Formalisation in Propositional Logic

Question 1

What does it mean for a connective to be truth-functional?

Answer 1

A connective is truth-functional if and only if the truth-value of the compound sentence cannot be changed by replacing a direct sub-sentence with another sentence having the same truth-value.

Question 2

What are the five formalisation steps?

Answer 2

1. Check if the sentence can be reformulated in a natural way as a sentence built up from one or more sentences with a truth-functional connective. If this is not possible, then the sentence should be put in brackets and not analysed any further.
2. If the sentence can be reformulated in a natural way as a sentence built up from one or more sentences with a truth-functional connective; do so.
3. If that truth-functional connective is not one of the standard connectives in Table 3.2, reformulate the sentence using the standard connectives.
4. Enclose the whole sentence in brackets, unless it is a negated sentence, that is, a sentence starting with ‘it is not the case that’.
5. Apply the procedure, starting back at 1. to the next sub-sentence(s) (that is, to the sentence(s) without the standard connective of step 3).

Question 3

What are the three steps to translate the logical form of the English sentence into a sentence of L1?

Answer 3

1. Replace standard connectives by their respective symbols.
2. Replace every English sentence by a sentence letter and delete the brackets surrounding the sentence letter. Use different sentence letters for distinct sentences and the same sentence letter for multiple occurrences of the same sentence.
3. Give a list (the ‘dictionary’) of all sentence letters in the resulting L1-sentence together with the respective sentences they have replaced.

Question 4

What is the scope of a connective?

Answer 4

The scope of an occurrence of a connective in a sentence Φ is (the occurrence of) the smallest sub-sentence of Φ that contains this occurrence of the connective.

Question 5

How are tautologies, contradictions, consistency redefined in terms of propositional formalisation in L1?

Answer 5

(i) An English sentence is a tautology if and only if its formalisation in propositional logic is logically true (that is, iff it is a tautology).
(ii) An English sentence is a propositional contradiction if and only if its formalisation in propositional logic is a contradiction.
(iii) A set of English sentences is propositionally consistent if the set of all their formalisations in propositional logic is semantically consistent.

Question 6

When is an argument propositionally valid?

Answer 6

An argument in English is propositionally valid if and only if its formalisation in L1 is valid.