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It interprets only if as expressing in the metalanguage that the sentences in the database represent the only knowledge that should be considered when drawing conclusions from the database. In first-order logic (FOL) with the standard semantics, the same English sentence would need to be represented, using if and only if , with only if ...
Venn diagram of (true part in red) In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement "if and only if" (often abbreviated as "iff " [1]), where is known as the antecedent, and the consequent.
This statement expresses the idea "' if and only if '". In particular, the truth value of p ↔ q {\displaystyle p\leftrightarrow q} can change from one model to another. On the other hand, the claim that two formulas are logically equivalent is a statement in metalanguage , which expresses a relationship between two statements p {\displaystyle ...
Example 2 For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number that is both even and prime. Example 3 Consider thunder, the sound caused by lightning. One says that thunder is necessary for lightning, since lightning never occurs without thunder. Whenever there is lightning, there is ...
A simple example, applied to two of the above illustrations, is the following: Let the letters 'P', 'Q', and 'S' stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as:
The first of these sentences is a basic zero conditional with both clauses in the present tense. The fourth is an example of the use of will in a condition clause [4] (for more such cases, see below). The use of verb tenses, moods and aspects in the parts of such sentences follows general principles, as described in Uses of English verb forms.
Equivalence (if and only if): , , , , (prefix) in which is the most modern and widely used, and is commonly used where is also used. For example, the meaning of the statements it is raining (denoted by p {\displaystyle p} ) and I am indoors (denoted by q {\displaystyle q} ) is transformed, when the two are combined with logical connectives:
Here is an example of an argument that fits the form conjunction introduction: Bob likes apples. Bob likes oranges. Therefore, Bob likes apples and Bob likes oranges. Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction.