2.5.4
Logic Part 4: Deductive Logic as Connected Assertions

If I were to state ‘Although it is sunny, I am presently indoors.’ you can see that I am conjoining two separate ideas. These ideas might be expressed separately to describe the same situation. ‘It is sunny’. ‘I am presently indoors’.  Of course, I could also join the ideas with the use of ‘and’ to produce the composite sentence ‘It is sunny and I am indoors’. In English, sentences that use either ‘and’ or ‘although’ have the same logical meaning.

The use of the words ‘although’ or ‘and’ join two or more ideas that in themselves could be said to each to have a truth value, which is either true or false in many forms of logic.  When we combine meanings of expressions with the words  ‘although’ and ‘and’  we are conveying a particular type of logical relationship between the ideas expressed in the sentence. Both of these words (‘although and ‘and’) are said to express the concept of logical conjunction. We can see this meaning is indeed logically distinctive we compare the following two sentences:

  1. I will give you a house AND a car  (which could be written ‘I will give you a car’ and ‘I will give you a house’)
  2. I will EITHER give you a house OR a car (used in the logical sense of either but not both)

You instinctively (or logically) know by understanding the meaning and use of the connectives ‘and’ and ‘either … or’ that you would own more if the first rather than the second sentence were true.

If I lied to you about the first part of sentence 1 so that it was false you would only get the car. The whole expression would be false because you would not be getting both. If however, I lied about the house in sentence 2, overall the sentence could have the same meaning because you would still receive the car.  In order for sentence 2 to be false, I would need to lie about both parts of the promise. In other words, when using the Logical Connective ” either … or”  to generate falsity both parts of the composite assertion need to be false. By contrast when we use of  ‘AND’ only one part needs to be false to render the whole statement false.

We can go further than the description given above and consider all of the combinations of false and true and so construct a ‘truth table’ for both of these connectives. For the ‘AND‘ connective (also called logical conjunction) the ‘truth table’ is

Assertion A
( I will give you a house)
Assertion B
(I will give you a car)
 Logical Value of the combined sentence
(house AND car)
True True True
True False False
False True False
False False False

Of course, we can extend the ways in which connectives extend the way composite assertions create meaning by extending the types of logical connections that we admit. So if I were to use the words “either, but not both” in a sentence I would be invoking the logical concept of exclusive disjunction otherwise known as the ‘exclusive OR’, which is equivalent to XOR ‘logic gate‘ in computer science.

The truth table for the XOR function “either, but not both” is very obviously:

Assertion A
( I will give you a house)
Assertion B
(I will give you a car)
 Logical Value of the combined
sentence
True True False
True False True
False True True
False False False

For the logical connective described as the Inclusive ‘OR’ ( also called disjunction) a different set of results are generated. This is because if either  A or B or both are true then the combined statement is true. The truth table for the Inclusive ‘OR’ is given below

Assertion A
( I will give you a house)
Assertion B
(I will give you a car)
 Logical Value of the combined
sentence
True True True
True False True
False True True
False False False

Notice that although we are looking at the meaning of logical connections in composite assertions we are not looking at the underlying meaning or implication of the word ‘true’ when applied to an individual statement or part of a composite statement. In constructing the truth table we must start with the truth or falsity of the individual parts of the assertion. Nevertheless, we can reach the intuitive understanding that by constructing the truth table we have understood that logic plays a role in shaping our understanding of the world. If for example, in the early twentieth century a court action was pursued against you for ‘breach of promise‘ with respect to what you said about a car and house in the above example it could matter a great deal whether or not you had used the word ‘and’ or ‘or’ in your promise. The problem with the use of ordinary language to express logical relationships is that it is often not as precise as symbolic expressions of formal logic that have a precisely defined meaning through their truth tables.

From my perspective, and quoting from a relevant article in the Stanford Encyclopedia of Philosophy concerning the intuitionist basis of logic, a logical “explanation is an account of what one knows when one understands and correctly uses the logical connectives”

This way of thinking can be extended to encompass expressions with words ‘implies’ or ‘therefore’ or the expressions ‘if … then’, “only if” (logical implication). Indeed we can pick out a larger set of words and phrases that cover a wider range of logical meanings and have associated truth tables. In addition, some of these connectives (AND, OR, XOR, NOT, NAND, NOR, XNOR ) are implemented as ‘logic gates’ in electronic circuits used in computing.

Steve Campbell
Glasgow, Scotland
2017, 2019

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