Proposition: Defined in logic and language
Propositions are key in logic and semantics, representing core ideas like “The sky appears blue.” They are essential in differentiating truth values across various languages and contexts.
By Brad Nakase, Attorney
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Introduction
A proposition is a fundamental idea in logic, semantics, philosophy of language, and related disciplines. It is frequently defined as the main source of fact or untruth. Another way to describe proposition definition is as the type of object that statements indicate. For example, the statement “The sky appears blue” expresses the idea that the sky looks blue. Essential to remember is that propositions aren’t language statements in and of themselves.
For example, although the 2 sentences aren’t the same, the English phrase “Snow is white” and the German phrase “Schnee ist weiß” indicate the same notion. As with different propositional attitudes, propositions may additionally be defined as the subjects of belief.
The idea that the sky appears blue, for example, is what one perceives if they have that belief. An idea is another way to conceptualize a proposition: “An idea or statement which individuals can evaluate or argue whether it’s real” is the meaning of a proposition according to the Collins Dictionary.
Traditionally, propositions are frequently represented as functions that connect a truth value to a conceivable world. A function that, provided the input of the real world, would yield the value of truth T, but, given a different universe where the sky is green, it would give F. This function, for example, can be used to simulate the proposition which states that the sky is blue. Other formalizations, such as the organized propositions perspective, have been put out as alternatives.
Logic, philosophy of language, linguistics, and associated fields have all relied heavily on propositions over their histories. In fact, David Lewis once said that “the idea we connect with the term ‘proposition’ could represent something of a muddle of opposing desiderata.” Some scholars have questioned whether a coherent description of propositionhood is conceivable. Many similar concepts have been referenced by the phrase, which is frequently used in a broad sense.
Historical Application
- Aristotle
A phrase that confirms or rejects an argument of a subject, maybe with the assistance of a copula, is known called a categorical proposition in Aristotelian reasoning. “All humans are mortal” and “Socrates is a human” are two examples of Aristotelian propositions. The first case has the subject “humans,” the predicate “mortal,” and the copula “are.” In contrast, the second case has the subject “Socrates,” the predicate “a human,” and the copula “is.”
- By the positivists who employ logic
A logical sentence or closed formula is sometimes used to associate propositions in order to differentiate them from the expressions of open formulas. Propositions constitute “statements” in this regard that convey the truth. The intellectual movement known as logical positivism endorsed this idea of a proposition.
Certain philosophers contend that in addition to declarative speech and actions, every (or some) other types of speech and actions have propositional substance. As an illustration, yes or no inquiries pose statements and probe their veracity. However, certain signs may contain declarative statements of propositions without even being linguistic or forming a phrase (traffic signs, for example, transmit a clear meaning that can only be true/false).
Additionally, beliefs and other intentional dispositions like hopes, preferences, and wishes are referred to as propositions. Sayings like “I wish I had a new vehicle” or “I wonder if it’s going to snow” (or if “it shall snow” actually happens) are examples. We refer to desires, beliefs, doubts, and so forth that take on this kind of content as propositional attitudes.
- Russell
According to Bertrand Russell, propositions are organized things made up of attributes and objects. The Russellian explanation maintains the ability to distinguish between two statements that are true in any number of worlds or states of things, which is a significant departure from Ludwig Wittgenstein’s perspective, which holds that a proposition is a collection of potential worlds or states of things in which it is true. For example, there is a difference in a Russellian perspective between the propositions “two combined with two becomes four” and “three in addition to three becomes six.” Nonetheless, all mathematical facts (as well as all other essential truths) belong to the exact same set, which is the set of every potential world, if propositions constitute sets of prospective worlds.
In connection with the mind
Proposition definition is mostly examined in terms of their compatibility with propositional attitudes in regard to the mental. Propositional dispositions are essentially behaviors that one can adopt around a proposition (such as “it is pouring,” “the snow is white,” and so on) that are typical of cultural psychology (believing, desire, etc.).
A “that sentence” (e.g., “Jane perceives that it’s raining”) commonly follows traditional psychology beliefs in English propositions. Cognitive states are frequently understood to consist mostly of propositional attitudes in psychology and philosophy of consciousness. As the “mental substance” of the mindset, the propositions are typically described. The premise “it’s raining” is what Jane would mentally hold, for instance, if she were in a state of perceiving that it is raining. They are also referred to as intentional states of mind since they are focused on something, specifically propositions.
Particularly challenging to explain are non-mentalist conceptions of propositions, like the logical positivist and Russellian perspectives discussed above, as well as Gottlob Frege’s theory that propositions are actually Platonist units—that is, that they exist in an abstracted, non-physical domain. This is why some modern perspectives on propositions have concluded that they are insane. While propositions cannot correspond to specific thoughts since specific thoughts aren’t transferable, they can represent categories of cognitive processes or characteristics of ideas (which can be the same for different minds).
Whether propositions are external or internal to the agent, or if they are entities that are dependent on the mind has also been a topic of discussion in modern philosophical discussions concerning propositions and propositional attitudes. Further information can be found in the philosophy of thought section on externalism versus internalism.
Logic-based treatment
The Logic of Aristotle
As previously mentioned, a proposition in the logic of Aristotle is a specific type of statement (a declaratory sentence) that, if necessary with the aid of a copula, confirms or rejects an argument of a subject. Propositions from Aristotle include “Socrates is a human” and “All men are human.”
