1. The conditional and the bi-conditional

2. Some new connectives…
So far in this course we have been working with just three connectives: ‘∧’, ‘∨’ and ‘~’. This is, in fact, more than enough (you can actually get by with just ‘~’ and ‘∨’ if you want to) – but it is standard to include two more connectives in most accounts of sentence logic. So we will follow that lead.
These connectives are the conditional and the biconditional.

3. The conditional
Let’s start with the conditional. We briefly touched on it earlier in this course, but let’s refresh our memories…
The conditional is usually represented with the symbol ‘⊃’ or the symbol ‘→’.
It is supposed to be a rough analogue to the English language connective ’if…then…’

4. The conditional
The first part of the conditional (after the ‘if’) gets called the antecedent and the second half (after the ‘then’) gets called the consequent.
E.g. for ‘if the litmus paper turns red, then it’s an acid’, we get ‘the litmus paper turns red’ as the antecedent, and ‘it’s an acid’ as the consequent.

5. The conditional
The truth table that defines the conditional looks like this: P Q P ⊃ Q t t t t f f f t t f f t

6. The conditional
It is only false in case 2, that is, when the antecedent is true and the consequent is false. Otherwise it is true.
The rationale for using this as an analogue of the English language ‘if…then…’ runs roughly as follows. The only way to be sure that an ‘if…then…’ clause is false is if the antecedent is true and the consequent false. For how could it be the case that ‘if P then Q’ if you have P but not Q!

7. The conditional
So the conditional is false when the antecedent is true and the consequent is false. Otherwise it’s not false – and so it must be true!

8. The conditional
This is a pretty shoddy argument. It starts with the idea of what it would take to know that a conditional is false, and then elides that with the question of whether or not a conditional is false.
And it assumes that in natural languages everything that is not false must be true – which is a questionable assumption. (Some things may be neither, some people even think that some things may be both!)

9. The conditional
Moreover, using the English language connective ‘if…then…’ as truth functional in the same fashion as the logical connective ‘⊃’ can lead you to some strange places.
Consider: ‘if Callum MacRae is going to be the next President of the USA, then it will rain tomorrow’.

10. The conditional
Now, if we assume that it is false that I will be the next President of the USA, then this sentence comes out as true! Because look at the truth table for the conditional – in every case in which the antecedent is false, the whole sentence comes out true.
But that can’t be right. Even though it’s false that I will be the next President of the USA, that doesn’t mean that if I was going to be the next President then it would rain tomorrow.

11. The conditional
So as it turns out, ‘⊃’ is not a great analogue for the English language ‘if…then…’
But remember – in sentence logic we’re looking for that balance of simplification and preservation of original meaning. And for our current purposes, it will often suit our purposes enough to serve as an adequate stand-in. Moreover, as we will soon see, it has some interesting properties that render it of independent interest to us as logicians.

12. The conditional
So we will stick with the conditional. And we will even often use it to transcribe the English language ‘if…then…’ But don’t forget that there are some serious problems with it – don’t fall into the trap of seeing it as a perfect analogue of our English language conditional.
Interestingly, the quest for a more accurate logical analogue to the English conditional has led to some very exciting developments in recent decades… (Complex modal logics, even paraconsistent logic.)

13. Logic and natural language
There’s much more that could be said here about the relationships between natural language and formal logic. I won’t go over these issues now, because Teller does a great job of overviewing them in this section of the textbook.
But it’s good to know that there are some interesting philosophical questions here for you to think about.

14. The conditional
So, with this detour through philosophical logic aside, we can now get back to defining the conditional. We’ve seen the truth table already.
The best way to remember the truth table is to remember that a conditional is false only when the antecedent is true and the consequent is false, otherwise it is true. Or alternately, a conditional is true whenever the antecedent is false or the consequent is true – otherwise it is false.

15. The conditional
Note that conditionals are not symmetric, like conjunctions or disjunctions. P ⊃ Q is not logically equivalent to Q ⊃ P. (That’s why it is useful to have two different names for the antecedent and the consequent.)

