t-LOGIC FOR TEENS

Teens easily learn essentials of standard statement logic by this mathtivity, which merely manipulates letters of the alphabet to achieve the resultsof logic. (Years ago, a game, "Wiffenproof", was devised and promulgated by THE MENSA SOCIETY. But it uses confusing alphabetic letters for OPERATORS and OPERANDS. And, elsewhere, I've declared my opposition to GAMES in learning.)

I call this "t-logic", in keeping with my extension (described in detail, elsewhere) of logico-mathematics to embrace both t-systems and o-systems. What I call t-logic allows NO DISTINCTIONS OF ORDER a.k.a. DEGREE. It is separated from o-logic which DOES RECOGNIZE ORDER a.k.a. DEGREE, ALONG WITH TYPE a.k.a. KIND.

Briefly (in t-logic), we take, as GNOMON (A.K.A. UNIT), any statement (a.k.a. declarative sentence, a.k.a. assertion, a.k.a. proposition), which is potentially verifiable as true or false. (For example, "It is raining" and "The streets are wet" are statements. But "The women of Planet Sappho cannot hum" is not a statement, being unverifiable.) Next, we consider the combinators of these GNOMONS (STATEMENTS), namely,

  1. conjunction ("and"),
  2. disjunction ("or"),
  3. conditional ("if _, then _),
  4. biconditional ("If _ and only if _"),
  5. negation ("not _").

The first four are binary, that is, operate on two simple or compound assertions at a time, while the fifth is unary, that is, operates on a single simple or compound assertion at a time.

I use the vowels, A, E, I, O, U, to denote these operators: "A" for "and", "E" for "equivalence or biconditional", "I" for "conditional", "O" for "or", and "U" for "negation" ("undoes"). Also, the consonants denote simple or noncompound positive statements, that is, without NEGATION.

This formulation allows use of Polish prefix notation (as introduced by J. Lukasiewicz -- that L is supposed to be slashed but HTML protests, "I don't do slashes!") Note: A calculator uses "reverse or postfix Polish": put in the numbers and then the operator.

I'll now formulate univalency for t-statement logic and illustrate the general idea -- except for a final step -- by stating a definitive rule for a woof (wff, well-formed-formula). But I want to use "wff" in a more general way, so I shall speak about a "t-wff", a "well-formed formula in t-statement logic". (Elsewhere, I discuss the o-wff, which requires MORE OPERATORS.)

DEFINITION:

  1. Whatever is denoted by a consonant is a t-wff.
  2. Whatever is denoted by A, E, I, O, followed by two t-wff's, is a t-wff.
  3. Whatever is denoted by U followed by a t-wff is a t-wff.

Now this seems gingerpeachy. It shows what is in the system. But it "leaves the fence open so that a predator or a pretender could wander or sneak in"! We need a closure rule:

     4. Nothing is a t-wff unless it is designated by subrules 1-3.

The first three parts of the definitional rule, taken together, constitute that "packaging" denoted "a t-wff". The last part is an instance of a closure rule, a notion unique to good logico-mathematical language.

The definitional rules given for t-logic can be easily demonstrated for modus ponents (a.k.a. validity of asserting the precedent), the most famous of logical proof rules.

Let P, Q be two statements, that is, declarative sentences capable of verification as true or false.

  1. "If P, then Q" denotes the conditional, "If statement P is so, then statement Q is so". In Polish Prefix, this is (with "I" for CONDITIONAL): IPQ.
  2. "(If P, then Q) and P" denotes this conditional assertion conjuncted ("anded") with the statement that P is so; or AIPQP.
  3. "If ((If P, then Q) and P), then Q" denotes that the premise, "(If P, then Q) and P", implies its consequent, "then Q" is so; or IAIPQPQ.

Example: "If it is raining, then the streets are wet; it is raining; then the streets are wet".

The standard way of PROOF is to CHECK on paper, which involves UNDERLINING THE WFFS. I'll SPACE the terms, for convenience.

IAIPQPR Þ I A I P Q P Q (by RULE 1).

IAIPQPR Þ I A I P Q P Q Þ I A I P Q P Q (by RULE 2 ON I)

IAIPQPR Þ I A I P Q P Q Þ I A I P Q P Q Þ I A I P Q P Q (by RULE 2 on A)

IAIPQPR Þ I A I P Q P Q Þ I A I P Q P Q Þ I A I P Q P Q Þ IAIPQPQ (by RULE 2 on I)

Hence, IAIPQPQ = IAIPQPQ. QED.

On paper, you can omit the chain of steps above by UNDERLINING an UNDERLINE, but you can't easily show that on the computer.

Are you ready for a t-wff-TEST?.

OK. WOOF. UNIVALENT. But what about VALIDITY? We've seen that ANY CONSONANT REPRESENTS A t-wff, or DECLARATIVE SENTENCE WHOSE TRUTH IS VERIFIABLE.

Yet surely, for example, "It is raining" can't be a VALIDITY! No.

But, consider, "If it is raining at the Washington Monument, DC, on July 4, 1976, it is raining at the Washington Monument, DC, on July 4, 1976". THAT IS A VALIDITY. NOTHING IN LANGUAGE ITSELF CAN RENDER IT INVALID. Hence, in saying it, YOU CANNOT BE SAYING AN UNTRUTH.

Ok. But how do we get from here to there? From UNIVALENCY to VALIDITY? Ah, that's the next step: t-TABLES.

t-Logic corresponds to PECKING ORDER FOR PRESCHOOL KIDS and to FACTOR LATTICES for MIDDLE ELEMENTARY SCHOOL KIDS, as you can see.