As the Monte Python intro goes, “And now for something completely different!” This is an excerpt from “ESSAY 2: How We Believe; How Science Works” in my web-book, “Truth Cannot Contradict Truth”. This section covers “Rational Inquiry.” The other sections cover “Faith and Reason,” “How Science Works,” “The Limits of Science—What Science Can’t Do.” And forgive this shameless plug, but I’m trying to get some readership and comments (please oblige).
“’Contrariwise,’ continued Tweedledee, ‘if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.’”
—Lewis Carroll, “Through the Looking-Glass”
Syllogisms and Venn Diagrams
One way of knowing other than by faith/revelation is by deduction, drawing conclusions from propositions we believe to be true (“premises”), using logical procedures first set up by Aristotle–going from the general to the specific. Here’s an example (with apologies to Gelett Burgess), a “syllogism”:
Major Premise: All cows are purple.
Minor Premise: This animal is a cow.
Conclusion: This animal is purple.
If you know the premises to be true, then the conclusion is true. If the premises aren’t generally true (as in this example), then the conclusion may or may not be true. For example, you could paint a cow purple, or it could be a mutation.
Note the difference between the syllogism above and the one below:
Major Premise: All cows are purple.
Minor Premise: This animal is purple.
Conclusion: This animal is a cow.
There are animals that are not cows that are purple, for example, purple frogs; therefore the conclusion is false. Note: if the Major Premise was stated as “Only cows are purple,” then the conclusion would be true. This kind of logical fallacy is called “affirming the consequent.”
Venn diagrams can help us understand logic problems, as shown in the diagram above, where several types of logical propositions are classified according to the corresponding Venn diagram.¹
“Complicated” Deductive Logic, “Sorites”
Charles Dodgson (better known as Lewis Carroll, author of the Alice books), an Oxford academic, gave us many amusing and complicated puzzles that mixed his love of nonsense and logic. (These “polysyllogisms” are termed “sorites.”) Some of Carroll’s logical puzzles were exceedingly complicated, involving many statements and logical variables. Here’s a relatively simple one:
1. All babies are illogical.
2. Nobody is despised who can manage a crocodile.
3. Illogical persons are despised.
We combine 1 and 3 to give
4. All babies are despised.
Then 4 and the contrapositive² of 2 can be combined to yield
5. No baby can manage a crocodile.
Here’s another example by Alex Bellos, given in the Guardian, with its solution:
“The only people in the cereal cafe are from Stoke.
Every person would make a great Uber driver, if he or she is not allergic to gluten.
When I love someone, I avoid them.
No one is a werewolf, unless they have orange skin and blond hair.
No one from Stoke fails to Instagram their breakfast.
No one ever asks me whether I prefer Wills to Harry, except the people in the cereal cafe.
People from Thanet wouldn’t make great Uber drivers.
None but werewolves Instagram their breakfast.
The people I love are the ones who do not ask me whether I prefer Wills to Harry.
People with orange skin and blond hair are not allergic to gluten.”
Such complicated problems often may require computer methods for their solution. Some of the statements (premises) may be redundant and some may be contradictory. The type of analysis required to make sure that none of the statements are contradictory, so that no paradoxes will occur, is a sub-discipline in mathematical logic, “satisfiability theory”.
Certainly the deductive method should yield propositions which are either true or false. Not so! There are uncertainties, paradoxes, in logical deduction, which is the topic we’ll look at next
Some Logical Paradoxes
“A paradox, a paradox, a most ingenious paradox!”
–Gilbert & Sullivan, The Pirates of Penzance
Can deductive logic always yield an unambiguous true or false set of propositions? In his very fine book, Labyrinths of Reason, William Poundstone gives examples of logical paradoxes for which it is difficult to make a truth judgment. Perhaps the most famous of these is the Cretan Liar paradox (see Star Trek, Fooling the Androids Episode):
Epimenides the Cretan says, ‘All Cretans are liars.'”
Question: Is this statement true or false? If it is true, then Epimenides is a liar, but if Epimenides is a liar, how can his statement be true?
There is also the barber paradox,
The barber is a man in town who shaves all those, and only those, men in town who do not shave themselves.
Question: Who shaves the barber? If the barber doesn’t shave himself, according to the statement he does shave himself…
Both paradoxes invoke self-reference, whence the paradox. Bertrand Russell attempted to deal with the problem of self-reference by his “Theory of Types,” which sets up a hierarchy of statements, i.e. statements about statements, statements about (statements about statements), etc.
Let’s look next at another rational path, induction or inductive reasoning. The results aren’t as sure, but it still is a basic route to judgment in law, science and everyday life.
“The deductive method is the mode of using knowledge, and the inductive method the mode of acquiring it.”
