Logic Tutorial, Part 7.
Tautological Implications and Inferences

(c) 2017 by Barton Paul Levenson

Now that we have the conditional and biconditional operators (→, ↔), we can state some very useful equivalences using them. Rules using the conditional are "tautological implications," while those using the biconditional are "tautological equivalences." The first tautological implication is called Modus Ponens:

[(p → q) ∧ p] → q

This enables us to set up a nice syllogism:

M: If the Earth is about to blow up, we're all doomed.
m: The Earth is about to blow up.
C: We're all doomed.

p → q
p
———
∴     q

As opposed to putting all of a sequence of logical statements on the same line, putting them one to a line is putting them in "argument format." The underlining separates premises from conclusion; the ∴ is not so much a "conclusion operator" as a smug little "so there" symbol. Logicians aren't above the awful vice of pride.

Here's another useful tautology, Modus Tollens:

[(p → q) ∧ ¬q] → ¬p

A syllogism using it runs:

It's useless to object that there might be other reasons you didn't get an A. We're assuming the major premise to be true. It might have been clearer if it were phrased, "If I love sociology, nothing can prevent me from getting an A," or "If I love sociology, I will definitely get an A, whatever the circumstances." If it is true, and if the minor premise is also true, then the conclusion above is logically implied. Remember the distinction between validity and truth.

Here's a useful tautology which is easy to remember, the one called Simplification:

(p ∧ q) → p

and

(p ∧ q) → q

I.e., if both statements are true, then obviously either one taken by itself is true. Duh. Thinking of an example and writing it in English and in logical calculus is left as an exercise for the reader.

The next one, Addition, is a little more subtle:

p → (p ∨ q)

I.e., if we know p is true, then either p is true or q is true. This works because the ∨ ("or") operator yields a true conclusion if either one of its operands is true.

Note that the above argument is not a syllogism, since it has only one premise. Note also that this does not work in reverse! (p ∨ q) does not always yield p! (Check with a truth table). The statement that p → (p ∧ q) doesn't work, either. Note that in that last one, the "or" was changed to an "and."

The next rule is called the disjunctive syllogism:

[(p ∨ q) ∧ (¬p)] → q

[(p ∨ q) ∧ (¬q)] → p

Or in English, if either case A, or case B, must be true, and we know one is false, the other must be true. Obviously.

Here's an example:

The last rule is Transitivity:

[(p → q) ∧ (q → r)] → (p → r)

This one is pretty intuitive. If condition A implies condition B, and B implies C, then we can just say A implies C. These complicated rows of symbols say something pretty obvious if you just think them out.

Example:

The tautological implications we've just gone through all use the conditional operator (→). Another set of rules, "tautological equivalences," use the biconditional (↔).

We've already seen the Law of Double Negation:

p ↔ ¬(¬p)

Commutativity is another no-brainer:

(p ∧ q) ↔ (q ∧ p)

And, in summary, there's a tautological equivalence for every logical equivalence we covered in part 2. The whole set--the laws of logical inference--is summarized in this table:

*There has got to be a better name for the seventh law listed above than "Switcheroo Law." If anyone knows it, please e-mail me at levenson1960@gmail.com and let me know. Thanks!

Lastly, here's a table of logical operators in the propositional calculus. The last one is needed only when doing multi-line arguments in argument format:

So now we know simple statements can be symbolized with letters, and we have available to us seven logical operators and thirteen laws of logical inference. The next part of this tutorial deals with constructing sustained arguments.