The problem when trying to talk about semiotics is that every single semiotician defines and re-defines terms differently from the others. The easiest way to understand it, though, is probably from a historical perspective.
The “father” of semiotics was the linguist Ferdinand de Saussure. When talking about language he realised that he needed a new concept to refer to the smallest units of meaning. “Words” are no good: some expressions are made up of several words but can’t be analysed in terms of their parts. He thus came up with the notion of “sign.” A sign, for Saussure, had two dimensions: a mental image you meant to communicate (the signifed) and something physical you used to do so, like a sound or a shape (the signifier).
Once you break down a text into its signs, you can then analyse the resulting structure; that’s the “syntagm.” However, you can also look at how signs relate to other similar signs; the “paradigm.” For instance, in the sentence “He wore shoes” the relationship between “he” and “shoes” is syntagmatic and the relationship between “shoes” and “sandals” is “paradigmatic” – you can substitute “shoes” for “sandals.”
After Saussure, other thinkers realised that signs don’t need to be limited to speech. Everything has potential meaning. Thus, you could use semiotics to analyse anything. One of the more famous semioticians is Roland Barthes, who studied, among many other things, fashion. If you look at what a person is wearing, you can talk about the syntagm of their dress, the relationship between their shirt, their tie and their trousers. You can also talk about paradigms – the relationship between jackets and sweaters, and between the great many different designs of sweaters. Saussure was interested in how you could substitute the words “shoes” for “sandals,” but Barthes was interested in how you could literally substitute shoes for sandals.
Another important semiotician is the philosopher Charles Sanders Peirce, who defined signs differently from Saussure. Because his interests were broader than linguistic, he was interested in the relationship between a mental image (the interpretant), a object used to convey meaning (the sign) and the actual thing being signified (the object.) If you define signs that way, you can then talk about “symbols,” signs that have arbitrary relationships between sign and object, and “icons,” where signs share something with the objects.
For instance, the English word “tree” – the shape you see on your screen share no characteristics with actual trees. It’s a symbol. A photograph of a tree, however, is an icon, and so is the Chinese character for “tree”: 木. It’s a stylised drawing of a tree.
This is really the basics, but if you wish to go further it’s important to remember the following:
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Semiotics is merely a set of tools that, if used properly, can allow you dissect the meaning of something finely. Semiotics is not an ideology, and it is of absolutely no help when it comes to finding out what something means. It’s a collection of knives, scissors and scalpels that may allow you to find out how something means something.
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The overwhelming majority, if not the near totality of semiotic writing is crap. It is crap because the writers forgot about the point above.
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Jargon. I think you already noticed, but semioticians like to make words up. Also, they like to change the meaning of existing words. And, they don’t agree with each other. I already wrote that, but really that’s also a big reason for point #2.
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Nevertheless I think that sometimes, yes, a scalpel is the right tool for the job and sometimes, yes, semiotics analysis can be enlightening. I mentioned Roland Barthes and I think he’s a good example of someone who had something to say first and then used semiotics to say it more precisely. Also see this paper, it’s both hilarious and fascinating. That’s semiotics done right:
Signs of Life beyond Earth: A Semiotic Analysis of Interstellar Messages