What is This?

This is a supplement to section 2.2 of the Logic Text. You should read that before you read this. Section 2.2 discusses the various ways we express conjunctions, disjunctions, and biconditionals in English.

Review

Conjunctions:

• ‘and’ sometimes occurs in coordination with ‘both’. In this case, ‘both’ introduces one of the conjuncts and ‘and’ the other.
• there are other words in English that are equivalent to ‘and’:
  • ‘but’, ‘although’, ‘even though’

Disjunctions:

• ‘or’ sometimes occurs in coordination with ‘either’. In this case, ‘either introduces one of the disjuncts and ’or’ the other.
• ‘unless’ is another way of saying ‘or’.
• for the purposes of symbolization, we will assume that ‘or’ in English is inclusive.

Biconditionals:

• ‘\(P\) if and only if \(Q\)’ is equivalent to ‘\(P\) if \(Q\), and \(P\) only if \(Q\)’
• ‘just in case’ and ‘exactly on the condition that’ are other ways of expressing the biconditional. (Compare the role here of ‘just’ when it comes before a variant of ‘if’ with the role of ‘only’ when it comes before a variant of ‘if’)

Test Your Understanding

Symbolize each of the following, using your own scheme of abbrevation:

1. It rains unless I bring my umbrella.

   \(R{\mathord{\vee}}U\)

   Remember: ‘unless’ is just another way of saying ‘or’.

2. I bring my umbrella but it is sunny.

   \(U{\mathord{\wedge}}S\)

   Remember: ‘but’ is just another way of saying ‘and’.

3. Although I read the syllabus, I didn’t know the requirements

   \(S{\mathord{\wedge}}R\)

   ‘Although’ is just another way of saying ‘and’.

4. The Redbirds won exactly on the condition that their opponents lost.

   \(R{\mathord{\leftrightarrow}}P\)

   ‘exactly on the condition that’ is another way of saying ‘if and only if’

5. I bring my umbrella only on the condition that it rains.

   \(U{\mathbin{\rightarrow}}R\)

   Don’t confuse ‘only on the condition that’, which is another way of saying ‘only if’, and so expresses the conditional, with ‘exactly on the condition that’, which is used to express the biconditional.

6. I bring my umbrella but it doesn’t rain.

   \(U{\mathord{\wedge}}{\mathord{\sim}}R\)

   ‘but’ is another way of saying ‘and’. The ‘not’ here negates the smallest sentence it is a part of.

7. It is not the case that both the Redbirds win and their opponents win.

8. Both it is not the case that the Redbirds win and their opponents win.

   In the (7), the negation comes before the ‘both’; in (8), it comes after the ‘both’. So

   7. \({\mathord{\sim}}(R{\mathord{\wedge}}P)\)
   8. \(({\mathord{\sim}}R{\mathord{\wedge}}P)\) (Or, in informal notation, \({\mathord{\sim}}R{\mathord{\wedge}}P\).)

9. Either I bring my umbrella but it doesn’t rain, or I don’t bring my umbrella but it rains.

   This sentence is significantly more difficult to symbolize, and takes us beyond section 2.2. Try to apply the principles you learned in chapter (1). ‘Either…or’ works kind of like ‘If…then’. Commas indicate the bigger break. Aside from ‘it is not the case that’, negations negate the smallest sentence they are a part of.

   \((U{\mathord{\wedge}}{\mathord{\sim}}R){\mathord{\vee}}({\mathord{\sim}}U{\mathord{\wedge}}R)\)