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Defining tilde

In this video, York student Joseph shows how we define the connective 'tilde' using a truth-table.
The next truth table we’ll look at is for ‘tilde’ corresponding to ‘not’ or ‘it is not the case that’. Tilde does not connect two sentences, rather it goes in front of one sentence. So, in the truth table for tilde, we’ll need just one sentence, called ‘P’. Where ‘P’ is true, ‘tilde P’ is false, and where ‘P’ is false, ‘tilde P’ is true. And it’s that simple.

This video explains how we can define the meaning of the sentence-connective tilde using a truth-table. Below, we look at little more closely at its character.

Key points about tilde

First, tilde differs from ampersand in that, where ampersand is a two-place connective (we need to plug two sentences/sentential clauses into ‘&’ to make a grammatical sentence), tilde is a one-place connective: we only plug one sentence/sentential clause into ‘~’ to make a grammatical sentence.

Secondly, like ampersand, tilde is a truth-functional sentence connective. The truth-value of a tilde sentence is fixed in all cases by the truth-value of the sentence plugged into it. If a sentence ‘P’ is false, then ‘~P’ is true. If a sentence ‘P’ is true, then ‘~P’ is false.

Thirdly, as already suggested, it looks like tilde corresponds closely in meaning to ‘not’/‘it’s not the case that’ in English. For example, if ‘It’s raining’ is false, then ‘It’s not the case that it’s raining’ is true. If ‘It’s raining’ is true, then ‘It’s not the case that it’s raining’ is false.

Finally, as noted with ampersand, strictly speaking (if we were being really picky), when we’re defining the meaning of tilde we shouldn’t use sentence letters like P. Instead, we should use something else (we use Greek letters) to show any sentence could be plugged in to the connective. So the defining truth-table for tilde would look like this:

Figure 1. The defining truth-table for tilde

(This last point is little pedantic—many presentations just use sentence-letters to give the defining truth-table—but it’s worth being aware of the distinction.)

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Logic: The Language of Truth

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