Deductively valid arguments are arguments in which the premises guarantee the truth of the conclusion. If the premises the true, the conclusion must also be true. Deductive arguments have a formal structure, which can be put into symbolic form. Deductive arguments are either valid or invalid based on their structure or form, so they are sometimes called formal arguments. We looked at some basic deductively valid argument forms: modus ponens, modus tollens, disjunctive syllogism and dilemma. We also looked at two formal fallacies; denying the antecedent and asserting the consequent.

You can put any set of premises and any conclusion into a valid deductive argument and know that the conclusion is true if the premises are. If Socrates is human and hemlock is fatal to all human beings, it follows as a matter of logic that Socrates will die after drinking the hemlock. If Socrates drinks the hemlock and seems no the worse for wear, then one of our premises must be false. Either he's not really a human being - maybe he's a god in human disguise or an alien - or it's not true that hemlock is fatal to all human beings. Maybe he didn't really drink the hemlock. Maybe some human beings have an immunity to hemlock. Whatever the case, we know that one of our premises must be false, because if the premises were true, the conclusion would have to be. If the premises are not true, nothing can be deduced about the truth of the conclusion.

Inductively strong arguments only guarantee the truth of the conclusion with a certain degree of probability. If the premises are true, then the conclusion is probably true, perhaps with 99.999999999% certainty, but there is a logical possibility that it's false. Why not just go with deductive arguments then? Take the following example: "If the defendant had the means, motive and opportunity to commit the crime, then he is guilty. He had the means, motive and opportunity to commit the crime, therefore he is guilty." That's modus ponens and a deductively valid argument. But how do you know the second premise is true? The prosecutor gains nothing by casting his argument in deductive form, because what matters is the strength of inductive evidence, not the formal structure of his argument. It's more natural to argue, "The defendant had the means, motive and opportunity to commit the crime, therefore he is guilty to a reasonable doubt. Most arguments are inductive in nature and attempting to stuff them into a formal deductive structure is counter-productive.

Logicians reserve the terms valid and invalid for deductive arguments but refer to inductive arguments as strong or weak. Like deductive arguments, they can go wrong in two ways: weak premises or faulty logic. If one of the premises is false or doubtful, then the conclusion doesn't logically follow. Or, all of the premises may be true and well grounded in evidence, but they still don't support the conclusion. Suppose someone argues, "The rich hardly pay any taxes, therefore any tax reform plan should exclude them. This might be a good argument if the premise was true, but it's not. For example, the top 20% pay 95% of all Federal income taxes, whereas the bottom 40% pay nothing or get money back. The flaw in this argument is a faulty premise. On the other hand, suppose someone argues, "Margaret Sanger, the founder of Planned Parenthood, was a white supremacist and promoted abortion as a means of keeping keeping the 'less desirable' races from breeding. Abortion is a eugenics project." The premise is actually true in this case, but doesn't support the conclusion. Just because Planned Parenthood was founded on eugenicist principles doesn't mean it operates on those principles today. To conclude something about a thing based on its origin is called the genetic fallacy.

Argument by Analogy: This is a very common form of argument, especially in moral and political issues. The argument begins with a case, sometimes hypothetical, which everyone agrees has some quality. Next, the proponent says that this case is like the case at hand about which he is trying to convince you. Finally, the proponent argues that because these cases are similar, the quality from the first case must carry over to the second. Utilitarian philosopher Peter Singer says suppose you are on your way to a lecture and see a child drowning in a small pond, but stopping to save the child would ruin your new shoes and make you late. He suggests that not saving the child under those conditions would be monstrous. But then, he argues so is going about our busy lives and chasing trivial pursuits while children in the third world die of preventable diseases and malnutrition. Just as we are obligated to help the fictitious drowning child, we are obligated to help the "drowning" children in impoverished countries.

Reductio Ad Absurdum: In this form of argument, you start with your opponent's view, assume for the sake of argument that it's true, then show it implies something totally implausible that no one could possibly believe. For example, suppose someone's a strong Second Amendment advocate and says there should be no restrictions on firearms because it's the last line of defense against a tyrannical government. But if you believe that, you believe that people should be able to have weapons of the type that could defend against a military takeover. That in turn implies that people should be able to have fully automatic machine guns, bazookas and anti-aircraft guns. But it's ridiculous that someone be able to walk into a store and purchase military armaments, capable of inflicting casualties on a massive scale, so there must be some restrictions on firearms, whatever the Founding Fathers' motivations for the Second Amendment. You might see this as an opposite approach to argument by analogy. In argument by analogy, you start with a case that everyone agrees with and try to show that your argument is just like it; in reductio ad absurdum, you take your opponent's case and show that it is like a case no one agrees with.


