by Brian Bosse, Copyright April 10, 2010, all rights reserved. 730 views
Definition: A declarative sentence is a sentence that asserts something or makes a statement.
The propositional calculus concerns itself with declarative sentences only as opposed to interrogatives, exclamatory, and imperative statements. Also, we will make an important distinction between a proposition and a declarative sentence. Consider the following three sentences…
1. ‘4’ is an even number.
2. ‘2+2’ is an even number.
3. The square root of 16 is an even number.
Even though sentences 1 through 3 above are different sentences, they all express the same thing. What these three different sentences express is the same proposition. Consider the following three sentences…
4. God is good.
5. Dios es bueno.
6. θεος εστιν αγαθος.
Once again, sentences 4 through 6 are different sentences all expressing the same thing. As such, we make a distinction between a sentence and that which it expresses. This is the motivation for our next definition…
Definition: A proposition is the referent of a declarative sentence.
Within the propositional calculus we will use certain capital letters called ‘sentence letters’ to stand for sentences. For example, we might assign the letter ‘P’ to sentence 4 above. As such, in the propositional calculus, when we see the letter ‘P’ we know it represents the sentence ‘God is good.’ So, if we say, “P is a true statement,” then we are really saying, “‘God is good’ is a true statement.” This is the motivation for the next definition and convention.
Definition: Sentence letters are symbols representing declarative sentences.
Convention: Capital letters ‘P’ through ‘Z’ are sentence letters.
Symbolic Sentences
1. Sentence letters are symbolic sentences.
2. If φ is a symbolic sentence, then so is ¬φ.
Informally, ‘¬’ is an abbreviation for “it is not the case that.” It can be thought of as the negation of the symbolic sentence following it.
3. If φ and ψ are symbolic sentences, then so is (φ → ψ).
Informally, ‘→’ indicates a conditional, an “if…then…” sentence. Given the symbolic sentence (φ → ψ), ‘φ’ is called the antecedent of the conditional (the ‘if’ part), and ‘ψ’ is called the consequent of the conditional (the ‘then’ part).
4. If φ and ψ are symbolic sentences, then so is (φ ∧ ψ).
Informally, ‘∧’ indicates a conjunction (“and”).
5. If φ and ψ are symbolic sentences, then so is (φ ∨ ψ).
Informally, ‘∨’ indicates a disjunction (“or”).
6. If φ and ψ are symbolic sentences, then so is (φ ↔ ψ).
Informally, ‘↔’ indicates a bi-conditional, an “if and only if” sentence.
7. Nothing else is a symbolic sentence within the propositional calculus.