1.2 Propositions

Contrariwise, 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

“We don’t need no education.” - Roger Waters

Logic is concerned with Propositions.

A proposition is a declarative sentence that is (or at least could be) true or false.

Example

Here are four propositions:

  1. Two plus three equals five.
  2. There are more students enrolled in CS 81 this semester than last.
  3. 2016 was music’s best year ever.
  4. I have chocolate and I have vanilla.

Some propositions are always true (1) or always false, while others (2) may be true sometimes and false at other times. We might not all agree with (3), but it certainly would be reasonable for someone to argue for its truth or falsehood Links to an external site.. And as we will see later, some propositions like (4) can be viewed as combinations of two or more sub-propositions.

In contrast, the following sentences are not propositions.

  1. Is there milk in the fridge?
  2. Please pass the salt.
  3. Come here!

Human Languages Can Be Misleading

One complaint that students often have when studying modern formal logic is that it is packed with all sorts of specialized symbols and abstract formulas. Why is all this necessary?

The problem with trying to study logic using English (or any other human language) is that natural language is too slippery, full of exceptions and ambiguities.

For example, we could study the following inference and decide that it makes complete sense:

    Assumption 1: All dogs are mammals.
    Assumption 2: Dogs are sitting on my lawn.
    Conclusion:   Mammals are sitting on my lawn.

but other arguments with (apparently) the same structure have more dubious conclusions:

    Assumption 1: All dogs are mammals.
    Assumption 2: Dogs are good pets.
    Conclusion:   Mammals are good pets.

Similarly, this argument seems fine:

    Assumption 1: This dog is a husky.
    Assumption 2: This dog is mine.
    Conclusion:   This dog is my husky.

but if we change just a single common noun, we end up with plausible premises and an implausible conclusion:

    Assumption 1: This dog is a father.
    Assumption 2: This dog is mine.
    Conclusion:   This dog is my father.

Of course, the conclusion is correct if we take “is my father” to mean “it’s mine, and it’s a father,” (just as “is my husky” means “it’s mine, and it’s a husky.”), but if we interpret the conclusion with its conventional meaning (as understood by native speakers of English) then the conclusion is false even if the assumptions are true.

These are examples of what Aristotle called the fallacy of equivocation, where a word or phrase is used in two different ways within a single argument. (Fallacies are arguments that look logical at first glance, but are not valid.)

An even more blatant example of equivocation is the following:

    Assumption: Steelworkers belong to unions.
    Conclusion: Steelworkers are unionized.
    And so:     Steelworkers are electrically neutral.

taking advantage of the fact that the English word “unionized” has two meanings: (1) belonging to a union, and (2) not having gained or lost any electrons (i.e., being un-ionized).

Being Formal

Formal logic is logic performed using an artificial language designed with mathematical precision:

Working abstractly lets us focus on whether arguments (or patterns of argument) are correct, without linguistic pitfalls.

Well-Formed Formulas of Propositional Logic

Definition

The Well-Formed Formlulas (WFFs) Propositional Logic are defined inductively/recursively:

  1. An uppercase roman letter (possibly with primes and subscripts) is a WFF, e.g., \(P\), \(Q\), \(P'\), \(R_5\), …
  2. \(\top\) and \(\bot\) are WFFs.
  3. If \(\cal A\) stands for some WFF, then \(\lnot {\cal A}\) is a WFF.
  4. If \(\cal A\) and \(\cal B\) stand for WFFs, then \(({\cal A}\land{\cal B})\) is a WFF.
  5. If \(\cal A\) and \(\cal B\) stand for WFFs, then \(({\cal A}\lor {\cal B})\) is a WFF.
  6. If \(\cal A\) and \(\cal B\) stand for WFFs, then \(({\cal A}\to {\cal B})\) is a WFF.

If a sequence of symbols cannot be created by applying the above rules some finite number of times, it is not a Well-Formed Formula.

Example

The following are all WFFs: - \(P\) - \((Q\land \lnot\bot)\) - \(\lnot\lnot(\lnot\lnot P\to P)\) - \(((P'\land Q')\lor(P''\to Q''))\)

We can justify the second example as follows: \(Q\) is a WFF because it’s an uppercase letter; \(\bot\) is a WFF by the second rule so \(\lnot\bot\) is a WFF by the third rule; thus \((Q\land\lnot\bot)\) is a WFF by the fourth rule.

Conversely, the following are not WFFs as defined above (because there’s no way to construct these exact sequences of characters via the given rules):

We can justify the second example as follows: the only way for \((P\lnot \land \bot)\) to be a valid WFF is to show \(P\lnot\) and \(Q\) are both WFFs (and then apply the fourth rule for building WFFs), but there’s no rule that will let us conclude \(P\lnot\) is a WFF.

The third example is more subtle. There is no rule that lets us build a sequence of symbols as a WFF that contains \(\land\) but that does not contain parentheses, i.e., the closest legal WFF would be \((P\land Q)\). However, shortly we will relax these rules and let us write \(P\land Q\) as an informal “shorthand” for the correct, fully-parenthesized sequence \((P\land Q)\).

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Next: 1.3 Truth and Falsehood