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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 ...
The truth value of an arbitrary sentence is then defined inductively using the T-schema, which is a definition of first-order semantics developed by Alfred Tarski. The T-schema interprets the logical connectives using truth tables, as discussed above. Thus, for example, φ ∧ ψ is satisfied if and only if both φ and ψ are satisfied.
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 ...
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.
A less trivial example of a redundancy is the classical equivalence between and . Therefore, a classical-based logical system does not need the conditional operator " → {\displaystyle \to } " if " ¬ {\displaystyle \neg } " (not) and " ∨ {\displaystyle \vee } " (or) are already in use, or may use the " → {\displaystyle \to } " only as a ...
For example, "Snow is white" (in English) and "Schnee ist weiß" (in German) are different sentences, but they say the same thing, so they express the same proposition. Another definition of proposition is: Two meaningful declarative sentence-tokens express the same proposition, if and only if they mean the same thing. [citation needed]