7 - Basic Logic
Logic has more to do with everyday life than brain teasers and puzzles, though it is very often subsumed or ignored. We apply logic, to some degree, to every action we take, and could benefit from being more deliberate in our approach.
Some Basic Concepts
The author considers some basic concepts of logic:
Consistency
Consistency among logical statements exists only when they can all be true: a man can be both happy and married, as one does not generally prevent the other. They are inconsistent when one precludes the other: the man can be either 30 or 35 years old, but not both. We generally give little attention to consistency, and pause when we notice inconsistencies and attempt to resovle them.
Truth or falseness does not factor into consistency. "Paris in Italy" and "Only twenty people live there" are consistent with each other - one does not make the other false - but both of them are false for other reasons. Statements that are demonstrably true are consistent, and only when a statement is not demonstrable do we begin to wonder whether it might be inconsistent.
Statements that are broad, or speak to the extremes (all/none, everyone/nobody, always/never) activate the inconsistency switch. They are very seldom true and, when true, generally cannot be confirmed, so we are instantly aware that there must be exceptions and the statement must be false.
Entailment
Entailment implies that one statement logically follows from another, often where the second is broader in scope than the first. If the phrase "I am wearing lipstick" is true, it follows that "I am wearing makeup" is also true, lipstick being a specific subclass of makeup, It is generally true that an entailment is broader or more vague than the statement it follows.
One important notion is that a true statement cannot have false consequences, but a false statement can have true consequences. As such, the statement "a bomb exploded in London" cannot be true if its entailment "there was an explosion in England" is false.
There is also the instance of phrases presented as if they are entailments, but which do not logically follow their precedent. "This cake tastes good" and "Herman speaks Latin" do not follow one another logically, in either order.
A less silly example might be the sales of shark-fin cures on the basis that sharks don't get cancer. It is implied to follow, but doesn't actually follow, that eating a creature that is immune to a disease grants us the same immunity. It does not follow (not to mention that sharks actually do get cancer in their cartilage).
Logical Equivalence
If one statement entails a second, and the second entail the first, then the two are logical equivalents. "A lion runs faster than an elk" and "an elk runs slower than a lion" communicate the exact same truth-value, and are logically equivalent.
It is also not possible for one of them to be true and the other to be false.
Logical Connectives
Logical connectives indicate the way that statements can be linked together to describe more complex statements.
Conjunction
A logical conjunction pairs two statements and presumes both to be true: Protons are positively charged and electrons are negatively charged.
A conjunction can also tie two concepts to the same qualifier: Picasso and Rembrandt are painters.
If either statement is false, the conjunction is false - but it does not prove the other statement to be false: apples are fruit and cheese is fruit is false because cheese is not fruit (but apples are still fruit).
Disjunction
Disjunction suggests that one statement is true and the other is false: "The visitor is a man or the visitor is a woman." It is a choice of one or the other, but not both.
In practice, there are few situations in which a true disjunction occurs, "both" and "neither" are often the case - such as when someone suggests your car is not working because "either the battery is dead or you have blown a fuse" - though most people don't split hairs.
(EN: There is a sales ploy in which you sucker a mark into buying something by using a disjunctive structure to suggest his choice is to buy one thing or another, hoping he will forget he has the option to buy neither.)
There's some punctiliousness about grammar as well - such as saying "white glue or tape" indicates the tape must be white - but this seems excessive.
Negation
The author mentions negation, which seems a bit like disjunction, in which you a choice is presented as exclusive: you may choose one or the other but not both.
The difference, perhaps, is that in negation, the act of making a choice negates the other alternative. It is not about its truth or falseness beforehand, but as a consequence of a choice that is being made.
It's also suggested that a statement and its negation are an "exhaustive and exclusive" set - there can be no other alternatives, and one being right makes the other wrong.
The Conditional
A conditional statement requires the first condition to be met before the second can even be considered. "If you are a member then you can get a discount" means that you must be a member to get the discount (it will not be offered to a person who is not a member) - but the second bit is not necessarily true (becoming the member makes you eligible for the discount, but doesn't guarantee you will receive it).
Computer programs make heavy use of conditionals because what they do depends on their evaluation of the input they receive or the data to which they have access. Likewise, procedural documents and decision trees are often cascading if/then conditions. Laws also entail conditions under which doing a certain thing under specific circumstances is permissible or prohibited.
The Biconditional
A biconditional sets up two statements as being conditions of one another, examples of which are difficult to find outside of the abstract world of mathematics.
Consider the statements "it is a triangle" and "it is a polygon with three sides" - if either of these is true, than the other is also true.