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This proof is taken from Chapter 10, section 4, 5 of Mathematical Logic by H.-D. Ebbinghaus. As in the most common proof of Gödel's First Incompleteness Theorem through using the undecidability of the halting problem, for each Turing machine there is a corresponding arithmetical sentence , effectively derivable from , such that it is true if and only if halts on the empty tape.
Here A, "this sentence is true", refers to the overall sentence, while B is "Germany borders China". So, assuming A is the same as assuming "If A, then B". Therefore, in assuming A, we have assumed both A and "If A, then B". Therefore, B is true, by modus ponens, and we have proven "If this sentence is true, then 'Germany borders China' is true ...
The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a necessary consequence of its premises. An argument that is not valid is said to be "invalid". An example of a valid (and sound) argument is given by the following well-known syllogism:
This resolution technique uses proof by contradiction and is based on the fact that any sentence in propositional logic can be transformed into an equivalent sentence in conjunctive normal form. [4] The steps are as follows. All sentences in the knowledge base and the negation of the sentence to be proved (the conjecture) are conjunctively ...
The Polish logician Alfred Tarski identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on the logical form of the sentences: (2) The relation is a priori, i.e., it can be determined with or without regard to empirical evidence (sense experience); and (3) The logical consequence ...
Propositional logic, as currently studied in universities, is a specification of a standard of logical consequence in which only the meanings of propositional connectives are considered in evaluating the conditions for the truth of a sentence, or whether a sentence logically follows from some other sentence or group of sentences.
One can also prove a theorem by proving the contrapositive of the theorem's statement. To prove that if a positive integer N is a non-square number, its square root is irrational, we can equivalently prove its contrapositive, that if a positive integer N has a square root that is rational, then N is a square number.
The predicate calculus goes a step further than the propositional calculus to an "analysis of the inner structure of propositions" [4] It breaks a simple sentence down into two parts (i) its subject (the object (singular or plural) of discourse) and (ii) a predicate (a verb or possibly verb-clause that asserts a quality or attribute of the object(s)).
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