# Implication vs sufficient condition vs hypothetical reasoning

I was bit confused to clarify the difference between them because "if-then" are used a lot in everyday life. So for example we have a car which is full function and someone says if i turn the key the car will start . My question is if the "if i turn the key" is a sufficient condition for car to start because we know already that is full function or just an hypothetical syllogism or an implication. General speaking if someones turn the key of a car (A) doesnt guarantee that the car will start (B). So A isnt a sufficient condition for B but can occassionaly be like the example above? I mean can we have "ocassionaly" sufficient conditions ?

Sufficent and necessary conditions usually are translated with the conditional.

See : Jan von Plato, Elements of Logical Reasoning (Cambridge UP, 2013), pag 10 :

The two sentences "if A, then B" and "B if A" seem to express the same thing. Natural language seems to have a host of ways of expressing a conditional sentence that is written A → B in the logical notation. Consider the following list :

From A, B follows; A is a sufficient condition for B; A entails B; A implies B; B provided that A; B is a necessary condition for A; A only if B.

It sound a bit strange to say that B is a necessary condition for A means A → B. When one thinks of conditions as in A → B, usually A would be a cause of B in some sense or other, and causes must precede their effects. A necessary condition is instead something that necessary follows, therefore not a condition in the causal sense.

The link with "hypothetical reasoning" is obvious; the rule of Conditional Proof licenses us to move from : we have a proof of B from (assumption) A to : we have a proof of A → B.

"If A then B" does not mean that A causes B. Whether in natural language or symbolic logic, if A and B are logical true-or-false propositions, then "if A then B" means that it is not the case that A is true and B is false. (Although this is usually given as a definition, it can be derived from other well-known principles of logic.)

For logical propositions P and Q, the following are all considered to be logically equivalent:

• If P then Q
• P implies Q
• Q if P
• P only if Q
• P is sufficient for Q
• Q is necessary for P

To prove that P implies Q we can do either of the following:

• Assume P and subsequently prove Q
• Assume ~Q and subsequently prove ~P
• Prove ~[P and ~Q]
• Prove ~P
• Prove Q

To disprove that P implies Q, we need only prove P and ~Q.

All conditional reasoning (the if . . . Then . . ) does not imply a necessary or sufficient relationship. Sometimes a conditional argument or premises to an argument meet the sufficient or necessary relationship.

If some life forms on Earth have no brain, then the Philadelphia Eagles are the reigning superbowl champions. [Clearly there is no relationship between the antecedent and the consequent to guarantee if any part is true.]

If Jesus dies for the sins of mankind, then everyone is capable of being saved by the blood of Christ Jesus. [Don't comment on the subject matter of religion as I am demonstrating logical form here. Please don't get caught up in reading sentences and what is being said. Logic us about a pattern of reasoning, but the point is here the relationship in this example is necessary if the proposition is true.]

If the NY Jets win the next superbowl, then I will eat my hat. [This example expresses that even if the first part is true I still may renege on my part of eating my hat. The point here is the proposition is not truth functional but a rhetorical intent. I am expressing the Jets won't make it as far as a Superbowl this coming season. So if one responds that this is invalid the message has gone over the head of the receiver.]

If you score at least a grade of 65 on the final exam you will pass the class. [There are many ways to fail a class as well. I scored a 100 on the final exam but I entered into a fight where I shoot another student 10 seconds after I submitted the exam. Another school shooting in the making. This will make the conditional false by truth table. I met the grade requirement but still did not pass the class. Sufficient means there could be other alternatives other than what is given. Sufficient does not guarantee the consequent but expresses how the consequent could be true if the specifications are met.]

If you don't clean up you room, I will murder you in the first degree. [Do I need to say this is not to be taken literally? My mother said those words to me & I did not always clean my room despite the threatening tone. I am still alive. Again this is a rhetorical effect and not to be translated into Mathematical logic.]

So be clear when you are referring to logic there are many types. Logic is not always the same logic. Mathematical logic has a lot of things that Philosophy does not have included in it's teaching of logic. The purposes for both subject having a logic component don't match. Math is about validity whereas Philosophy (at least used to be) more about Soundness. Mathematical logic has it's own agenda and different from what Philosophy used to teach. You will find some Mathematical logic is not realistic & applicable in many work places, but a trivial ability. In academia Mathematical has a place.