Introduction

The following is one of the shortest possible refutations of Special Relativity. 
 

Reductio ad Absurdum

1. By the definitions of the terms "acceleration" and "deceleration", if a body is accelerating rectilinearly its velocity increases, while if it is decelerating rectilinearly, its velocity decreases. 

 
2. And according to Special Relativity, the greater the velocity of a clock, the slower it ticks — and vice versa.

 
3. So an accelerating clock should gradually tick slower and slower as time passes, while a decelerating clock should gradually tick faster and faster as time passes.

 
4. However, according to Special Relativity, a single object can be both accelerating and decelerating simultaneously. 

 
5. Thus, according to Special Relativity, a single clock can both be gradually ticking slower and slower as time passes, and also  gradually ticking faster and faster as time passes!

 
6. This is logically impossible.

 
7. Therefore either it is false that the greater the velocity of a clock, the slower it ticks — and vice versa; or it is false that the same object can be accelerating and decelerating simultaneously … either of which disproves the Special Theory of Relativity.

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* Some people might not get just how this can be, and so here's a simple explanation: Imagine two objects, A and B, receding from each other rectilinearly at uniform velocity; now send a third object, C, from A toward B at a rate of acceleration high enough to ensure that C eventually will arrive at B. Then during its trip from A to B, the single object C will be accelerating with respect to A and simultaneously decelerating with respect to B.