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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Logic | 3/11 | https://en.wikipedia.org/wiki/Logic | reference | science, encyclopedia | 2026-05-05T03:56:39.118518+00:00 | kb-cron |
Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion. For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars is a red planet". For most types of logic, it is accepted that premises and conclusions have to be truth-bearers. This means that they have a truth value: they are either true or false. Contemporary philosophy generally sees them either as propositions or as sentences. Propositions are the denotations of sentences and are usually seen as abstract objects. For example, the English sentence "the tree is green" is different from the German sentence "der Baum ist grün" but both express the same proposition. Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, philosophical naturalists usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions. These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like the symbols displayed on a page of a book. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted. Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as psychologism. It was discussed at length around the turn of the 20th century but it is not widely accepted today.
==== Internal structure ==== Premises and conclusions have an internal structure. As propositions or sentences, they can be either simple or complex. A complex proposition has other propositions as its constituents, which are linked to each other through propositional connectives like "and" or "if...then". Simple propositions, on the other hand, do not have propositional parts. But they can also be conceived as having an internal structure: they are made up of subpropositional parts, like singular terms and predicates. For example, the simple proposition "Mars is red" can be formed by applying the predicate "red" to the singular term "Mars". In contrast, the complex proposition "Mars is red and Venus is white" is made up of two simple propositions connected by the propositional connective "and". Whether a proposition is true depends, at least in part, on its constituents. For complex propositions formed using truth-functional propositional connectives, their truth only depends on the truth values of their parts. But this relation is more complicated in the case of simple propositions and their subpropositional parts. These subpropositional parts have meanings of their own, like referring to objects or classes of objects. Whether the simple proposition they form is true depends on their relation to reality, i.e. what the objects they refer to are like. This topic is studied by theories of reference.
==== Logical truth ====
Some complex propositions are true independently of the substantive meanings of their parts. In classical logic, for example, the complex proposition "either Mars is red or Mars is not red" is true independent of whether its parts, like the simple proposition "Mars is red", are true or false. In such cases, the truth is called a logical truth: a proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true under all interpretations of its non-logical terms. In some modal logics, this means that the proposition is true in all possible worlds. Some theorists define logic as the study of logical truths.
==== Truth tables ==== Truth tables can be used to show how logical connectives work or how the truth values of complex propositions depends on their parts. They have a column for each input variable. Each row corresponds to one possible combination of the truth values these variables can take; for truth tables presented in the English literature, the symbols "T" and "F" or "1" and "0" are commonly used as abbreviations for the truth values "true" and "false". The first columns present all the possible truth-value combinations for the input variables. Entries in the other columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression "
p
∧
q
{\displaystyle p\land q}
" uses the logical connective
∧
{\displaystyle \land }
(and). It could be used to express a sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables,
p
{\displaystyle p}
("yesterday was Sunday") and
q
{\displaystyle q}
("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are
¬
{\displaystyle \lnot }
(not),
∨
{\displaystyle \lor }
(or),
→
{\displaystyle \to }
(if...then), and
↑
{\displaystyle \uparrow }
(Sheffer stroke). Given the conditional proposition
p
→
q
{\displaystyle p\to q}
, one can form truth tables of its converse
q
→
p
{\displaystyle q\to p}
, its inverse (
¬
p
→
¬
q
{\displaystyle \lnot p\to \lnot q}
), and its contrapositive (
¬
q
→
¬
p
{\displaystyle \lnot q\to \lnot p}
). Truth tables can also be defined for more complex expressions that use several propositional connectives.
=== Arguments and inferences ===