Syntactic characterization
Statements in a formal tongue are frequently referred to as propositions in contemporary logic. As formalized syntactic objects in this use, propositions may be analyzed without reference to the semantic significance that semantics would assign them. In one text, these concepts are typically not interchangeable with propositions; other names for them include sentences, statements, well-formed formulas, formulas, and statement forms.
Various symbol types are used to start a formal language. Variables, operators, predicate (or connection) symbols, function signs, propositional constants, and quantifiers are a few examples of these types. (Grouping symbols like delimiters, which are purely illogical, are frequently inserted for language convenience.)
To create strings that will be ascribed truth values, symbols are combined together using recursive rules. The operators, quantifiers, and function & predicate symbols must all be combined with additional strings according to the requirements. And then a string having a particular shape is a proposition. A proposition’s shape is determined by the kind of logic it uses.
The language of propositional, statement, or sentential logic consists solely of propositional and operator constants as symbols. The propositions within this language are built by recursively assigning operators to propositions; propositional constants are regarded as atomic propositions, while compound (or composite) propositions are made up of several propositions. Applying the associated concatenation principle is all that needs to be said in this context.
Quantificational, predicate, and n-order logics are logics that use variables, quantifiers, predicate & function symbols, and operators as elements in their respective languages. In these logical systems, the propositions are more intricate. Generally, one begins by providing the following proposition definition of a term:
- A variable
- Function symbol multiplied by the no. of words needed to satisfy the arity of the function symbol.
For instance, if x, y, & z are variables and + represents a binary function, the expression x+(y+z) can be represented in a variety of ways using the symbols. A proposition may be characterized by what follows after a term has been specified:
- A predicate symbol used to indicate how many phrases its arity requires
- The operator utilized in relation to the no. of propositions mandated by its arity
- A quantifier is used in relation to a claim.
A proposition would be ∀x,y,z [(x = y) → (x+z = y+z)] if = represents a binary predicate character and ∀ represents a quantifier. These logics have more expressive capacity because of the more intricate arrangement of the propositions, which enables them to distinguish between inferences more precisely.
Semantic characterization
The standard semantic understanding of propositions is that they are indicator functions that take a conceivable world as input and output the truth value. The claim that the sky looks blue, for instance, could be expressed as a function 𝑓 in which 𝑓(𝑤) = 𝑇 for all worlds 𝑤, if any, in which the sky looks blue, and 𝑓(𝑣) = 𝐹 for all worlds 𝑣, if any, in which it’s not. The indicator function, also referred to as the characteristic group of the proposition, can be used to describe a proposition correspondingly with the reverse representation of 𝑇. The claim that the sky looks blue, for example, could be represented as the set {𝑤,𝑤′} if 𝑤 & 𝑤′ are the sole worlds in which it is blue.
Inquisitive propositions & structured propositions are two of the many variations and substitute ideas of propositionhood that have been put forth. When a proposition has constituents, it is referred to be a structured proposition in general. Presuming a structured conception of propositions, one can differentiate between three types of propositions: general propositions, which are not specific to any individual, particularized propositions, which are specific to a person but don’t include that person as a constituent, as well as singular propositions, also known as Russellian propositions, derived from Bertrand Russell.
Arguments against propositions
The following are some efforts to offer a practical proposition definition:
The identical proposition is expressed by two valid declarative phrases if both of them have the same meaning.
This provides a synonymy proposition definition. For instance, although “Schnee ist weiß” (which is in German) & “Snow looks white” (in English) are distinct words, they convey the same idea. A different way to express proposition definition is:
A pair of significant descriptive sentence-tokens can only represent an idea if and when their meanings are identical.
With respect to the aforementioned definitions, two sentences or sentence tokens may seem to convey the same idea while having distinct truth values. Examples of this would be “I am Spartacus,” which was said by both Spartacus & John Smith, as well as “It’s Wednesday,” which is said on both Wednesdays and Thursdays. The confusion in everyday language that causes false equivalency between the statements is reflected in the above instances.
The words “I am Spartacus,” said by Spartacus, signify that the speaker is, in fact, Spartacus. When said by John Smith, it’s an untrue statement about another speaker. “I am Spartacus” has distinct meanings depending on what you mean when you use the word “I.”
When two phrases represent an identical proposition but have distinct truth values, this is an associated issue. It is possible that both Socrates & Plato said, “I’m a philosopher.” While accurate in both cases, the statement has a different meaning.
In predicate reasoning, these issues are resolved by creating a variable that represents the problematic phrase, allowing Socrates or Plato to be used in place of Z in the sentence “Z is a philosopher,” demonstrating the distinction between the statements “Socrates was a philosopher” and “Plato was a philosopher.” Likewise, “I am Spartacus” transforms into “Z is Spartacus,” with Z being substituted with phrases that signify Spartacus & John Smith.
Stated otherwise, the aforementioned difficulties can be avoided if sentences are precisely constructed so that words have clear definitions.
Many linguists and philosophers contend that no proposition definition is ever precise enough to be of any service. They see it as nothing more than a false notion that ought to be eliminated from semantics and philosophy. The unpredictability of translation, according to W. V. Quine, who acknowledged the presence of sets in math, meant that propositions could not be meaningfully discussed and should instead be replaced with sentences. On the opposite hand, P. F. Strawson supported the utilization of the word “statement.”
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