16. The conditional
We can now add a couple of extra laws of logical equivalence concerning the conditional.
There’s the law of the conditional (C): X ⊃ Y is logically equivalent to ~(X ∧ ~Y), and thus, by De Morgan’s law, to ~X ∨ Y.
And the law of contraposition (CP): X ⊃ Y is logically equivalent to ~Y ⊃ ~X.

17. The conditional and validity
The conditional is connected in interesting ways with the concept of validity.
You may have noticed there is a bit of similarity in their definitions. Remember that a valid argument is true just so long as it is impossible that the premises are true and the conclusion is false.
And note that the conditional is true whenever it is not the case that the antecedent is true and the consequent false.

18. The conditional and validity
With the concept of logical truth in hand we can connect the two: ‘X ⊃ Y’ is a logical truth if and only if the argument ‘X, therefore Y’ is valid.
That’s because if ‘X ⊃ Y’ is a logical truth, then there is no case in which it is false. Which is to say, there is no case in which X is true and Y is false. And that’s just what it means to say that the argument is valid!

19. The biconditional
As well as the conditional we’re going to add another connective, called the biconditional.
The idea behind the biconditional is pretty simple – a biconditional is true whenever both the components have the same truth value (i.e. whenever they are either both true or both false).
The symbol used is ‘≡’, or sometimes ‘⇔’.

20. The biconditional
The truth table for the biconditional looks like this: P Q P ≡ Q t t t t f f f t f f f t

21. The biconditional
We can usefully think of the biconditional as like the conditional operating in both directions. (That’s where the name comes from: bi- conditional.) ‘X ≡ Y’ is logically equivalent to ‘(X ⊃ Y) ∧ (Y ⊃ X)’.
Unlike the conditional, and like conjunction and disjunction, the biconditional is symmetric

22. The biconditional
The biconditional is an analogue of the English language phrase ‘…if and only if…’.
As a transcription of natural language it works much better than the conditional does!

23. The biconditional
There’s another law of logical equivalence for us – the law of the biconditional (B): ‘X ≡ Y’ is logically equivalent to ‘(X ⊃ Y) ∧ (Y ⊃ X)’.
Moreover, since the biconditional is closely related to the conditional, and since the conditional has interesting properties vis-à-vis validity, we should expect the biconditional to have a similar such property.

24. The biconditional
And sure enough we can say that ‘X ≡ Y’ is a logical truth if and only if X and Y are logically equivalent.
That’s because ‘X ≡ Y’ is a logical truth just in case it is true in all cases. I.e. just in case there is no case in which X is true and Y is not or vice versa. Which is just to say that X and Y are true in all and only the same cases. Which is just the same as saying that they are logically equivalent!

25. Transcription Ok, so that’s most of the technicalities out of the way.
I just want to finish with a bit of advice about transcription. We’ve covered the reasons for transcribing the conditional as ‘if…then…’ (which is far from perfect, but it’ll do) and the reasons for transcribing the biconditional as ‘if and only if’.

26. Transcription
But what about the case of ‘unless’? Let’s take an example: unless there’s a miracle, college will be open tomorrow.
How should we transcribe that?

27. Transcription
Well, let’s think about what the English sentence means. If we say ‘unless there’s a miracle, college will be open tomorrow’ I’m saying that the only way that college won’t be open is if there’s a miracle. So if it’s not the case that college is open, we can definitely say that there has been a miracle.
So I’m saying if it’s not the case that (college is open), then (there’s a miracle). Which we could formalize as ~Q ⊃ P.

28. Transcription
‘Unless’ can often crop up in various places within a sentence. It is almost always best to use a conditional to transcribe it, but you have to think carefully about how to do so.
The best method is to just think carefully about what the sentence means, and try to paraphrase it in terms of ‘ifs’ and ‘thens’.

29. Examples
Transcribe: ‘AFC Wimbledon won yesterday, unless my uncle lied.’ ‘Unless something strange has happened, my name is Callum MacRae.’