Symbology is basically the magic of Star Ocean: Till The End of Time, it basically gives the character the ability to damage using non-physical attacks or manuvers. For characters like Sophia, they learn Symbology from their basic skillset - Meaning they level up and learn the abilities themselves. However, other characters like Fayt, don't learn magic themselves. Some Symbology like 'Healing,' so they have to equip a certain set of abilites, like 'Common Support Symbols.'

For a list of Symbology, look at each character's 'Skill Page' on the navigation - under 'Magical Skills,' No Symbology means they learn most or all of their Symbology from items.