How to Think: Introduction to Logic, Lecture 5 with David Gordon - Mises Academy
Lecture 5The Categorical Syllogism (II)
Dictum de Omni et Nullo(Maxim of All and None)●Whatever is affirmed (or denied)universally of any subject is therebyaffirmed (or denied) of every logical partof that subject.●The Scholastics thought that this wasthe fundamental principle of reasoning
Dictum Continued●Example: All men are elephants●Socrates is a man●Therefore, Socrates is an elephant●What is affirmed universally of men isaffirmed of a part of the class of men, i.e., Socrates
Still More Dictum●How do we know that the dictum is true?●We directly grasp its truth●If it is denied, there is a violation of the Law ofContradiction.●You would affirm both an A proposition andan O proposition about the same subject, e.g., “All men are elephants” and “Some man isnot an elephant”.
More of Guess What?●In the dictum, the universal subject istaken as a whole, not as a collection ofindividuals.●The A proposition is not the result of anenumeration
“Reasoning” By Enumeration●Suppose we argue this way:●All students in this course are interested inlogic●Andreas is a student in this course●Therefore, Andreas is interested in logic●One could know that the first premise is trueonly by checking to see whether everystudent in this course was interested in logic.But if we did this, we would already know thatAndreas was interested in logic.●The argument would result in nothing new.
Fallacy of Division●The dictum de omni shouldn’t beconfused with the fallacy of division●In this fallacy, what applies to aconjunction is wrongly taken to apply tothe parts of the conjunction●The conjunction can be implicit
Diamond-Water Paradox●The famous diamond-water paradox is anexample of the fallacy of division●Before the “marginalist revolution”, mosteconomists thought that subjective valuescouldn’t explain prices.●The reason they thought that was thisargument:●Water is more valuable than diamonds●If subjective value explains prices, then watershould have a higher price than diamonds●Diamonds have a higher price than water.
Structure of the Paradox●We have derived by a valid argumentthat water has a higher price thandiamonds.●But this conclusion is false●Therefore, at least one of the premisesmust be false●This is similar to a reductio argument,but is not the same. “Water has a higherprice than diamonds” is false but not a
So What’s Wrong With theParadox?●The paradox commits the fallacy of division●“Water is more valuable than diamonds” istrue only for the total quantities of water anddiamonds●There is thus an implicit conjunction in thispremise●It doesn’t then follow that any particularquantity of water will be more valuable thanany particular quantity of diamonds
Fallacy of Composition●In this fallacy, what applies to a part of aconjunction is wrongly applied to thewhole conjunction●Mill’s proof of the principle of utilityappears to commit this fallacy
Mill’s Proof●“No reason can be given why the generalhappiness is desirable except that eachperson. . .desires his own happiness. This,however, being a fact, we have not only allthe proof which the case admits of, but allwhich is possible to require, that happiness isa good: that each person’s happiness is agood to that person, and the generalhappiness, therefore, a good to the aggregateof all persons.” (Mill, Utilitarianism. Chapter 4)
Finding the Fallacy●The fallacy is here: “and the generalhappiness, therefore, a good to theaggregate of all persons”●This doesn’t follow from each person’staking his own happiness to be a good●You don’t have to hold that there is agood to the “aggregate of all persons”.
Another Fallacy in Mill’sPassage●The same paragraph from Mill alsosays: “The only proof capable of beinggiven that an object is visible, is thatpeople actually see it. The only proofthat a sound is audible is that peoplehear it. . .in like manner, I apprehend,the sole evidence it is possible to provethat anything is desirable, is that people
Equivocation●Mill’s argument by analogy can be used toshow that if something is actually desired, thisshows it is capable of being desired.●Mill uses the analogy to show that ifsomething is desired, then it ought to bedesired.●Mill thus equivocates on “desirable”●There have been defenses of Mill that acquithim of both fallacies. There is a vast literature
Paradox of Thrift●A famous example that uses ( not commits)the fallacy of composition is the paradox ofthrift.●If a single person tries to save, his savingswill increase.●It doesn’t follow that if everyone tries to save,total savings will go up.●Keynes used this argument but it goes backto Bernard Mandeville.●Note that from the fact that it doesn’t followthat total savings will go up that it is false that
Back to the Dictum●The Scholastics held that the First Figureshows the dictum most clearly●In the first figure, the major premise has themiddle term as the subject in the majorpremise and the middle term as predicate inthe minor premise.●No Misesians are Keynesians●All faculty at the Mises Institute are Misesians●No faculty at the Mises Institute areKeynesians●This is a valid mood (EAE) of the first figure. It
Immediate Inference●The second, third, and fourth figures ofthe syllogism can be reduced to the firstfigure●We aren’t going to show how to do this,but the process depends on immediateinference.●Without introducing a new term, we canderive a new proposition from a givenproposition.
Some Types of ImmediateInference●Conversion●We interchange the subject and predicate.From “All men are mortal”, we get “Somemortals are men.”●Two rules for conversion●Both propositions must have the samequality, i.e., both affirmative or both negative●If a term isn’t distributed in the originalproposition, it can’t be distributed in the newproposition.
More on Conversion●In the example, we can’t convert to “Allmortals are men” because “mortals” isn’t distributed in the original●We can convert “No Misesians toKeynesians” to “No Keynesians toMisesians”. An E proposition excludestwo classes from each other, so it workseither way.
Obversion●In obversion, we keep the same subject butchange the predicate into its contradictory.●To do this, we have to change the quality ofthe proposition, i.e., affirmative to negative ornegative to affirmative●The obverse of “All men are mortal” is “Nomen are not-mortal”.●There are many other types of immediateinference, but we won’t go into them here.
Mill’s Criticism of theSyllogism●Mill thought that the major premise ofthe syllogism begs the question. Wecouldn’t know “All men are mortal’unless we had examined each personand found that he was mortal. But thenwe would already know Socrates wasmortal and wouldn’t need the syllogismto prove it. We couldn’t learn anything
The Attack on Mill Continued●We have have already seen what is wrongwith Mill’s argument. He is taking thesyllogism to be an argument fromenumeration.●The major premise of the syllogism isn’t anenumeration. It is about the necessaryfeatures of the concept of the middle term●Mill didn’t believe in conceptual necessities.He thought that “2 + 2 = 4” is an empiricalgeneralization.
Progressive Syllogisms●A Progressive Syllogism reasons fromcause to effect●Rent control will produce a shortage ofapartments●The City Council has just imposed rentcontrol●We will have a shortage of apartments.
Regressive Syllogisms●A regressive syllogism reasons from effect tocause●A business cycle, in certain conditions, mustresult from an expansion of bank credit●We are now in a business cycle in theseconditions.●This business cycle has resulted from anexpansion of bank credit
Example of a Fallacy●“This dichotomy [defended by freemarket economists] between marketsand states---between trade and rules---is false and hides more than it reveals.Market exchange, and especially long-distance trade, cannot exist withoutrules imposed from somewhere.” DaniRodrik, The Globalization Paradox (
The Fallacy Revealed●Market exchange, and especially long-distance trade, cannot exist withoutrules imposed from somewhere.”●It doesn’t follow from the necessity ofrules that these rules have to beimposed or that we must have a state.