1.1 Truth Value vs Truth

What did they say? What did they really say? Here’s my favorite trick for thinking critically. I was first introduced to this technique by my sixth grade teacher, Mrs. Ravitz. You probably saw this if you studied geometry.

Does this statement make sense?

2+2=4

Of course it does. Is it true? I think we would all agree it is.

Does this statement make sense?

12-x4=18+

No. It violates the rules of a valid math expression: it has bad grammar and makes no sense. It is nonsense. To discuss whether it is true or false has no meaning.

Does this statement make any sense?

2+4=7

Sure it does. Its grammar is perfect, its vocabulary simple and familiar, its syntax unimpeachable. But is it true? No. Its truth value is FALSE.

In grade school, you spent very little time examining valid math expressions that were false. In my case, we spent about three hours on it when learning what “=” meant. But every valid expression has a truth value: it may be true or false. Thus,

2+4=7 is a valid expression with truth value FALSE.

2+4<>7 is a valid expression with truth value TRUE. (“<>” means “is not equal to”.)

Do you see this? If not, quit reading this blog right now. It is foundational to everything I write.

Here are some analogous examples from speech. Does this statement make sense?

If I do not eat then I will die.

Of course it does. Is it true? If you had to render an opinion without access to any other information, you will likely say “Yes, it is true.”

Does this statement make any sense?

;Donald quack was prxgql my .yesterday blimp ? yellow

No. It violates rules of grammar, vocabulary, spelling and syntax. It is nonsense, and discussing its truth value has no meaning. Try this one; it’s one of my favorites:

If you love me, then you will do it.

It is a valid statement; it makes sense; but what of its truth value?

Statements of the form “If-Then” are called conditionals. The If sets the condition under which the Then is true. A popular notation is

p -> q

which reads “If p then q”. You can flip the parts around a few ways to create variants of a conditional.

~p -> ~q reads “If not p then not q” and is called the converse of the original statement.

q -> p reads “If q then p” and is called the inverse.

~q -> ~p reads “If not q then not p”. Note this is the inverse of the converse, or the converse of the inverse if you prefer. It has its own name: the contrapositive.

Now here is a wonderful fact about conditional statements that you may take for granted or prove to yourself as homework or look up elsewhere:

For any valid conditional statement, the statement and its contrapositive will have the same truth value. Its inverse and converse will have the same truth value.

Wow! Imagine that! If you can’t we will run through some simple examples in a moment. But the implication is that if you can fit some crazy assertion you heard on the news into the “If-Then” form, you can do a little flipping around to help you figure out if the statement was true, false, or invalid — nonsense.

Consider the statement “They always cancel games for rain.” Let’s assume this is true. We can put this in the form of a conditional:

“If it is raining, then the game is cancelled.” Statemement, TRUE.

“If it is not raining, then the game is not cancelled.” Converse, FALSE (why?).

“If the game is cancelled, then it is raining.” Inverse, FALSE (why?).

“If the game is not cancelled, then it is not raining.” Contrapositive, TRUE.

Note the statement and its contrapositive have the same truth value: they are both TRUE. The Inverse and converse also have the same truth value: they are both FALSE. Why are they false? Because for all we know there may be lots of other reasons to cancel a game besides rain: flood, fire, diarrhea, zombie attack, etc. But because the four forms fit our expected pattern of truth value, we can stay confident that the original statement was valid.

Let’s look at my favorite:

“If you love me, then you will do it.” Statement, uncertain truth value (I, at least, am suspicious).

“If you do not love me, then you will not do it.” Converse

“If you will do it, then you love me.” Inverse

“If you will not do it, then you do not love me.” Contrapositive.

The statement is powerful and persuasive. It calls the audience to a decision. It clearly implies, “The time has come to prove your love for me with your actions.” It clearly implies its contrapositive: “If you refuse me, then I will know you do not love me.” There is a heavy assumption behind this: that love is proven by action, and that the lack of action is proof of lovelessness. (Did you catch the statement/contrapositive riff in that last sentence?) I find it difficult to assign a truth value to any of the four statements, but I tend toward FALSE, because it seems to me our action or inaction could be informed by many motivations — not just love.

Look at this closing example by way of contrast. The “q” has a “not” in it, so watch carefully how we transform the statement:

“If you love me, you will not manipulate me.” Statement, TRUE in my book; maybe false in yours.

“If you do not love me, you will manipulate me.” Converse, FALSE, because clearly manipulation is not inevitable.

“If you will not manipulate me, then you love me.” Inverse, FALSE, because I could refrain from manipulation for many reasons and yet despise you.

“If you will manipulate me, then you do not love me.” Contrapositive, TRUE in my book.

The statement and its contrapositive have matching truth value; the inverse and converse have matching truth value. We can be confident in the validity of the position taken in the original statement.

This simple analysis sheds light on “What are they saying? What are they really saying?” I will use this technique from time to time in posts to this blog, and I will refer back to this post by reference.

Mind your p's and q's
Mind your p’s and q’s
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9 Comments Add yours

  1. Micah Mundinger says:

    If you love Me -> you will obey My commandments

    also:

    Parents often manipulate their kids out of love.

    Obviously, these are side-line to the (completely valid) point of the post, but…

    Like

    1. Wow! What a profound observation. I agree: I was thinking of things horny teens say to manipulate each other; you remind me that Christ told us the self-same thing. I demonstrated that this assertion is wobbly because I might obey you for a variety of reasons–or conform to the teachings of Christ for reasons other than love. I never thought of Christ or God as manipualtive, probably because I came to love, trust and respect them before I got hit up with the proposition. Perhaps trust and respect make all the difference in a relationship.

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  2. A is not ~A

    Is this true?

    Liked by 1 person

    1. I’ll bite! I would say Yes, this is true–usually. It reads “A is not Not-A”. If A=”It is raining”, then it reads “‘It is raining’ is not the same thing as ‘It is not raining'”, and most people would agree this is true.

      HOWEVER! In the field of number theory, Gödel’s Theorem demonstrates that for every system of arithmetic you can imagine, no matter how simple or complex, there will always exist valid expressions (ones that conform to all the grammar rules) for which you CANNOT prove the truth value. Gödel proved that there will always be some expression p for which you can prove: p=~p
      That is, the truth value of p is indeterminate.

      What this means to me is that all math has little holes in it. It is not perfect and pure. There is the slightest trace of ambiguity and doubt even in 2+2=4. And 2+2=5.

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  3. Give an actual example ( not from maths) where A is not ~A is untrue.

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    1. I doubt I can. In maths, Godel had to range very far afield to identify vastly complex, obscure expressions of indeterminate truth value. Perhaps the equivalent from human narrative would be the seemingly endless proofs for the existence/non-existence of God.
      Following are some examples I can think of from the human narrative that hint at this idea that it may be difficult or impossible to assess the truth value of a valid expression; though this is similar to paradox:
      The kidnapper says, “Meet my demands or the hostage will die, and it will be your fault.”
      The white-collar criminal says to the detective, “If you arrest me, the good work I do will go unfinished, and you will be responsible for the negative consequence.”
      God says to Abraham, “Take Isaac, your son whom you love and through whom I have promised to bless all the earth, to a place I will show you and sacrifice him to me there.”

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  4. How does any of it disprove that A is not~A?

    There are billions of statements whose truth value no human knows. But that does not disprove that
    A is not ~A

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    1. I can only refer you to “Gödel, Escher, Bach: An Eternal Golden Braid” by Douglas Hofstadter.

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