This philosophical problem of logic considers whether the ‘if A then B’ construct is a good match for the use of the conditional in everyday language. In classical logic, as stipulated by Frege in 1879, ‘if A then B’ is considered false if A is true but B is false. In all other cases, it is considered true. The question then arises if this is an accurate way of capturing the normal use of the conditional.
In particular, some have argued that because the logical construct is automatically true when A is false, we could come up with all sorts of unusual logical truths, for example ‘if I win the lottery today, world poverty will be eradicated’. We have good reason to suspect the antecedent is false (I am not going to win the lottery), which means under the rules of classical logic the ‘if then’ statement would be true. However, this does not sound likely. In this case, we are unable to find the necessary equivalence between ‘if’ in natural language and -> in logic.
One possible solution to bring the logic conditional more in-line with natural language is to introduce the idea of probability. In natural language, we tend to use the ‘if’ construct when we are not completely certain of a proposition and wish to hedge our bets that it may or may not be true. So rather than analysing a logic conditional on a strict true / false basis, we could assign the probability of the ‘if then’ being true if both A and B are true. For example if A has a 50% change of occurring, and B has a 50% chance of occurring given A has already happened, then the overall probability of B would be 0.50 x 0.50 = 25%.
More formally, the idea of probability can be expressed as follows:
Pr(if A then B) = Pr(B|A).
The probability of ‘if A then B’ depends on the probability of B given A is also true. Since the result is rarely going to be 100% (completely true) or 0% (completely false), Lewis and Edginton have both argued that this does not fit well with the structure of classical logic, which requires everything to be true or false. Most ‘if A then B’ statements would be neither true nor false, which means that ‘if A then B’ is not a valid proposition at all. Philosophers who argue along these lines could be called ‘non-propositionalists’. They argue that ‘if A then B’ is a conditional assertion at most. It can only be assessed if A is already true. If A is false or indeterminate, then the condition has not been satisfied, so the proposition is neither true nor false.
The ‘conditional assertion’ view does not totally agree with natural language however. There are two possible ways we could argue against the views of Lewis and Edginton. Firstly, there is the question of negation. If someone makes a sentence ‘if it rains tomorrow, the trains will run late’, you could negate this in one of two ways, both of which would be valid: (1) you could argue that it won’t rain tomorrow (the antecedent is wrong), or (2) you could agree that it will rain tomorrow, but the trains will run on time (the antecedent is right, but the consequent is wrong). The conditional assertion view cannot cater for this distinction between the two different types of negatives.
The second argument against the conditional assertion view is when the consequent comes naturally as a result of the antecedent, such as ‘if John doesn’t accept the price if you offer it to him, you shouldn’t sell him the car’. The non-propositionalist would argue that this is a conditional assertion, yet in natural language, it sounds stronger than this.
If you support either of these arguments, you would be claiming that the Lewis / Edginton view is incorrect, which means that ‘if A then B’ should be considered a valid proposition. This returns us back to the question of probability and whether we should accept this as a valid workaround to the correct reading of ‘if A then B’.
The logical ‘if A then B’ construct can also be criticised along the following lines. If B is true, then ‘if A then B’ is automatically true, regardless of whether A is true or false. This means you could create a logical statement with an absurd antecedent, but with a likely true consequent: ‘if I perform a rain dance outside, rain will fall within the next week’.
The rules of classical logic also indicate that ‘if p -> q’, then if (p & r) -> q’. Provided p and q are sensible enough, ‘r’ could be anything you wanted, for example ‘if I put icing on the cake and I add arsenic to the ingredients, people will be happy to eat the cake’.
Another problem with the logical construct is that it requires two free-standing propositions A and B. However in English, especially when using counterfactuals, proposition B must be read in light of proposition A. For example: ‘if Germany had won the war, school children would have been required to learn German’. The consequent in this example cannot be mapped to a standalone proposition. ‘School children would have been required to learn German’ does not make any sense as a standalone sentence. More generally, as we’ll see further below when discussing subjunctive conditions, whenever the antecedent is a hypothetical situation, it does not make any sense to assign a truth value to it.
The use of the conditional in natural language also implies some sort of causal connection between A and B, whereas in classical logic, there is no requirement for A and B to be connected in any way. This means the following is a valid argument: ‘if England win the football today, it will be a sunny day tomorrow’. In normal language, this would not be a very persuasive argument as there is no obvious connection between the two propositions.