Huh. If "reduce the strength of the piece" doesn't mean "the object fails at a lower input force," what does it mean, exactly?
The trouble is the word "strength" in this context means material strength. It's convenient to talk about an "object's strength" as the amount of stress an object can take before failure. But objects don't have strength, objects have materials that have strength. Talking about an object's strength is a convenience, but it doesn't have any physical meaning unless you are actually talking about the material.
For example, say we have two swords made of the same material. Both swords are the same shape, only one has a crack. It is convenient for us to say that the sword with a crack has "less strength" than the sword without a crack because it will fail under less load.
But strength isn't a property of an object's geometry. Strength is a property of a material. And since they are made of the same material, they therefore must have the same strength.
Which brings me back to why a cracked object will fail sooner... not because it has lower strength (a material property), but because it has higher stress (in this case, a property dependent on the object's geometry).
Take a look at this pic to see why the geometry causes high stress:
This picture represents an object in tension. The red lines are 'force lines' showing where the object is supporting the tensile force. When the lines are bunched up, the stress in that area is very high.
As you can see, the crack acts as a discontinuity. It forces a large portion of the load to be carried near the crack's edges, which have a very small surface area. High force and low area = big stress.
The key here is that the material strength anywhere on the object is the same as it always has been (Not exactly true, of course.. a crack may have cold worked or corroded, etc.). But it will still fail because it has more stress in a smaller area.. failure will begin at the crack because it is under high stress, making the crack bigger, making the stress higher, making the crack bigger, etc.
All that has nothing to do with the material properties (strength) and everything to do with the geometry of the object (stress).
So to answer your question, a crack does not "reduce the strength of the piece", it greatly magnifies the stress, which causes "the object [to fail] at a lower input force,". Maybe that's a trivial difference in a mathematical model, but they definitely aren't the same thing.