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Introduction to Logic and Probability
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
Two statements p and q are logically equivalent if they have identical truth tables, denoted by p↔q $ p\leftrightarrow q $ or p≡q $ p\equiv q $ , or simply p=q $ p=q $ . In other words, p and q are logically equivalent if and only if p→q $ p\rightarrow q $ and q→p $ q\rightarrow p $ . Other forms of equivalent propositions are p if and only if q, and a necessary and sufficient condition for p is q. The following example illustrates the logically equal statements (see Table 6.5).
Proving Mathematical Statements
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
There are a number of “logical equivalent tricks” that we can employ as we try to prove theorems going forward. The phrase “standard simplification” might better describe what we are trying to capture here since there is nothing particularly “tricky” that we are doing – we are simply proving a mathematical statement by, instead, proving a logically equivalent statement. We have already seen two examples of logically equivalent statements: the contrapositive and the contradiction statement.
Logic
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
Two propositions are said to be logically equivalent if they have identical truth values for every set of truth values of their components. Using P and Q to denote (possibly) compound propositions, we write P ≡ Q if P and Q are logically equivalent. As with tautologies and contradictions, logical equivalence is a consequence of the structures of P and Q.
A logic for best explanations
Published in Journal of Applied Non-Classical Logics, 2019
Jared Millson, Christian Straßer
Not just any sequent composed of literals should be allowed to qualify as the starting point of explanatory reasoning. For reasons already discussed, proper axioms must also be premise-consistent, material, and defeasible, yet not defeated. Naturally, the empty sequent should not be provable from the members of , for this would render any cut-free non-logical axiomatic extension of classical propositional logic inconsistent.16 The background set of any proper axiom ought to be substantive in the sense that it provides unique information regarding the context of inference. In order to permit logically equivalent proper axioms in a system that lacks tradition negation rules, we must ensure that literals can be removed from one side of the turnstile and replaced by their complement on the other – affecting a kind of closure under contraposition. Finally, in order to permute cut to the top of derivations, and hence to ensure cut-elimination, should be closed under the cut rule. We impose the following constraints on proper axioms to secure these conditions.