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Example: Examples of Propositions With a More Complex Syntax

(related to Definition: Interpretation of Propositions - the Law of the Excluded Middle)

We have introduced a special case for the syntax of $PL0$ as propositions corresponding to Boolean terms. Let us recall the examples of Boolean terms we have considered as $PL0$ so far:

• $x$,
• $(x)$,
• $\neg a$,
• \((\neg(\neg y))\),
• \(((x\Rightarrow(y\vee(1\wedge(\neg w))))\wedge 0)\),
• \(((a\Leftrightarrow b)\vee(0\Rightarrow y))\)

However, the Law of Excluded Middle also holds for any kind of more complex syntax, in which we allow the two-valued interpretation of $PL0$.

Let us first discuss some less formal example - the example of spoken human language (like English).

• Interrogative statements like "Are you his uncle?" or imperative statements like "Report to the colonel's office!" are syntactically correct statements of English language but are not propositions, because no truth value can be assigned to them.
• The truth value of a proposition can depend on the person making the judgment or on the time and circumstances in which the statement was made. For instance, "I'm cold" would mean something completely different when a child in winter says it, or when Socrates (who drank a poison by his hand) says it in the moment of his death.
• "He enjoys mathematics." is not a proposition, since the pronoun "he" prevents us from being able to decide whether the sentence is true or false.
• "Socrates was a man" is a proposition, since we can decide that the sentence is true.

Now, let us discuss some more formal examples - the example invoking the formal language of equations:

• "$1=0$" is a proposition since we assign the truth value "false" to it.
• "$1=1+0$" is a proposition since we assign the truth value "true" to it.
• "$x+2=5$" is not a proposition because the value of $x$ is not defined. If we said "$x+2=5$" where "$x=2$", then it would be a proposition, since we can state that it is false.

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References

Bibliography

1. Cryan D., Shatil S., Mayblin B.: "Logic. A Graphic Guide", Icon Books Ltd., London, 2001
2. Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016