Mistakes in reasoning can be classified into two kinds. Formal fallacies show a mistake which can be put in symbolic form and are mistakes in deductive logic. Deductive arguments are either valid or invalid, depending on whether their premises guarantee their conclusions. We looked at two formal fallacies, denying the antecedent and assorting the consequent, which masquerade as the valid argument forms modus ponens and modus tollens. Informal fallacies are mistakes in inductive reasoning and also often mimic logically sound forms of reasoning.

Appeal to authority: Appeals to experts are appropriate, especially when giving supportive evidence in an argument. However, it is tempting to forgo looking at the evidence oneself and to defer to experts. Accepting the conclusion of an authority without looking at any of the particulars of the evidence is usually problematic. When considering the use of an authoritative source of evidence in an argument, ask yourself the following five questions: 1. Are there even experts in this area at all? 2. Is this person or organization recognized by others in the field as having expertise? 3. Are they an expert on this particular subject? If the answer to any of these questions is no, then appealing to such a person or organization commits this fallacy.

Ad hominem: Literally "against the man." This is when you attack the character or motives of the person making the argument rather than the argument itself. The ad hominem abusive form is when you tar your opponent as a nazi, racist, sexist, homophobe, xenophobe, anti-immigrant, anti-woman, etc. instead of addressing his argument. Ad hominem circumstantial is when insinuate that the person has some nefarious or self-serving motive for believing something. It can also take the form of requiring some sort of experiential or moral standing in order have a view on a subject. So for example, someone might object to a man expressing a prolife position "because he can't get pregnant" or a person with a view on the military who has "never served." An argument is sound or unsound based on it's contents, not on who utters it. To see this is so, consider that any person making an argument could simply have an ally of the preferred identity or circumstances utter exactly the same words. Do the words mean something different because they came out of someone else's mouth? Do the ideas they represent change? Of course not. Chances are, if you think identity matters in logical arguments, you're trying to think with your emotions.

Guilty by Association: This is similar to the ad hominem argument, except you attack the person's argument based on who or what it is associated with. For example, someone might point out that Hitler was in favor of gun control. Just because Hitler favored it, doesn't mean it'd bad. Similarly, the fact that serial rapist Harvey Weinstein was a big Democratic donor or that white supremacist David Duke ran a Republican doesn't say anything about those parties as a whole.

Tu quoque: Literally "You, too." This argument defends attempts to distract from a problematic argument by pointing out that the other side "does it, too." For example, suppose someone brings up the topic of female "circumcision" in the Middle East as a violation of human rights. A defender of the societies who practice it might bring up male circumcision as practiced in the United States. Whether male circumcision is bad has no bearing on whether female "circumcision" is bad.

Lack of proportion: This is where two cases are compared which are very different in degree. In the above example, the procedure lops off the clitoris and removes or sews up the vaginal lips. An analogous case in men would be castration or cutting off the head of the penis. This is why this procedure is more commonly referred to as FGM or female genital mutilation, not circumcision. Circumcision is an unnecessary surgery on a child and does take away some sensation, but is not comparable to FGM.

Straw man: This is when you mischaracterize an opponent's view in order to more easily attack it. This is a old medieval combat metaphor. When practicing with weapons, instead of fighting a real opponent, you'd attack a suit of clothes stuffed as straw which would represent him. Those who commit this fallacy metaphorically impale a straw man who can't fight back and declare victory, while ignoring their real opponent. It's much easier to attack an absurd, exaggerated version of your opponent's argument than their real argument.

Red herring: This is when someone brings up a seemingly related but irrelevant topic in order to distract from the argument at hand. The name comes from a technique used to train hunting dogs. The dogs would be following some scent and the trainer would drag a ripe, pungent fish across the trail in order to to draw the dogs off the trail. Similarly, an opponent may try to take you "into the weeds" with some other issue which has no bearing on the subject. For example, during an abortion protest, someone might say "Why don't you help the children who are already born?" Whether children who are already born are receiving sufficient care is irrelevant to the topic of whether unborn children deserve consideration or have a right to life.

Ad Populum/Bandwagon: Everyone does or believes something, so you should do. For example, someone might argue that every modern, industrialized country has socialized medicine, so the U.S. should do. Or she might argue that no other modern, industrialized country has the prison population that the U.S. has, therefore we should have fewer prisoners. Sometimes this is used as an excuse for bad behavior. "There's nothing wrong with taking a few office supplies from work; everybody does it."

Slippery slope: This is the argument that a first step will lead to a cascading series of steps culminating in some catastrophe. For example, a person might argue that gun registration will lead to gun confiscation and then a police state like North Korea. Sometimes small steps do lead to cascading negative effects. Taxes and and the creation of government agencies are like this, as are new laws and regulation. But in the foregoing example, a lot has to be assumed to fill in the steps. For example, we've had gun registration for some time but there has been no mass confiscation, and parts of Europe, which have largely confiscated private firearms are not police states like North Korea.