# If-then syllogism and multiple conditionals

Can one condtional statement cancel another one ? For example i say if you get 85/100 at maths test i will gift you a car a but if you get 85/100 at maths and 50/100 at history i wont gift you anything . Does this claim make sense ?

• Usually, a more specific statement outweighs a more general statement. If this were part of a legal contract, the more specific statement would be regarded as modifying the general statement. – barrycarter Aug 28 '18 at 19:35

Not clear ...

This is not a syllogism.

Having said that, the statement:

"if A, then B and if (A and C), then not-B"

If the student does not get 85/100 at maths test, both conditionals are TRUE, and thus the conjunction is TRUE.

But also if A and B are both TRUE, the conjunction is still TRUE, provided that C is FALSE.

• yes but A isnt enought to guarantee B ? if a is true then no matter what b must follow. where i do it wrong ? – ado Aug 25 '18 at 21:10
• @ado Mauro Allegranza is saying, rightly, that the material implications, A -> B and (A & C) -> ~B, may both be satisfied by A being false. The two statements say that (1) A being true will guarantee that B is true, and (2) A and C both being true will guarantee that B is false. So since B cannot be both true and false, those two statements must entail that A cannot be true. – Graham Kemp Aug 26 '18 at 12:08
• Of course, that is assuming both statements are true. By contraposition: If it turns out that \$A\$ is true, then they both cannot be. – Graham Kemp Aug 26 '18 at 23:44
• What do you mean that the above is not a syllogism? I take it you mean the above is not a "categorical syllogism" as in Aristotelian logic. Many logic texts refer to conditional arguments as "hypothetical syllogisms". So it is a type of syllogism technically --just not a categorical syllogism. – Logikal Aug 28 '18 at 15:29
• @Logikal - see Syllogism : "a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true." – Mauro ALLEGRANZA Aug 28 '18 at 17:12

Conditionals like this are not contrary, because many, perhaps most, ordinary English conditionals allow for exception conditions. When we say, If A then B, there is usually an unstated "other things being equal". Adding an extra term to the antecedent is sometimes referred to as 'strengthening' of the conditional, and sometimes, confusingly as 'weakening'. Another example might be: "If I turn the key in the ignition the engine starts", which is good most of the time, but "If I turn the key in the ignition and the battery is flat then the engine does not start" is also good. If these conditionals were material implications, one would have to conclude that I don't turn the key, which of course is absurd. Fortunately, most conditionals in ordinary English are not material implications, so this doesn't matter. One approach to representing these conditionals is to use default logics, which allow for implications that are defeasible. Another option is to represent the logic of conditionals using marginal probabilities.

It doesn't make sense in some cases when, for example, you do not get 85% in math and you do get 50% in history. In this case, it is ambiguous whether you will get the car.

We are considering the logical statement:

(M => C) & (M & H => ~C) (assumed to be true)

where

M = 85% in math

H = 50% in history

C = getting a car

The truth table for this statement is here.

The 5th line is M = F, C = T, H = T, and (M => C) & (M & H => ~C) = T

The 7th line is M = F, C = F, H = T, and (M => C) & (M & H => ~C) = T

If the above statement is true (4th column = T) and you do not get 85% in math (1st column = F) and you get 50% in history (3rd column = T), then it is ambiguous whether you will get a car (2nd column = ?).