Working backwards

Definition

Working backwards is a problem-solving strategy that involves starting from the desired conclusion and tracing back through the steps needed to reach that conclusion. This method can help clarify the logical structure of an argument, making it easier to identify valid premises or necessary assumptions. It is particularly useful in formal systems of proof where you want to demonstrate that a conclusion follows from given premises.

5 Must Know Facts For Your Next Test

1. In working backwards, you begin with the conclusion you wish to prove and consider what premises or assumptions would logically lead to that conclusion.
2. This technique is especially effective in natural deduction systems, where establishing a conclusion can often depend on linking it to existing premises.
3. By breaking down the problem into smaller parts, working backwards helps in identifying which inference rules apply and how they relate to each step.
4. When working backwards, it’s crucial to keep track of assumptions made along the way, as these will impact the validity of the final argument.
5. This approach often leads to discovering alternative ways to reach a conclusion, enriching your understanding of the logical connections involved.

Review Questions

• How does working backwards help in constructing a proof within natural deduction?
  • Working backwards aids in constructing proofs by allowing you to start with the conclusion and trace back to see which premises must be true for that conclusion to hold. This process clarifies which inference rules apply at each step and helps ensure that all necessary assumptions are accounted for. By identifying these connections early on, it becomes easier to organize the proof logically and systematically.
• Compare working backwards with other problem-solving techniques used in logical proofs. What are its advantages?
  • Compared to other techniques like direct proof or proof by contradiction, working backwards offers unique advantages by emphasizing clarity in reaching conclusions. While direct proof focuses on building an argument forward from premises, working backwards ensures that all possible routes to a conclusion are considered, making it easier to spot gaps in logic. This method also encourages a deeper understanding of how conclusions relate to their foundational premises, which can enhance overall problem-solving skills in logic.
• Evaluate the effectiveness of working backwards as a strategy in natural deduction when faced with complex arguments. What challenges might arise?
  • Working backwards can be highly effective for unraveling complex arguments by breaking them down into manageable parts and clarifying the necessary connections. However, challenges may arise when assumptions become convoluted or when there are multiple potential conclusions stemming from the same premises. Additionally, if a step is overlooked or misinterpreted while tracing back, it can lead to invalid conclusions. Thus, while this strategy is powerful, it requires careful attention to detail and systematic reasoning throughout the process.

Related terms

Natural Deduction:

A formal proof system that allows for the derivation of conclusions from premises through the application of inference rules.

Inference Rules: Rules that dictate the valid steps one can take to derive conclusions from premises within a logical framework.

Proof by Contradiction:

A proof technique where one assumes the opposite of what is to be proven, leading to a contradiction, thereby confirming the original statement.