—Henry Mayhew, nineteenth century British Journalist,
Induction is generally regarded as proceeding from particular instances or events to a general conclusion. (I’m not referring in this context to the mathematical method of proof.) Here’s an example.
A naturalist notices that bees move their rear end back and forth in a special way—”dance”–after they have been gathering nectar from a certain group of flowers. The dance is the same for a given group of flowers. The naturalist concludes that this bee-dancing is a communication to other bees about the location of the flowers and receives a Nobel Prize. (We’ll see below that science is generally more than collecting data and making inferences.)
Here’s a neat video of bees doing the waggle dance.
There are methods of assessing inductive reasoning propositions by means of probability statements, strength of belief quantification, and by Bayesian probability analysis.
ABDUCTIVE REASONING—INFERENCE TO THE BEST EXPLANATION (IBE)
“When you have eliminated the impossible, whatever remains, however improbable, must be the truth.”
–Sherlock Holmes, The Sign of the Four, Chapter 6.
One of the best known examples of abductive reasoning is given in the quotation above and indicated by its commonly used name, “Inference to the Best Explanation”or “IBE.” IBE uses given data to infer the most likely explanation of a past event that could have caused the data. It is commonly used in the so-called “historical sciences” (geology, paleontology, cosmology) for which laboratory experiments aren’t in order.
Here’s an everyday example adapted from one given in Stephen Meyer’s book about Intelligent Design,The Signature in the Cell:
You look out your window and note that your driveway is wet; three possible explanations occur to you: it has rained, the sprinkler has been set so that it also wets the driveway, your car has been washed. You notice that neither the street nor your lawn are wet, so you conclude that the third explanation—your car has been washed—is the correct one. A pail of water beside your car is confirmatory evidence for that conclusion.
Some philosophers of science put down IBE as lacking certainty and leading to false conclusions. In the past theories proposed as best explanations have turned out to be duds: caloric fluid for heat, ether as a medium for electromagnetic waves. However, it should be kept in mind that these theories were disproved by additional empirical evidence: caloric fluid by Count Rumford’s cannon-boring experiments, the ether by the Michelson-Morley experiments.
“In a retroduction, the scientist proposes a model whose properties allow it to account for the phenomena singled out for explanation. Appraisal of the model is a complex affair, involving criteria such as coherence and fertility, as well as adequacy in accounting for the data. The theoretical constructs employed in the model may be of a kind already familiar (such as “mountain” and “sea” in Galileo’s moon model) or they may be created by the scientist specifically for the case at hand (such as “galaxy,” “gene,” or “molecule”).”
–Ernan McMullin, “A Case for Scientific Realism”
Retroductive reasoning is commonly used by scientists to explain phenomena by a familiar model. A very early example is that in which Galileo proposed that the moon had seas and mountains on it just as does the earth, in order to explain the differing patterns of light and dark on the moon at different orientations with respect to the sun.
An example from contemporary molecular physics is the model used to describe molecular vibrations (nuclei in a molecule vibrating back and forth): a weight attached to a spring, vibrating back and forth, the so-called “harmonic oscillator.”
¹To make this example concrete, let “S stand for “cows” and “P” for “purple animals.” Accordingly, the “SaP” proposition is “All cows are purple (animals);” the white space means some purple animals are not cows and the black space means there aren’t any cows that aren’t purple. The “SeP” proposition is “No cow is purple;” the black space where the two circles overlap shows that there aren’t any animals that are both purple and a cow. The “SiP” proposition is “Some cows are purple;” there are cows that are not purple (white space in the “S” circle) and cows that are purple (red space where the S and P circles overlap) and purple animals that are not cows (white space in the P circle). The SoP proposition is “Some cows are not purple (animals);” the red space of the S circle. You can see (I hope) that the major premise of both syllogisms corresponds to SaP, and since there is a white space in the P circle (purple animals), there are some purple animals that aren’t cows, so the second syllogism can’t be true.
²The contrapositive of a logical statement just switches the hypothesis (premise) and conclusion of the statement and negates both. We can write the statement symbolically as A>B, (meaning “if the hypothesis A is true, then the conclusion B is true,”); then the contrapositive of that statement would be written symbolically as NotB>NotA, (meaning that if the negation of B is true, then the negation of A is true). For example, going to the first syllogism above,
A; this animal is a cow (minor premise);
B: this animal is purple (conclusion);
would correspond to
“If this animal is a cow, then this animal is purple” (true if the major premise of the syllogism, “all cows are purple,” were true).
The contrapositive would be
NotB: this animal is not purple (contrapositive premise);
NotA: this animal is not a cow (contrapositive conclusion),
which corresponds to
“If this animal is not purple, then this animal is not a cow.”