Example of a difficult derivation
Roll your own derivations
You may have derivations of your own that you wish to try. Just type, paste, or drag and drop, them into the panel, select your derivation, and click 'Start from selection'.
[Often copy-and-paste won't work directly from a Web Page; however, usually drag-and-drop will work!]
You will need to use the correct logical symbols. Here they are
F ∴ F & G ∼ & ∨ ⊃ ≡ ∀ ∃ ∴
Review of Sentential Logic
You now have to tools to appraise sentential arguments.
Let us run through how these might be used with two examples.
Consider the argument
If no human action is free, then no one is responsible for what they do.
If no one is responsible for what they do, no one should be punished.
If no human action is free, no one should be punished.
First it should be symbolized
So that we can show certain arguments to be valid.
The focus of the course lies with the validity and invalidity of arguments. Now, invalidity can be established by counter-example (by producing an interpretation under which all the premises are true and the conclusion false, at the same time). But validity is a different matter. And the usual approach is to have rules of inference and to do derivations.
Skills to be acquired
Becoming familiar with common inference patterns and being able to use them via three new rules of inference and via rewrite rules. This helps with assessing ordinary everyday reasoning such as that found in the law, in newspapers, in advertisements, etc.
Bergmann The Logic Book Section 5.5
Example of a Harder Propositional Proof: One of De Morgan's Laws
Example of a Harder Propositional Proof
Or Elimination and Bi-Conditional Introduction
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Skills to be acquired:
Learning the Rules Or Elimination and the Introduction of the Biconditional.
Bergmann The Logic Book Section 5.1 and 5.4
Or Elimination, in the guise of Dilemma, also is a form of inference dating from antiquity.