the judgment of whether an inductive argument is strong or weak is a matter of opinion, and thus inherently subjective. The facts themselves are not what is being judged –only the sufficiency of our grounds for believing claims about them.

On one hand, arguments are supposed to be objective – something which is true is always true, for everyone. On the other hand, if person says “P exists because X,Y,Z”, while he personally has seen the evidence (x,y,z) for P, and another person says “P exists because X,Y,Z” and he has only read about X,Y,Z from second sources – their knowledge is actually very different. Where is that difference (crucial one) reflected in logic?

I think your question is rooted in the fact that we generally consider two distinguishably different subjects under the umbrella of logic. The first is the art of argument as it takes place in natural language. The second is an deliberately constructed system, first proposed by Aristotle, and later greatly extended and expanded by figures such as Boole and Frege, which is intended to capture some of the essential qualities of natural argument, but to do so in a way that has mathematical rigor, precision and surety.

In ordinary language some of the arguments we use are based on matters of fact, the perceived truth of which cannot help but be subjective. Other arguments are largely definitional and thus deductive –they express relations of ideas that follow directly and inevitably from agreed-upon definitions.

In formal logic, the only arguments allowed are of the second type, and they exist within a framework carefully constructed so as to eliminate all remaining ambiguities of context, meaning and truth value. Thus, they can truthfully be described as objective and sure to the same extent and with the same assurance as mathematical truths such as “2+2=4” –at least (and this is an important caveat!) within the agreed-upon framework.

Your example of [(x AND y AND z ) IMPLY p ], however, shows the effects of mixing the two types of argument in ordinary language. Let us assume that there is a valid deductive relationship between x,y and z and p. In that case, given x,y, and z, it would be objectively the case that p would also have to be true. But all arguments are essentially hypothetical statements. The conclusion is only as good as the premises. If we have reason to doubt x,y and z, then we have equal reason to doubt p (assume we have no independent reason to believe p).

In general, deductive arguments are “objective”, inductive arguments are “subjective.” But one should be aware of an additional opportunity for confusion stemming from the concept of “mathematical induction,” which, despite its name, is an objective and sure process for establishing mathematic truths. This concept encompasses all arguments of the form “f(0), f(n) IMPLIES f(n+1), therefore all f”. It is so-named because of superficial similarities to ordinary language induction, but it gains a rigor on its way into mathematics that moves it into the realm of surety.

I would take a standard textbook on math, where all the propositions are correct. Write down 99 correct mathematical statements. And then add “Zeus exists”, and compile a text. Then I would argue, that if we have a box, from which we sample randomly 99 balls and they are have the property of being black, we can think with good reason that the next one will be black. And therefore, since 99 of the math propositions in the texts are have the property of being correct, there is good reason to think that “Zeus exists” is also correct. It seems wrong somewhere. But where?

No matter how strong an inductive argument is, it cannot guarantee results the same way a deductive argument can. It is always theoretically possible for the premises of an inductive argument to be true and the conclusion to be false.

In the case you mention, one might make a strong inductive argument to the effect that since the first 99 statements in the book were true, the last would be as well –but that conclusion might still be false. Furthermore, if you are aware of how the book was constructed and you do not include that information in your argument, you are guilty of the fallacy of suppressed evidence, and the argument can no longer be considered “strong.” Clearly, writing a false statement in a book of all true statements does not magically make that last statement true, and your knowledge of that fact must therefore be counted as a factor when judging the strength of the argument.

That may seem hopelessly subjective, but unlike the mathematically precise and assured conclusions of deductive arguments, the conclusions of inductive arguments can never be divorced from the real-world vagaries of context and circumstance.

A circular argument is technically a valid argument. For every case in which the premises are true, the conclusion will be true. So what makes it a bad one?


Thanks for your question. Although we tend to focus on validity in logic, it is actually only the minimal baseline requirement for a “good” argument. Valid means that the conclusion is guaranteed to be true if the premises are true. But in order for an argument to be “sound” it needs also to have true premises. Otherwise, it might just be “vacuously valid” as in the case where the premises contradict each other, and thus can never be true simultaneously.

P and not P
Therefore X

is also a “valid” argument, no matter what P and X might be. But it can never be sound.
Your case, however is different.

Therefore P

The problem with the circular argument is not that it cannot be sound, but that it does not play the function of an argument –to convince us of things we did not believe before. A circular argument yields exactly and only what you put into it. It does not increase your understanding or store of knowledge. It is “bad” not because it is invalid, but because it is non-functional. If presented in a debate or philosophical paper, it is considered misleading and illegitimate because it claims to prove something that it presents as a given or as an assumption.