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Structure and truth-conditions

Here we see how structure in propositional logic determines truth-conditions, and start to explore using truth-tables to work out truth-conditions.
A finely structured external staircase going up several levels
© Image by MichaelGaida

The grammar of our formal language is designed so that every grammatical sentence in our language has exactly one structure: there’s just one way to see it as being built up, level by level, from wffs. And our rules for connectives give exactly one set of truth-conditions for each kind of wff. This means each grammatical sentence in our language has exactly one set of truth-conditions: so, no ambiguities due to structure.

Let’s see how the truth-conditions for a complex sentence are determined. (Here we’ll also see how we can use truth-tables to work out the truth-values of a complex sentence in each and every kind of possible situation.)

We’ll look at the sentence ‘((A & B) & ~C)’.

We can see this complex sentence as built up in stages, starting with the basic sentences ‘A’, ‘B’, ‘C’, and working up through sentential clauses, until we plug clauses into a final connective. This is the main connective of the sentence, and defines what sort of sentence it is. Let’s look at how this works.

At the first level, we have our basic sentences: ‘A’, ‘B’, ‘C’.

At the next level up, we have sentential clauses made by plugging only basic sentences into sentence connectives:

  • (A & B)
  • ~C

At the next level (which is also the final level for this complex sentence), these two sentential clauses are plugged into ‘&’, to give us the whole sentence:

  • ((A & B) & ~C)

The last connective to get involved was the second ampersand. This is the main connective of the sentence, and it defines what sort of sentence it is …

Sentences with ampersand as their main connective are called conjunctions. (This term is also applied to ‘and’-sentences in English and to related sentences in other natural languages.) This sentence is the conjunction of ‘(A & B)’ and ‘~C’. The two clauses plugged into the main connective of a conjunction are called its conjuncts.

Now, watch the video to see how to start constructing a truth-table for our example sentence.

This is an additional video, hosted on YouTube.

© University of York
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Logic: The Language of Truth

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